The Askey scheme of hyp ergeometric orthogonal p olynomials
and its q analogue
Ro elof Ko eko ek Ren e F Swarttouw
February
Abstract
We list the so called Askey scheme of hyp ergeometric orthogonal p olynomials In chapter we
give the de nition the orthogonality relation the three term recurrence relation and generating
functions of all classes of orthogonal p olynomials in this scheme In chapter we give all limit
relations b etween di erent classes of orthogonal p olynomials listed in the Askey scheme
In chapter we list the q analogues of the p olynomials in the Askey scheme We give their
de nition orthogonality relation three term recurrence relation and generating functions In
chapter we give the limit relations b etween those basic hyp ergeometric orthogonal p olynomials
Finally in chapter we p oint out how the classical hyp ergeometric orthogonal p olynomials of
the Askey scheme can b e obtained from their q analogues
Acknowledgement
We would like to thank Professor Tom H Ko ornwinder who suggested us to write a rep ort like
this He also help ed us solving many problems we encountered during the research and provided
us with several references
Contents
Preface
De nitions and miscellaneous formulas
Intro duction
The q shifted factorials
The q gamma function and the q binomial co e cient
Hyp ergeometric and basic hyp ergeometric functions
The q binomial theorem and other summation formulas
Transformation formulas
Some sp ecial functions and their q analogues
The q derivative and the q integral
ASKEY SCHEME
Hyp ergeometric orthogonal p olynomials
Wilson
Racah
Continuous dual Hahn
Continuous Hahn
Hahn
Dual Hahn
Meixner Pollaczek
Jacobi
Gegenbauer Ultraspherical
Chebyshev
Legendre Spherical
Meixner
Krawtchouk
Laguerre
Charlier
Hermite
Limit relations b etween hyp ergeometric orthogonal p olynomials
Wilson Continuous dual Hahn
Wilson Continuous Hahn
Wilson Jacobi
Racah Hahn
Racah Dual Hahn
Continuous dual Hahn Meixner Pollaczek
Continuous Hahn Meixner Pollaczek
Continuous Hahn Jacobi
Hahn Jacobi
Hahn Meixner
Hahn Krawtchouk
Dual Hahn Meixner
Dual Hahn Krawtchouk
Meixner Pollaczek Laguerre
Meixner Pollaczek Hermite
Jacobi Laguerre
Jacobi Hermite
Meixner Laguerre
Meixner Charlier
Krawtchouk Charlier
Krawtchouk Hermite
Laguerre Hermite
Charlier Hermite
q SCHEME
Basic hyp ergeometric orthogonal p olynomials
Askey Wilson
q Racah
Continuous dual q Hahn
Continuous q Hahn
Big q Jacobi
Big q Legendre
q Hahn
Dual q Hahn
Al Salam Chihara
q Meixner Pollaczek
Continuous q Jacobi
Continuous q ultraspherical Rogers
Continuous q Legendre
Big q Laguerre
Little q Jacobi
Little q Legendre
q Meixner
Quantum q Krawtchouk
q Krawtchouk
A ne q Krawtchouk
Dual q Krawtchouk
Continuous big q Hermite
Continuous q Laguerre
Little q Laguerre Wall
q Laguerre
Alternative q Charlier
q Charlier
Al Salam Carlitz I
Al Salam Carlitz I I
Continuous q Hermite
Stieltjes Wigert
Discrete q Hermite I
Discrete q Hermite I I
Limit relations b etween basic hyp ergeometric orthogonal p olynomials
Askey Wilson Continuous dual q Hahn
Askey Wilson Continuous q Hahn
Askey Wilson Big q Jacobi
Askey Wilson Continuous q Jacobi
Askey Wilson Continuous q ultraspherical Rogers
q Racah Big q Jacobi
q Racah q Hahn
q Racah Dual q Hahn
q Racah q Krawtchouk
q Racah Dual q Krawtchouk
Continuous dual q Hahn Al Salam Chihara
Continuous q Hahn q Meixner Pollaczek
Big q Jacobi Big q Laguerre
Big q Jacobi Little q Jacobi
Big q Jacobi q Meixner
q Hahn Little q Jacobi
q Hahn q Meixner
q Hahn Quantum q Krawtchouk
q Hahn q Krawtchouk
q Hahn A ne q Krawtchouk
Dual q Hahn A ne q Krawtchouk
Dual q Hahn Dual q Krawtchouk
Al Salam Chihara Continuous big q Hermite
Al Salam Chihara Continuous q Laguerre
q Meixner Pollaczek Continuous q ultraspherical Rogers
Continuous q Jacobi Continuous q Laguerre
Continuous q ultraspherical Rogers Continuous q Hermite
Big q Laguerre Little q Laguerre Wall
Big q Laguerre Al Salam Carlitz I
Little q Jacobi Little q Laguerre Wall
Little q Jacobi q Laguerre
Little q Jacobi Alternative q Charlier
q Meixner q Laguerre
q Meixner q Charlier
q Meixner Al Salam Carlitz I I
Quantum q Krawtchouk Al Salam Carlitz I I
q Krawtchouk Alternative q Charlier
q Krawtchouk q Charlier
A ne q Krawtchouk Little q Laguerre Wall
Dual q Krawtchouk Al Salam Carlitz I
Continuous big q Hermite Continuous q Hermite
Continuous q Laguerre Continuous q Hermite
q Laguerre Stieltjes Wigert
Alternative q Charlier Stieltjes Wigert
q Charlier Stieltjes Wigert
Al Salam Carlitz I Discrete q Hermite I
Al Salam Carlitz I I Discrete q Hermite I I
From basic to classical hyp ergeometric orthogonal p olynomials
Askey Wilson Wilson
q Racah Racah
Continuous dual q Hahn Continuous dual Hahn
Continuous q Hahn Continuous Hahn
Big q Jacobi Jacobi
Big q Legendre Legendre Spherical
q Hahn Hahn
Dual q Hahn Dual Hahn
Al Salam Chihara Meixner Pollaczek
q Meixner Pollaczek Meixner Pollaczek
Continuous q Jacobi Jacobi
Continuous q ultraspherical Rogers Gegenbauer Ultra spherical
Continuous q Legendre Legendre Spherical
Big q Laguerre Laguerre
Little q Jacobi Jacobi
Little q Legendre Legendre Spherical
Little q Jacobi Laguerre
q Meixner Meixner
Quantum q Krawtchouk Krawtchouk
q Krawtchouk Krawtchouk
A ne q Krawtchouk Krawtchouk
Dual q Krawtchouk Krawtchouk
Continuous big q Hermite Hermite
Continuous q Laguerre Laguerre
Little q Laguerre Wall Laguerre
Little q Laguerre Wall Charlier
q Laguerre Laguerre
q Laguerre Charlier
Alternative q Charlier Charlier
q Charlier Charlier
Al Salam Carlitz I Charlier
Al Salam Carlitz I Hermite
Al Salam Carlitz I I Charlier
Al Salam Carlitz I I Hermite
Continuous q Hermite Hermite
Stieltjes Wigert Hermite
Discrete q Hermite I Hermite
Discrete q Hermite I I Hermite
Bibliography
Index
Preface
This rep ort deals with orthogonal p olynomials app earing in the so called Askey scheme of hyp er
geometric orthogonal p olynomials and their q analogues Most formulas listed in this rep ort can
b e found somewhere in the literature but a handb o ok containing all these formulas did not exist
We collected known formulas for these hyp ergeometric orthogonal p olynomials and we arranged
them into the Askey scheme and into a q analogue of this scheme which we called the q scheme
This q scheme was not completely do cumented in the literature So we lled in some gaps in order
to get some sort of complete scheme of q hyp ergeometric orthogonal p olynomials
In chapter we give some general de nitions and formulas which can b e used to transform
several formulas into di erent forms of the same formula In the other chapters we used the most
common notations but sometimes we had to change some notations in order to b e consistent
For each family of orthogonal p olynomials listed in this rep ort we give the conditions on
the parameters for which the corresp onding weight function is p ositive These conditions are
mentioned in the orthogonality relations We remark that many of these orthogonal p olynomials
are still p olynomials for other values of the parameters and that they can b e de ned for other
values as well That is why we gave no restrictions in the de nitions As p ointed out in chapter
some de nitions can b e transformed into di erent forms so that they are valid for some values of
the parameters for which the given form has no meaning Other formulas such as the generating
functions are only valid for some sp ecial values of parameters and arguments These conditions
are left out in this rep ort
We are aware of the fact that this rep ort is by no means a full description of all that is known
ab out basic hyp ergeometric orthogonal p olynomials More on each listed family of orthogonal
p olynomials can b e found in the articles and b o oks to which we refer
In later versions of this rep ort we want to add recurrence relations for the monic p olynomials
in each case id est for the p olynomials with leading co e cient equal to We also hop e to include
more formulas containing quadratic transformations and we want to pay more attention to the
transformation q q
Comments on this version of the rep ort and suggestions for improvement are most welcome
If you nd errors or gaps or if you have suggestions for inclusion of more formulas on basic
hyp ergeometric orthogonal p olynomials please contact us and let us know
Ro elof Ko eko ek and Ren e F Swarttouw
Ro elof Ko eko ek Ren e F Swarttouw
Delft University of Technology Free University of Amsterdam
Faculty of Technical Mathematics and Informatics Faculty of Mathematics and Informatics
Mekelweg De Bo elelaan
CD Delft HV Amsterdam
The Netherlands The Netherlands
ko eko ek twi tudelft nl rene cs vu nl
De nitions and miscellaneous
formulas
Intro duction
In this rep ort we will list all known sets of orthogonal p olynomials which can b e de ned in
terms of a hyp ergeometric function or a basic hyp ergeometric function
In the rst part of the rep ort we give a description of all classical hyp ergeometric orthogonal
p olynomials which app ear in the so called Askey scheme We give de nitions orthogonality rela
tions three term recurrence relations di erential or di erence equations and generating functions
for all families of orthogonal p olynomials listed in this Askey scheme of hyp ergeometric orthogonal
p olynomials
In the second part we obtain a q analogue of this scheme We give de nitions orthogonality
relations three term recurrence relations di erence equations and generating functions for all
known q analogues of the hyp ergeometric orthogonal p olynomials listed in the Askey scheme
Further we give limit relations b etween di erent families of orthogonal p olynomials in b oth
schemes and we p oint out how to obtain the classical hyp ergeometric orthogonal p olynomials from
their q analogues
The theory of q analogues or q extensions of classical formulas and functions is based on the
observation that
q
lim
q !
q
Therefore the numb er q q is sometimes called the basic numb er
Now we can give a q analogue of the Po chhammer symb ol a which is de ned by
k
a and a a a a a k k
k
This q extension is given by
k
a q and a q a aq aq aq k
k
It is clear that
q q
k
lim
k
k
q !
q
In this rep ort we will always assume that q
For more details concerning the q theory the reader is referred to the b o ok by G Gasp er
and M Rahman
Since many formulas given in this rep ort can b e reformulated in many di erent ways we will
give a selection of formulas which can b e used to obtain other forms of de nitions orthogonality
relations and generating functions
Most of these formulas given in this chapter can b e found in
We remark that in orthogonality relations we often have to add some condition s on the
parameters of the orthogonal p olynomials involved in order to have p ositive weight functions By
using the famous theorem of Favard these conditions can also b e obtained from the three term
recurrence relation
In some cases however some conditions on the parameters may b e needed in other formulas
to o For instance the de nition of the Laguerre p olynomials has no meaning for negative
integer values of the parameter But in fact the Laguerre p olynomials are also p olynomials in
the parameter This can b e seen by writing
n
X
n
k
k
L x k x
n k
n
n k
k
In this way the Laguerre p olynomials are de ned for all values of the parameter
A similar remark holds for the Jacobi p olynomials given by We may also write see
section for the de nition of the hyp ergeometric function F
n n x
n
n
F P x
n
n
which implies the well known symmetry relation
n
P x P x
n n
Even more general we have
n
k
X
x n
k
n k P x
k n k
n
n k
k
From this form it is clear that the Jacobi p olynomials can b e de ned for all values of the parameters
and although the de nition is not valid for negative integer values of the parameter
We will not indicate these di culties in each formula
Finally we remark that in each recurrence relation listed in this rep ort except for for
the Chebyshev p olynomials of the rst kind we may use P x and P x as initial
conditions
The q shifted factorials
The symb ols a q de ned in the preceding section are called q shifted factorials They can
k
also b e de ned for negative values of k as
k
a q a q q q q k
k
k
aq aq aq
Now we have
n
n
q a
2
q n a q
n
n
aq q q a q
n n
where
n
n n
We can also de ne
1
Y
k
aq a q
1
k
This implies that
a q
1
a q
n
n
aq q
1
and for any complex numb er
a q
1
a q
aq q
1
where the principal value of q is taken
If we change q by q we obtain
n
n
2
a q a q a q a
n n
This formula can b e used for instance to prove the following transformation formula b etween
the little q Laguerre or Wall p olynomials given by and the q Laguerre p olynomials
de ned by
q q
n
L x q p x q jq
n
n
q q
n
or equivalently
q q
n
L x q p x q jq
n
n
n
q q q
n
By using it is not very di cult to verify the following general transformation formula
for p olynomials see section for the de nition of the basic hyp ergeometric function
n n n
q a b c abcq q a b c
q q q
def d e f d e f
where a limit is needed when one of the parameters is equal to zero Other transformation formulas
can b e obtained from this one by applying limits as discussed in section
Finally we list a numb er of transformation formulas for the q shifted factorials where k and
n are nonnegative integers
n
a q a q aq q
n k n k
n
a q aq q
k k
k
aq q a q
n n
a q
n
k
aq q k n
n k
a q
k
n
n n
2
a q a q q a q a
n n
n
n
n n
2
aq q a q q a q a
n n
n
n
a q q a aq q
n n
a b
n
bq q b q q b
n n
k
k
q a q
n
nk
2
a k n q a q
n k
n
a q q a
k
k
n
a q a q b q q b
n k n k
a b k n
n
b q b q a q q a
n k n k
k
q q
n
nk
k n
2
k n q q q
k
q q
n k
a q q
n
n nk
aq q a q q a
k k
k
a q q
n
n k
k n
a q q a
n
n
2 2
aq q q a k n
n k
a q q q
k
a q a q aq q
n n n
a q a q a q
n n n
a q a q aq q
1 1 1
a q a q a q
1 1 1
Formula can b e used for instance to show that the generating function
for the continuous q Legendre p olynomials is a q analogue of the generating function
for the Legendre p olynomials In fact we obtain see section for the de nition of the basic
hyp ergeometric functions
r s
1 1 1 1
i i i i
2 2 2 2
e q e e q e q q
i i
q e t q e t
q q
i i
q e q e
i i
q e t q e t
Now we use the q binomial theorem to show that this equals
i i
q e t q q e t q q q
1 1
i i
q e t q e t
i i
e t q e t q
1 1
If we let q tend to one we now nd by using the binomial theorem
1 1
i i i i
2 2
p
x cos F e t e t e t e t F
xt t
which equals
The q gamma function and the q binomial co e cient
The q gamma function is de ned by
q q
1
x
x q
q
x
q q
1
This is a q analogue of the well known gamma function since we have
lim x x
q
q "
Note that the q gamma function satis es the functional equation
z
q
z z
q q q
q
which is a q extension of the well known functional equation
z z z
for the ordinary gamma function For nonintegral values of z this ordinary gamma function also
satis es the relation
z z
sin z
which can b e used to show that
k k
lim q ln q k
!k
This limit can b e used to show that the orthogonality relation for the Stieltjes Wigert
p olynomials follows from the orthogonality relation for the q Laguerre p olynomials
The q binomial co e cient is de ned by
i h
q q n n
n
k n
n k q q q q k
q
k n k
q
where n denotes a nonnegative integer
This de nition can b e generalized in the following way For arbitrary complex we have
i h
k
q q
k
k
k
2
q
k q q
q
k
Or more general for all complex and we have
q q q q
1 1 q
q q q q
q q 1 1
q
For instance this implies that
q q n
n
n q q
n
q
Note that
lim
q "
q
For integer values of the parameter we have
k
k
k
k k
and when the parameter is an integer to o we may write
n n
k n n
k k n k
This latter formula can b e used to show that
n
n
n
n
n n
This can b e used to write the generating functions and for the Chebyshev p oly
nomials of the rst and the second kind in the following form
r
1
n
p
X
n t
R T x R xt R xt t
n
n
n
and
1
n
p
X
n t
p
R U x xt t
n
n
R R xt
n
resp ectively
Finally we remark that
n
h i
X
k
n
k
2
a q q a
n
k
q
k
Hyp ergeometric and basic hyp ergeometric functions
The hyp ergeometric series F is de ned by
r s
1
k
X
a a a a z
r k r
z F
r s
b b b b k
s s k
k
where
a a a a
r k k r k
Of course the parameters must b e such that the denominator factors in the terms of the series
are never zero When one of the numerator parameters a equals n where n is a nonnegative
i
integer this hyp ergeometric series is a p olynomial in z Otherwise the radius of convergence of
the hyp ergeometric series is given by
if r s
if r s
if r s
A hyp ergeometric series of the form is called balanced or Saalschutzian if r s
z and a a a b b b
s s
The basic hyp ergeometric series or q hyp ergeometric series is de ned by
r s
1
k
X
k
z a a q a a
r k r
s r k s r
2
q q z
r s
b b q q q b b
s k k s
k
where
a a q a q a q
r k k r k
Again we assume that the parameters are such that the denominator factors in the terms of the
n
series are never zero If one of the numerator parameters a equals q where n is a nonnegative
i
integer this basic hyp ergeometric series is a p olynomial in z Otherwise the radius of convergence
of the basic hyp ergeometric series is given by
if r s
if r s
if r s
The sp ecial case r s reads
1
k
X
z a a a a q
s s k
q z
s s
b b q q q b b
s k k s
k
This basic hyp ergeometric series was rst intro duced by Heine in Therefore it is some
times called Heine s series A basic hyp ergeometric series of this form is called balanced or
Saalschutzian if z q and a a a q b b b
s s
The q hyp ergeometric series is a q analogue of the hyp ergeometric series de ned by
since
a a
r 1
q q a a
r
s r
z lim q q z F
r s r s
b b
s 1
q "
b b q q
s
This limit will b e used frequently in chapter
We remark that
z a a a a
r r
q q z lim
r s r s
a !1
b b b b r a
s s r
k
s r
s r k
2
In fact this is the reason for the factors q in the de nition of the
basic hyp ergeometric series
The limit relations b etween hyp ergeometric orthogonal p olynomials listed in chapter of this
rep ort are based on the observations that
a a a a
r r
z z F F
r s r s
b b b b
s s
z a a a a a
r r r
lim F a z F
r s r r s
!1
b b b b
s s
z a a a a
r r
z F lim F
r s r s
!1
b b b b b b
s s s s
and
a z a a a a a
r r r r
z F lim F
r s r s
!1
b b b b b b
s s s s
The limit relations b etween basic hyp ergeometric orthogonal p olynomials describ ed in chapter
of this rep ort are based on the observations that
a a a a
r r
q z q z
r s r s
b b b b
s s
a a z a a a
r r r
q a z q lim
r r s r s
!1
b b b b
s s
z a a a a
r r
q q z lim
r s r s
!1
b b b b b b
s s s s
and
a z a a a a a
r r r r
q q z lim
r s r s
!1
b b b b b b
s s s s
Mostly the left hand sides of the formulas and o ccur as limit cases when some
numerator parameter and some denominator parameter tend to the same value
Finally we intro duce a notation for the N th partial sum of a basic hyp ergeometric series
We will use this notation in the de nitions of the discrete orthogonal p olynomials We also use it
in order to write generating functions for discrete orthogonal p olynomials in a compact way We
de ne
N
k
X
a a a a z
r k r
z F
s r
b b b b k
s s k
k
where N denotes the nonnegative integer that app ears in each de nition of a family of discrete
orthogonal p olynomials We also de ne
N
k
X
k
z a a q a a
r k r
s r k s r
2
q q z
s r
b b q q q b b
s k k s
k
As an example of the use of these notations we remark that the de nition of the quantum
q Krawtchouk p olynomials must b e understo o d as
N
n x n x
X
q q q q q q
k
k k
n n
q pq pq
N N
q q q q q
k k
k
In cases of discrete orthogonal p olynomials like the Racah Hahn dual Hahn and Krawtchouk
p olynomials we need another sp ecial notation for the generating functions In order to simplify
the notation we write the generating functions as pro ducts of truncated as ab ove p ower series
in t for which the N th partial sum equals the right hand side In these cases we use the notation
instead of the sign As an example of this notation and the one mentioned ab ove we note
that the generating function of the dual Hahn p olynomials must b e understo o d as follows
The N th partial sum of
N
X
x x x x
k k
k t t
t t e e F
N k N
k k
k
equals
N
X
R x N
n
n
t
n
n
The q binomial theorem and other summation formu
las
One of the most imp ortant summation formulas for hyp ergeometric series is given by the
binomial theorem
1
X
a a
n
n a
z F z z jz j
n
n
A q analogue of this formula is called the q binomial theorem
1
X
a q az q a
n 1
n
q z z jz j
q q z q
n 1
n
n
For a q with n a nonnegative integer we nd
n
q
n
q z z q q n
n
In fact this is a q analogue of Newton s binomium
n n
X X
n n n
k
k k n
z F z z z n
k k
k k
As an example of the use of these formulas we note that the generating function of the
dual Hahn p olynomials can also b e written as
x N x x
t F t F
N
In a similar way we nd for the generating function of the quantum q Krawtchouk
p olynomials
x N x x x N
q q t q q q
1
x x
q q t q t q q t
pq pq t q
1
Another example of the use of the q binomial theorem is the pro of of the fact that the generating
function for the continuous q ultraspherical or Rogers p olynomials is a q analogue of
the generating function for the Gegenbauer or ultraspherical p olynomials In fact we
have after the substitution q
i i i
q q q e t q e t q q e t q
1 1
i i
q e t q e t
i i i
e t q e t e t q
1 1
which tends to for q
i i i i
e t e t e t xt t x cos F e t F
which equals
The well known Gauss summation formula
c c a b a b
F Re c a b
c c a c b
and Vandermonde s summation formula
c b n b
n
n F
c c
n
have the following q analogues
c a b a c b c q c
1
q
c ab c a b c q ab
1
n n
b c q cq q b
n
q n
b c q c
n
and
n
b c q q b
n
n
q q b n
c c q
n
On the next level we have the summation formula
c a c b n a b
n n
F n
c a b c n c c a b
n n
which is called Saalschutz or Pfa Saalschutz summation formula A q analogue of this summa
tion formula is
n
a c b c q q a b
n
n q q
n
c a b c q c abc q
n
Finally we have a summation formula for the series
a c a c q
1
q
c a c q
1
As an example of the use of this latter formula we remark that the q Laguerre p olynomials
de ned by have the prop erty that
n q L
n
q q
n
Transformation formulas
In this section we list a numb er of transformation formulas which can b e used to transform
de nitions or other formulas into equivalent but di erent forms
First of all we have Heine s transformation formulas for the series
b c z az b q a b
1
q b q z
az c z q c
1
c b c bz q abc z b
1
q
bz c z q b
1
abc z q abz a c b c
1
q
c z q c
1
The latter formula is a q analogue of Euler s transformation formula
c a c b a b
c a b
z z z F F
c c
Another transformation formula for the F series which is also due to Euler is
z c a b a b
b
z z F F
z c c
This transformation formula is also known as the Pfa Kummer transformation formula
As a limit case of this one we have Kummer s transformation formula for the con uent hyp er
geometric series
c a a
z
z z e F F
c c
Limit cases of Heine s transformation formulas are
z
q z q c
c c z q
1
q cz
c z q
1
az q a z
1
q z q c
c az c z q
1
a c
q az
c z q
1
a c a a z q
1
q a q z
z c q c
1
az a c
q ac z q
1
c c
and
z az b q a b
1
q b q z
az z q
1
bz q b
1
q az
bz z q
1
If we reverse the order of summation in a terminating F series we obtain a F series in fact
we have
n
x n a n n
x F n F
a x a
n
If we apply this technique to a terminating F series we nd
b n b n c n
n
n
x F n x F
c b n c x
n
The q analogues of these formulas are
n n n n n
q z aq q q a q
q z q n
a a q z
n
and
n
q b
q z
c
n n n
n
cq q c q b q
n
n
n
2
q n q z
n
b q c q bz
n
A limit case of the latter formula is
n n
q q b q
n n
n q z q q b q z
n
n
bz b q
The next transformation formula is due to Jackson
n n n
bc q z q q b q b c
1
q z q q n
c c b cq z bc z q
1
Equivalently we have
n n
aq b q q q a c q a
1
q q q n
n
c b c b q q b
1
Other transformation formulas of this kind are given by
n
q b
q z
c
n
n n
b c q q q z c q bz
n
q q
n
bc q c q q
n
n n
b c q q b bc q z
n
q q n
n
c q bc q
n
or equivalently
n n
b q q a b q b c q
n
n
q q q a
n
c b q c q a
n
n
a c q bq q a
n
n
q a n
n
ac q c q c
n
Limit cases of these formulas are
n n n
q b bz q q b
n
q q n q z b
or equivalently
n n
q q q a b
n
q q q b q a
n
n
b q a
n n
bq q a
n
n q a
a
On the next level we have Sears transformation formula for a terminating balanced series
n
q a b c
q q
d e f
n
a e a f q q a b d c d
n
n
a q q
n n
d ae q af q e f q
n
a a b ef a c ef q
n
e f a b c ef q
n
n
q a e a f a b c ef
n
q q def abcq
n
a b ef a c ef a q
Sears transformation formula is a q analogue of Whipple s transformation formula for a ter
minating balanced F series
e a f a n a b c
n n
F
d e f e f
n n
n a d b d c
a b c d e f n F
d a e n a f n
Whipple s formula can b e used to show that the Wilson p olynomials de ned by are
symmetric in their parameters in the sense that the following di erent forms are all equal
W x a b c d W x a b d c W x a c b d W x d c b a
n n n n
Sears transformation formula can b e used to derive similar symmetry relations for the Askey
Wilson p olynomials de ned by
p x a b c d p x a b d c p x a c b d p x d c b a
n n n n
Finally we mention a quadratic transformation formula which is due to Singh
a b c d
a b c d
q q q q
1 1
2 2
a b q cd cdq
abq abq cd
which is valid when b oth sides terminate
If we apply Singh s formula to the continuous q Jacobi p olynomials de ned by
and and use Sears transformation formula formula twice and also formula
then we nd the quadratic transformation
q q
n
n
P xjq q P x q
n n
q q
n
Some sp ecial functions and their q analogues
The classical exp onential function exp z and the trigonometric functions sin z and cos z
can b e expressed in terms of hyp ergeometric functions as
z
z exp z e F
z
sin z z F
and
z
cos z F
Further we have the well known Bessel function J z which can b e de ned by
z
z
F J z
Applying this formula to the generating function of the Laguerre p olynomials we
obtain
1
X
p
x L
n
n t
2
xt t xt e J
n
n
These functions all have several q analogues The exp onential function for instance has two
di erent natural q extensions denoted by e z and E z de ned by
q q
1
n
X
z
q z e z
q
q q
n
n
and
n
1
X
2
q
n
q z E z z
q
q q
n
n
These q analogues of the exp onential function are related by
e z E z
q q
They are q extensions of the exp onential function since
z
lim e q z lim E q z e
q q
q " q "
If we set a in the q binomial theorem we nd for the q exp onential functions
q z jz j e z
q
z q
1
Further we have
q z z q E z
1 q
For instance these formulas can b e used to obtain other versions of a generating function for
several sets of orthogonal p olynomials mentioned in this rep ort
If we assume that jz j we may de ne
1
n n
X
e iz e iz z
q q
sin z
q
i q q
n
n
and
1
n n
X
z e iz e iz
q q
cos z
q
q q
n
n
These are q analogues of the trigonometric functions sin z and cos z On the other hand we
may de ne
E iz E iz
q q
Sin z
q
i
and
E iz E iz
q q
Cos z
q
Then it is not very di cult to verify that
e iz cos z i sin z and E iz Cos z i Sin z
q q q q q q
Further we have
sin z Sin z cos z Cos z
q q q q
sin z Cos z Sin z cos z
q q q q
The q analogues of the trigonometric functions can b e used to nd di erent forms of formulas
app earing in this rep ort although we will not use them
Some q analogues of the Bessel functions are given by
z q q z
1
q J z q
q q q
1
and
q z z q q
1
q J z q
q q q
1
These q Bessel functions are connected by
z
J z q q J z q jz j
1
They are q analogues of the Bessel function since
k
lim J q z q J z k
q "
These q Bessel functions were intro duced by F H Jackson in They are therefore referred
to as Jackson q Bessel functions Other q analogues of the Bessel function are the so called Hahn
Exton q Bessel functions
As an example we remark that the generating function for the little q Laguerre or
Wall p olynomials can also b e written as
n
1
X
p 2
t q q q q
1 1
n
2
xt p x q jq t J xt q
n
q q q q
1 n
n
or as
n
1
X
2 p
q q q
1
n
2
xt p x q jq t xt q E t J
n q
q q q q
1 n
n
The q derivative and the q integral
The q derivative op erator D is de ned by
q
f z f q z
z
q z
D f z
q
0
f z
Further we de ne
n n
D f n D f f and D f D
q
q q q
It is not very di cult to see that
0
lim D f z f z
q
q "
if the function f is di erentiable at z
An easy consequence of this de nition is
D f x D f x
q q
for all real or more general
n n n
D f x D f x n
q q
Further we have
D f x g x f q x D g x g x D f x
q q q
which is often referred to as the q pro duct rule This can b e generalized to a q analogue of Leibniz
rule
n
i h
X
n
n k k k n
D f q x D g x n D f x g x
q q q
k
q
k
As an example we note that the q di erence equation of the q Laguerre p olynomials
can also b e written in terms of this q derivative op erator as
n
q xD y x q q q x D y q x q q y q x y x L x q
q
q n
The q integral is de ned by
Z
1
z
X
n n
f z q q f t d t z q
q
n
This de nition is due to J Thomae and F H Jackson Jackson also de ned a q integral on
by
Z
1
1
X
n n
f q q f t d t q
q
n 1
If the function f is continuous on z we have
Z Z
z z
lim f t d t f t dt
q
q "
For instance the orthogonality relation for the little q Jacobi p olynomials can also b e
written in terms of a q integral as
Z
q x q
1
x p x q q jq p x q q jq d x
m n q
q x q
1
q q q q q q q
1 n
n
q q
mn
n
q q q q q q q
1 n
ASKEY SCHEME
OF
HYPERGEOMETRIC
ORTHOGONAL POLYNOMIALS
F
Wilson Racah
Continuous Continuous
F
Hahn Dual Hahn
dual Hahn Hahn
Meixner
F
Jacobi Meixner Krawtchouk
Pollaczek
Laguerre
F F
Charlier
F
Hermite
Chapter
Hyp ergeometric orthogonal
p olynomials
Wilson
De nition
W x a b c d n n a b c d a ix a ix
n
F
a b a c a d a b a c a d
n n n
Orthogonality When Re a b c d and non real parameters o ccur in conjugate pairs
then
1
Z
a ix b ix c ix d ix
W x a b c d W x a b c d dx
m n
ix
n a b n c d
n a b c d n
n mn
n a b c d
where
n a b n c d
n a b n a c n a d n b c n b d n c d
If a and a b a c a d are p ositive or a pair of complex conjugates o ccur with p ositive
real parts then
1
Z
a ix b ix c ix d ix
W x a b c d W x a b c d dx
m n
ix
a b a c a d b a c a d a
a
X
a a a b a c a d
k k k k k
W a k W a k
m n
k a a b a c a d
k k k k
k
a k
n a b n c d
n a b c d n
n mn
n a b c d
where
W a k W a k W a k a b c d W a k a b c d
m n m n
Recurrence relation
W x W x A C W x C W x A a x
n n n n n n n n
where
W x a b c d
n
W x W x a b c d
n n
a b a c a d
n n n
and
n a b c d n a b n a c n a d
A
n
n a b c d n a b c d
n n b c n b d n c d
C
n
n a b c d n a b c d
Di erence equation
n n a b c d y x B x y x i B x D x y x D x y x i
where
y x W x a b c d
n
and
a ix b ix c ix d ix
B x
ix ix
a ix b ix c ix d ix
D x
ix ix
Generating functions
1
n
X
W x a b c d t a ix b ix c ix d ix
n
t t F F
a b c d a b c d n
n n
n
1
n
X
W x a b c d t b ix d ix a ix c ix
n
t F t F
b d a c a c b d n
n n
n
1
n
X
W x a b c d t b ix c ix a ix d ix
n
t F t F
a d b c n b c a d
n n
n
t a b c d a b c d a ix a ix
a b c d
t F
a b a c a d t
1
X
a b c d
n
n
W x a b c d t
n
a b a c a d n
n n n
n
Remark If we set
b a
c d
and
ix x
in
W x a b c d
n
W x a b c d
n
a b a c a d
n n n
de ned by and take
N or N or N with N a nonnegative integer
we obtain the Racah p olynomials de ned by
References
Racah
De nition
n n x x
n N R x F
n
where
x x x
and
N or N or N with N a nonnegative integer
Orthogonality
N
X
x x x x x
R x R x
m n
x
x x x x
x
n n
n n n n
M
mn
n n n n
where
R x R x
n n
and
N N
if N
N N
N N
if N M
N N
N N
if N
N N
Recurrence relation
x R x A R x A C R x C R x
n n n n n n n n
where
R x R x
n n
and
n n n n
A
n
n n
n n n n
C
n
n n
hence
n N n N n n
if N
n N n N
n n n N n
A if N
n
n n
n n n n N
if N
n n
and
n n n N n N
if N
n N n N
n n n n N
if N
C
n
n n
n n n N n
if N
n n
Di erence equation
n n y x B x y x B x D x y x D x y x
where
y x R x
n
and
x x x x
B x
x x
x x x x
D x
x x
Generating functions
x x x x
t t F F
N
X
n n
n
R x t
n
n
n
n
x x x x
t t F F
N
X
n n
n
R x t
n
n
n
n
x x x x
t t F F
N
X
n n
n
R x t
n
n
n
n
t x x
F t
t
N
X
n
n
R x t
n
n
n
Remark If we set a b c d a d a d and x a ix
in the de nition of the Racah p olynomials we obtain the Wilson p olynomials de ned by
R a ix a b c d a d a d
n
W x a b c d
n
W x a b c d
n
a b a c a d
n n n
References
Continuous dual Hahn
De nition
S x a b c n a ix a ix
n
F
a b a c a b a c
n n
Orthogonality When a b and c are p ositive except p ossibly for a pair of complex conjugates
with p ositive real parts then
1
Z
a ix b ix c ix
S x a b c S x a b c dx
m n
ix
n a b n a c n b c n
mn
If a and a b a c are p ositive or a pair of complex conjugates with p ositive real parts then
1
Z
a ix b ix c ix
S x a b c S x a b c dx
m n
ix
a b a c b a c a
a
X
a a a b a c
k k k k
k
S a k S a k
m n
k a a b a c
k k k
k
a k
n a b n a c n b c n
mn
where
S a k S a k S a k a b c S a k a b c
m n m n
Recurrence relation
S x S x A C S x C S x A a x
n n n n n n n n
where
S x a b c
n
S x S x a b c
n n
a b a c
n n
and
A n a b n a c
n
C n n b c
n
Di erence equation
ny x B x y x i B x D x y x D x y x i y x S x a b c
n
where
a ix b ix c ix
B x
ix ix
a ix b ix c ix
D x
ix ix
Generating functions
1
X
S x a b c a ix b ix
n
c ix n
t t F t
a b a b n
n
n
1
X
S x a b c a ix c ix
n
n b ix
t t t F
a c n a c
n
n
1
X
S x a b c b ix c ix
n
a ix n
t t F t
b c b c n
n
n
1
X
S x a b c a ix a ix
n
n t
t t e F
a b a c n a b a c
n n
n
References
Continuous Hahn
De nition
a c a d n n a b c d a ix
n n
n
p x a b c d i F
n
a c a d n
Orthogonality
1
Z
a ix b ix c ix d ix p x a b c d p x a b c d dx
m n
1
n a c n a d n b c n b d
mn
n a b c d n a b c d n
where
Re a b c d c a and d b
Recurrence relation
a ix p x A p x A C p x C p x
n n n n n n n n
where
n
p x a b c d p x p x a b c d
n n n
n
i a c a d
n n
and
n a b c d n a c n a d
A
n
n a b c d n a b c d
n n b c n b d
C
n
n a b c d n a b c d
Di erence equation
n n a b c d y x B x y x i B x D x y x D x y x i
where
y x p x a b c d
n
and
B x c ix d ix
D x a ix b ix
Generating functions
1
X
a ix b ix c ix d ix
n
p x a b c d t it it F F
n
n
1
X
p x a b c d d ix a ix
n
n
t it it F F
a c b d b d a c
n n
n
1
X
p x a b c d c ix a ix
n
n
it F it F t
b c a d a d b c
n n
n
t a b c d a b c d a ix
a b c d
t F
a c a d t
1
X
a b c d
n
n
p x a b c d t
n
n
a c a d i
n n
n
Remark Since the generating function is divergent this relation must b e seen as an
equality in terms of formal p ower series
References
Hahn
De nition
n n x
n N F Q x N
n
N
Orthogonality
N
X
x N x
Q x N Q x N
m n
x N x
x
n
n n
n N
mn
N n N
n n
Recurrence relation
xQ x A Q x A C Q x C Q x
n n n n n n n n
where
Q x Q x N
n n
and
n n N n
A
n
n n
n n n N
C
n
n n
Di erence equation
n n y x B x y x B x D x y x D x y x
where
y x Q x N
n
and
B x x N x
D x x x N
Generating functions
N
X
N x x N
n
n
Q x N t t t F F
n
n
n
n
N
X
N x x N
n
n
t t F F Q x N t
n
N N N n
n
n
t x
t F
N t
N
X
n
n
Q x N t
n
n
n
Remarks If we interchange the role of x and n in we obtain the dual Hahn p olynomials
de ned by
Since
Q x N R n N
n x
we obtain the dual orthogonality relation for the Hahn p olynomials from the orthogonality relation
of the dual Hahn p olynomials
N
X
N N n
n n
Q x N Q y N
n n
n
n n
n N
n
xy
x y f N g
N x x
N x x
For x N the generating function can also b e written as
N
X
N x x N
n
n
t t F F Q x N t
n
n
n
n
References
Dual Hahn
De nition
n x x
n N R x N F
n
N
where
x x x
Orthogonality
N
X
N N x
x x
R x N R x N
m n
x
x x
x N
x
mn
n N n
n N n
Recurrence relation
x R x A R x A C R x C R x
n n n n n n n n
where
R x R x N
n n
and
A n N n
n
C n n N
n
Di erence equation
ny x B x y x B x D x y x D x y x y x R x N
n
where
x x N x
B x
x x
x x x N
D x
x x
Generating functions
N
X
x N x
n
n x
R x N t t t F
n
n N
n
N
X
N x x
n
N x n
t t F R x N t
n
n
n
N
X
N x N x
n n
n x
R x N t t F t
n
N n N
n
n
N
X
R x N x x
n
t n
t e F t
N n
n
Remarks If we interchange the role of x and n in the de nition of the dual Hahn
p olynomials we obtain the Hahn p olynomials de ned by
Since
R x N Q n N
n x
we obtain the dual orthogonality relation for the dual Hahn p olynomials from the orthogonality
relation for the Hahn p olynomials
N
X
N n n
R x N R y N
n n
N n n
n
x
x x
x N
x y f N g
xy
N N x
x x
For x N the generating function can also b e written as
N
X
N x x
n
n N x
R x N t t t F
n
n
n
For x N the generating function can also b e written as
N
X
N x N x
n n
x n
t t F R x N t
n
N N n
n
n
References
Meixner Pollaczek
De nition
n ix
n
i in
e P x e F
n
n
Orthogonality
1
Z
x
e j ix j P x P x dx
m n
1
n
and
mn
sin n
Recurrence relation
n P x x sin n cos P x n P x
n
n n
Di erence equation
i i
e ix y x i i x cos n sin y x e ix y x i
where
x y x P
n
Generating functions
1
X
i ix i ix n
e t e t P x t
n
n
1
X
P x ix
n
n i t
t e t F e
in
e
n
n
References
Jacobi
De nition
n n x
n
F P x
n
n
Orthogonality
Z
x x P x P x dx
m n
n n
and
mn
n n n
Recurrence relation
n n
xP x P x
n n
n n
n n
P P x x
n
n
n n n n
Di erential equation
00 0
x y x x y x n n y x y x P x
n
Generating functions
1
p
X
n
xt t P x t R
n
R R t R t
n
1
X
P x x t x t
n
n
t F F
n n
n
x t
t F
t
1
X
n
n
P x t
n
n
n
x t
t F
t
1
X
n
n
x t P
n
n
n
R t R t
F F
1
p
X
n n
n
P x t R xt t arbitrary
n
n n
n
Remarks The Jacobi p olynomials de ned by and the Meixner p olynomials given by
are related in the following way
c
n
n x
M x c P
n
n
n c
The Jacobi p olynomials are also related to the Gegenbauer or ultraspherical p olynomials
de ned by by the quadratic transformations
1 1
n
2 2
P C x x
n
n
n
and
1 1
n
2 2
x xP C x
n
n
n
References
Sp ecial cases
Gegenbauer Ultraspherical
De nition The Gegenbauer or ultraspherical p olynomials are Jacobi p olynomials with
and another normalization
1 1
n n x
n n
2 2
C x P F x
n
n
n
n
Orthogonality
Z
1
n
2
C x C x dx and x
mn
m n
f g n n
Recurrence relation
n xC x n C x n C x
n
n n
Di erential equation
00 0
x y x xy x n n y x y x C x
n
Generating functions
1
X
n
x t C xt t
n
n
1
1
p
X 2
R xt
n
n
R C x t R xt t
n
n
n
1
X
x t x t C x
n
n
t F F
n n
n
1
X
C x x t
n
n xt
t e F
n
n
R t R t
F F
1
p
X
n n
n
C xt t arbitrary x t R
n
n n
n
1
X
x t
n
n
xt F C x t arbitrary
n
xt
n
n
Remarks The case needs another normalization In that case we have the Chebyshev
p olynomials of the rst kind describ ed in the next subsection
The Gegenbauer or ultraspherical p olynomials de ned by and the Jacobi p olynomials
given by are related by the quadratic transformations
1 1
n
2 2
C x P x
n
n
n
and
1 1
n
2 2
xP C x x
n
n
n
References
Chebyshev
De nitions The Chebyshev p olynomials of the rst kind can b e obtained from the Jacobi
p olynomials by taking
1 1
2 2
x P x n n
n
T x F
n
1 1
2 2
P
n
and the Chebyshev p olynomials of the second kind can b e obtained from the Jacobi p olynomials
by taking
1 1
2 2
P n n x x
n
U x n n F
n
1 1
2 2
P
n
Orthogonality
Z
n
mn
1
2
T x T x dx x
m n
n
mn
Z
1
2
x U x U x dx
mn m n
Recurrence relations
xT x T x T x T x and T x x
n n n
xU x U x U x
n n n
Di erential equations
00 0
x y x xy x n y x y x T x
n
00 0
x y x xy x n n y x y x U x
n
Generating functions
1
X
xt
n
T x t
n
xt t
n
r
1
p
X
n
n
xt t R R xt T x t R
n
n
n
1
X
T x x t x t
n
n
F F t
n
n
n
1
X
x t T x
n
xt n
e F t
n
n
R t R t
F F
1
p
X
n n
n
xt t arbitrary T x t R
n
n
n
n
1
X
x t
n
n
xt F T x t arbitrary
n
xt n
n
1
X
n
U x t
n
xt t
n
1
p
X
n n
q
xt t U x t R
n
n
R R xt
n
1
X
x t x t U x
n
n
F t F
n
n
n
1
X
U x x t
n
n xt
t e F
n
n
R t R t
F F
1
p
X
n n
n
U x t R xt t arbitrary
n
n
n
n
1
X
x t
n
n
U x t arbitrary xt F
n
xt n
n
Remarks The Chebyshev p olynomials can also b e written as
T x cos n arccos x
n
and
sin n
x cos U x
n
sin
Further we have
U x C x
n
n
x denotes the Gegenbauer or ultraspherical p olynomial de ned by in the where C
n
preceding subsection
References
Legendre Spherical
De nition The Legendre or spherical p olynomials are Jacobi p olynomials with
n n x
P x P x F
n
n
Orthogonality
Z
P x P x dx
mn m n
n
Recurrence relation
n xP x n P x nP x
n n n
Di erential equation
00 0
x y x xy x n n y x y x P x
n
Generating functions
1
X
n
p
P x t
n
xt t
n
1
X
x t x t P x
n
n
F F t
n
n
1
X
x t P x
n
xt n
F e t
n
n
R t R t
F F
1
p
X
n n
n
xt t P x t R arbitrary
n
n
n
1
X
x t
n
n
P x t arbitrary xt F
n
xt n
n
References
Meixner
De nition
n x
M x c F
n
c
Orthogonality
1
n
X
c n
x
x
c M x c M x c and c
m n mn
x c
n
x
Recurrence relation
c xM x c c n M x c
n n
n n c M x c nM x c
n n
Di erence equation
n c y x c x y x x x c y x xy x y x M x c
n
Generating functions
1
x
X
t
n
x n
t M x c t
n
c n
n
1
X
x c M x c
n
t n
e F t t
c n
n
Remarks The Meixner p olynomials de ned by and the Jacobi p olynomials given by
are related in the following way
c
n
n x
M x c P
n
n
n c
The Meixner p olynomials are also related to the Krawtchouk p olynomials de ned by
in the following way
p
x N K x p N M
n n
p
References
Krawtchouk
De nition
n x
n N F K x p N
n
p N
Orthogonality
N
n
n
X
p n N
x N x
p p p K x p N K x p N
mn m n
N p x
n
x
Recurrence relation
xK x p N p N n K x p N
n n
p N n n p K x p N n p K x p N
n n
Di erence equation
ny x p N x y x p N x x p y x x p y x
where
y x K x p N
n
Generating functions
N
x
X
N p
n N x
K x p N t t t
n
n p
n
N
X
t x K x p N
n
t n
F e t
N p n
n
Remarks The Krawtchouk p olynomials are self dual which means that
K x p N K n p N n x f N g
n x
By using this relation we easily obtain the so called dual orthogonality relation from the orthog
onality relation
x
p
N
X
N
p
n N n
p p K x p N K y p N
xy n n
N
n
n
x
where p and x y f N g
The Krawtchouk p olynomials are related to the Meixner p olynomials de ned by in the
following way
p
x N K x p N M
n n
p
For x N the generating function can also b e written as
N
x
X
p N
n N x
K x p N t t t
n
n p
n
References
Laguerre
De nition
n
n
L x x F
n
n
Orthogonality
1
Z
n
x
x e L x L x dx
mn
m n
n
Recurrence relation
n L x n x L x n L x
n n n
Di erential equation
00 0
xy x x y x ny x y x L x
n
Generating functions
1
X
xt
n
t exp L x t
n
t
n
1
X
x L
n
n t
t xt e F
n
n
1
X
xt
n
n
L x t arbitrary F t
n
t
n
n
Remarks The de nition of the Laguerre p olynomials can also b e written as
n
X
n
k
k
L x k x
n k
n
n k
k
In this way the Laguerre p olynomials can b e de ned for all Then we have the following
connection with the Charlier p olynomials de ned by
n
a
x n
C x a L a
n
n
n
The Laguerre p olynomials de ned by and the Hermite p olynomials de ned by
are connected by the following quadratic transformations
1
n n
2
H x n L x
n
n
and
1
n n
2
H x n xL x
n
n
In combinatorics the Laguerre p olynomials with are often called Ro ok p olynomials
References
Charlier
De nition
n x
C x a F
n
a
Orthogonality
1
x
X
a
n a
C x a C x a n a e a
m n mn
x
x
Recurrence relation
xC x a aC x a n a C x a nC x a
n n n n
Di erence equation
ny x ay x x a y x xy x y x C x a
n
Generating function
1
x
X
t C x a
n
t n
e t
a n
n
Remark The de nition of the Laguerre p olynomials can also b e written as
n
X
n
k
k
k x L x
n k
n
n k
k
In this way the Laguerre p olynomials can b e de ned for all Then we have the following
connection with the Charlier p olynomials de ned by
n
a
x n
C x a L a
n
n
n
References
Hermite
De nition
n n
n
H x x F
n
x
Orthogonality
1
Z
p
2
x n
e H x H x dx n
mn m n
1
Recurrence relation
H x xH x nH x
n n n
Di erential equation
00 0
y x xy x ny x y x H x
n
Generating functions
1
X
H x
n
n
exp xt t t
n
n
1
n
X
p
t n
t e cos x H x t
n
n
n
1
n t
X
p
e
n
p
sin x H x t t
n
n
t
n
1
X
2
H x
n
t n
e cosh xt t
n
n
1
X
2
H x
n
n t
t e sinh xt
n
n
1
X
x t
n
n
t F H x t
n
t n
n
1
X
x t xt
n
n
p
F H x t
n
t n
t
n
1
X
H x x t xt t
n
n
t exp
3
t n
2
t
n
where denotes the largest integer smaller than or equal to
Remarks The Hermite p olynomials can also b e written as
n
k n k
X
H x x
n
n k n k
k
where denotes the largest integer smaller than or equal to
The Laguerre p olynomials de ned by and the Hermite p olynomials de ned by
are connected by the following quadratic transformations
1
n n
2
H x n L x
n
n
and
1
n n
2
H x n xL x
n
n
References
Chapter
Limit relations b etween
hyp ergeometric orthogonal
p olynomials
Wilson ! Continuous dual Hahn
The continuous dual Hahn p olynomials can b e found from the Wilson p olynomials de ned by
by dividing by a d and letting d
n
W x a b c d
n
S x a b c lim
n
d!1
a d
n
where S x a b c is de ned by
n
Wilson ! Continuous Hahn
The continuous Hahn p olynomials de ned by are obtained from the Wilson p olynomials
by the substitution a a it b b it c c it d d it and x x t in the de nition
of the Wilson p olynomials and the limit t in the following way
W x t a it b it c it d it
n
lim p x a b c d
n
n
t!1
t n
Wilson ! Jacobi
The Jacobi p olynomials given by can b e found from the Wilson p olynomials by substituting
q
c it d it and x t x in the de nition a b
of the Wilson p olynomials and taking the limit t In fact we have
W x t it it
n
lim P x
n
n
t!1
t n
Racah ! Hahn
If we take N and let in the de nition of the Racah p olynomials we obtain
the Hahn p olynomials de ned by Hence
lim R x N Q x N
n n
!1
The Hahn p olynomials can also b e obtained from the Racah p olynomials by taking N
in the de nition and letting
lim R x N Q x N
n n
!1
Another way to do this is to take N and N in the de nition of the
Racah p olynomials and then take the limit In that case we obtain the Hahn p olynomials
given by in the following way
lim R x N N Q x N
n n
!1
Racah ! Dual Hahn
If we take N and let in then we obtain the dual Hahn p olynomials from
the Racah p olynomials So we have
lim R x N R x N
n n
!1
And if we take N and let in then we also obtain the dual Hahn
p olynomials
lim R x N R x N
n n
!1
Finally if we take N and N in the de nition of the Racah
p olynomials and take the limit we nd the dual Hahn p olynomials given by in the
following way
lim R x N N R x N
n n
!1
Continuous dual Hahn ! Meixner Pollaczek
The Meixner Pollaczek p olynomials given by can b e obtained from the continuous dual
Hahn p olynomials by the substitutions x x t a it b it and c t cot in the
de nition and the limit t
x t it it t cot S
n
P x lim
n
t!1
t
n
sin
n
Continuous Hahn ! Meixner Pollaczek
By taking x x t a it c it and b d t tan in the de nition of
the continuous Hahn p olynomials and taking the limit t we obtain the Meixner Pollaczek
p olynomials de ned by
p x t it t tan it t tan
n
P x lim
n
t!1
it
n
i
cos
n
Continuous Hahn ! Jacobi
The Jacobi p olynomials de ned by follow from the continuous Hahn p olynomials by the
substitution x xt a it b it c it and d it
n n
in division by t and the limit t
xt it it it it p
n
P x lim
n
n n
t!1
t
Hahn ! Jacobi
To nd the Jacobi p olynomials from the Hahn p olynomials we take x N x in and let
N We have
x P
n
lim Q N x N
n
N !1
P
n
Hahn ! Meixner
If we take b N c c in the de nition of the Hahn p olynomials and let
N we nd the Meixner p olynomials given by
c
x b N lim Q N M x b c
n n
N !1
c
Hahn ! Krawtchouk
If we take pt and p t in the de nition of the Hahn p olynomials and let t
we obtain the Krawtchouk p olynomials de ned by
lim Q x pt p t N K x p N
n n
t!1
Dual Hahn ! Meixner
To obtain the Meixner p olynomials from the dual Hahn p olynomials we have to take
and N c c in the de nition of the dual Hahn p olynomials and let N
c
N M x c x N lim R
n n
N !1
c
Dual Hahn ! Krawtchouk
In the same way we nd the Krawtchouk p olynomials from the dual Hahn p olynomials by setting
pt p t in and let t
lim R x pt p t N K x p N
n n
t!1
Meixner Pollaczek ! Laguerre
The Laguerre p olynomials can b e obtained from the Meixner Pollaczek p olynomials de ned by
by the substitution x x and letting
1 1
x
2 2
lim P x L
n
n
!
Meixner Pollaczek ! Hermite
p
cos in the de nition of the Meixner Pollaczek If we substitute x sin x
p olynomials and then let we obtain the Hermite p olynomials
p
cos H x n x
n
2
lim P
n
!1
sin n
Jacobi ! Laguerre
The Laguerre p olynomials can b e obtained from the Jacobi p olynomials de ned by by
letting x x and then
x
lim P L x
n n
!1
Jacobi ! Hermite
The Hermite p olynomials given by follow from the Jacobi p olynomials de ned by
by taking and letting in the following way
H x x n
n
2
P lim
1
n
n
!1
n
2
Meixner ! Laguerre
If we take and x c x in the de nition of the Meixner p olynomials and
let c we obtain the Laguerre p olynomials
x L x
n
c lim M
n
c!
c
L
n
Meixner ! Charlier
If we take c a a in the de nition of the Meixner p olynomials and let we
nd the Charlier p olynomials
a
lim M x C x a
n n
!1
a
Krawtchouk ! Charlier
The Charlier p olynomials given by can b e found from the Krawtchouk p olynomials de ned
by by taking p N a and let N
a
N C x a x lim K
n n
N !1
N
Krawtchouk ! Hermite
The Hermite p olynomials follow from the Krawtchouk p olynomials de ned by by setting
p
x pN x p p N and then letting N
s
n
p
N H x
n
s
lim pN x K p p N p N
n
n
N !1
n
p
n
n
p
Laguerre ! Hermite
The Hermite p olynomials de ned by can b e obtained from the Laguerre p olynomials given
by by taking the limit in the following way
n
n
2
1
2
lim L x H x
n
n
!1
n
Charlier ! Hermite
If we set x a x a in the de nition of the Charlier p olynomials and let a
we nd the Hermite p olynomials de ned by In fact we have
1 n
n
2 2
a C x a a H x lim a
n n
a!1
SCHEME
OF
BASIC HYPERGEOMETRIC
ORTHOGONAL POLYNOMIALS
Askey Wilson
Big
Continuous Continuous
dual q Hahn q Hahn
q Jacobi
q Meixner
Al Salam
Big
Continuous Little
q Jacobi
q Jacobi
q Laguerre
Chihara
Pollaczek
Continuous Continuous Little
q Laguerre
big q Hermite q Laguerre q Laguerre
Stieltjes
Continuous
q Hermite
Wigert
SCHEME
OF
BASIC HYPERGEOMETRIC
ORTHOGONAL POLYNOMIALS
q Racah
Big
q Hahn Dual q Hahn
q Jacobi
Quantum
A ne Dual
q Meixner q Krawtchouk
q Krawtchouk q Krawtchouk
q Krawtchouk
Al Salam Al Salam
Alternative
q Charlier
q Charlier
Carlitz I Carlitz I I
Discrete Discrete
q Hermite I q Hermite I I
Chapter
Basic hyp ergeometric orthogonal
p olynomials
Askey Wilson
De nition
n n i i n
q abcdq ae ae a p x a b c djq
n
q q x cos
ab ac ad ab ac ad q
n
The Askey Wilson p olynomials are q analogues of the Wilson p olynomials given by
Orthogonality When a b c d are real or o ccur in complex conjugate pairs if complex and
max jaj jbj jcj jdj then we have the following orthogonality relation
Z
w x
p
p x a b c djq p x a b c djq dx h
m n n mn
x
where
1 1
i
2 2
e q h x h x h x q h x q
1
w x w x a b c djq
i i i i
ae be ce de q h x a h x b h x c h x d
1
with
1
Y
k k i i
h x xq q e e q x cos
1
k
and
n n
abcdq q abcdq q
n 1
h
n
n n n n n n n
q abq acq adq bcq bdq cdq q
1
If a and b c d are real or one is real and the other two are complex conjugates
max jbj jcj jdj and the pairwise pro ducts of a b c and d have absolute value less than one
then we have another orthogonality relation given by
Z
w x
p
p x a b c djq p x a b c djq dx
m n
x
X
w p x a b c djq p x a b c djq h
k m k n k n mn
k
k
aq a
where w x and h are as b efore
n
k k
aq aq
x
k
and
k
k
a q a q a ab ac ad q q
1 k
w
k
q ab ac ad a b a c a d q a q ab q ac q ad q q abcd
1 k
Recurrence relation
xp x A p x a a A C p x C p x
n n n n n n n n
where
n
a p x a b c djq
n
p x
n
ab ac ad q
n
and
n n n n
abq acq adq abcdq
A
n
n n
a abcdq abcdq
n n n n
a q bcq bdq cdq
C
n
n n
abcdq abcdq
q Di erence equation
h i
1 1 1 1
2 2 2 2
w x aq q D bq cq dq jq D y x
q q
w x a b c djq y x y x p x a b c djq
n n
where
w x a b c djq
p
w x a b c djq
x
n n n
q q abcdq
n
and
f x 1 1
q
i i i
2 2
D f x e f q e x cos with f e f q
q q
x
q
If we de ne
n n
q abcdq az az ab ac ad q
n
q q P z
n
n
ab ac ad a
then the q di erence equation can also b e written in the form
n n n
q q abcdq P z
n
A z P q z A z A z P z A z P q z
n n n
where
az bz cz dz
A z
z q z
Generating functions
1
i i i i
X
p x a b c djq ce de ae be
n
i i n
q e t q e t t x cos
cd ab cd q q ab
n
n
1
i i i i
X
p x a b c djq ae ce be de
n
n i i
t x cos q e t q e t
ac bd q q ac bd
n
n
1
i i i i
X
p x a b c djq ae de be ce
n
i i n
q e t q e t t x cos
ad bc ad bc q q
n
n
Remark The q Racah p olynomials de ned by and the Askey Wilson p olynomials
given by are related in the following way If we substitute a q b q
i x
c q d q and e q in the de nition of the Askey Wilson
p olynomials we nd
1 1 1 1 1 1 1 1 1 1 1 1 1
n
2 2 2 2 2 2 2 2 2 2 2 2 2
q p x q q q q jq
n
R x jq
n
q q q q
n
where
1 1 1 1 1 1
x x
2 2 2 2 2 2
x q q
References
q Racah
De nition
n n x x
q q q q
q q n N R x jq
n
q q q
where
x x
x q q
and
N N N
q q or q q or q q with N a nonnegative integer
Since
k
Y
x x j j
q q q x q q
k
j
it is clear that R x jq is a p olynomial of degree n in x
n
Orthogonality
N
x
X
q q q q q q
x
R x R x h
m n n mn
x
q q q q q q q
x
x
where
R x R x jq
n n
and
n
q q q q q q q q q
n 1
h
n
n
q q q q q q q q q q q
1 n
This implies
N n N N
q q q q q q q q q
N n
N
if q q
n N N N
q q q q q q q q q
N n
N N n
q q q q q q q q q
n N
N
h
if q q
n
n N
q q q q q q q q q
N n
N n N
q q q q q q q q q
n N
N
if q q
n N
q q q q q q q q q
N n
Recurrence relation
x x
q q R x
n
A R x A C R x C R x
n n n n n n n
where
n n n n
q q q q
A
n
n n
q q
n n n n
q q q q q
C
n
n n
q q
q Di erence equation
w x B x y x
n n n
q q q w x y x y x R x jq
n
where
f x f x f x
x
q q q q q q
x
w x
x
q q q q q q q
x
and B x as b elow This q di erence equation can also b e written in the form
n n n
q q q y x B x y x B x D x y x D x y x
where
y x R x jq
n
and
x x x x
q q q q
B x
x x
q q
x x x x
q q q q q
D x
x x
q q
Generating functions
x x x x
q q q q
x x
q q t q q t
q q
N
X
q q q
n
n
R x jq t
n
q q q
n
n
x x x x
q q q q
x x
q q t q q t
q q
N
X
q q q
n
n
R x jq t
n
q q q
n
n
x x x x
q q q q
x x
q q t q q t
q q
N
X
q q q
n
n
R x jq t
n
q q q
n
n
Remark The Askey Wilson p olynomials de ned by and the q Racah p olynomials given
by are related in the following way If we substitute abq cdq adq
x i
ad and q a e in the de nition of the q Racah p olynomials we nd
x a cos
and
n
a p x a b c djq
n
a cos abq cdq adq ad jq R
n
ab ac ad q
n
References
Continuous dual q Hahn
De nition
n i i n
q ae ae a p x a b cjq
n
q q x cos
ab ac ab ac q
n
Orthogonality When a b c are real or one is real and the other two are complex conjugates
and max jaj jbj jcj then we have the following orthogonality relation
Z
w x
p
p x a b cjq p x a b cjq dx h
m n n mn
x
where
1 1
i
2 2
e q h x h x h x q h x q
1
w x w x a b cjq
i i i
ae be ce q h x a h x b h x c
1
with
1
Y
k k i i
x cos xq q e e q h x
1
k
and
h
n
n n n n
q abq acq bcq q
1
If a and b and c are real or complex conjugates max jbj jcj and the pairwise pro ducts
of a b and c have absolute value less than one then we have another orthogonality relation given
by
Z
w x
p
p x a b cjq p x a b cjq dx
m n
x
X
w p x a b cjq p x a b cjq h
k m k n k n mn
k
k
aq a
where w x and h are as b efore
n
k k
aq aq
x
k
and
k
k
k
a q a q a ab ac q
1 k
k
2
w q
k
q ab ac a b a c q a q ab q ac q q a bc
1 k
Recurrence relation
xp x A p x a a A C p x C p x
n n n n n n n n
where
n
a p x a b cjq
n
p x
n
ab ac q
n
and
n n
A a abq acq
n
n n
C a q bcq
n
q Di erence equation
h i
1 1 1
2 2 2
w x aq q D bq cq jq D y x
q q
n n
q q w x a b cjq y x y x p x a b cjq
n
where
w x a b cjq
p
w x a b cjq
x
and
1 1
f x
q
i i i
2 2
e f q e x cos with f e f q D f x
q q
x
q
If we de ne
n
ab ac q q az az
n
q q P z
n
n
ab ac a
then the q di erence equation can also b e written in the form
n n
q q P z A z P q z A z A z P z A z P q z
n n n n
where
az bz cz
A z
z q z
Generating functions
1
i i
X
p x a b cjq ae be ct q
n 1
i n
q e t t x cos
i
ab e t q ab q q
1 n
n
1
i i
X
bt q p x a b cjq ae ce
1 n
i n
q e t t x cos
i
ac e t q ac q q
1 n
n
1
i i
X
p x a b cjq be ce at q
n 1
i n
q e t t x cos
i
bc e t q bc q q
1 n
n
References
Continuous q Hahn
De nition
i n n n i i
ae p x a b c d q q abcdq ae ae
n
q q x cos
i i
abe ac ad abe ac ad q
n
Orthogonality When c a and d b then we have if a and b are real and max jaj jbj
or if b a and jaj
Z
w cos p cos a b c d q p cos a b c d q d h
m n n mn
where
i
e q
1
w x w x a b c d q
i i i i
ae be ce de q
1
1 1
2 2
h x q h x h x h x q
i i i i
h x ae h x be h x ce h x de
with
1
Y
k k i i
h x xq q e e q x cos
1
k
and
n n
abcdq q abcdq q
n 1
h
n
n n i n n n n n i
q abq e acq adq bcq bdq cdq e q
1
Recurrence relation
i i
xp x A p x ae a e A C p x C p x
n n n n n n n n
where
i n
ae p x a b c d q
n
p x
n
i
ac ad abe q
n
and
i n n n n
abe q acq adq abcdq
A
n
i n n
ae abcdq abcdq
i n n n i n
ae q bcq bdq cde q
C
n
n n
abcdq abcdq
q Di erence equation
i h
1 1 1 1
2 2 2 2
bq cq dq q D y x q D w x aq
q q
w x a b c d q y x y x p x a b c d q
n n
where
w x a b c d q
p
w x a b c d q
x
n n n
q q abcdq
n
and
1 1 f x
q
i i i
2 2
with f e f q e f q e x cos D f x
q q
x
q
Generating functions
i i i i
ce de ae be
i i
q e t q e t
i i
cde abe
1
n
X
p x a b c d q t
n
x cos
i i
abe cde q q
n
n
i i i i
ae ce be de
i i
q e t q e t
ac bd
1
X
p x a b c d q
n
n
t x cos
ac bd q q
n
n
i i i i
be ce ae de
i i
q e t q e t
bc ad
1
X
p x a b c d q
n
n
t x cos
ad bc q q
n
n
References
Big q Jacobi
De nition
n n
q abq x
q q P x a b c q
n
aq cq
Orthogonality
aq
Z
a x c x q
1
P x a b c q P x a b c q d x
m n q
x bc x q
1
cq
q a c ac q abq q
1
aq q
aq bq cq abc q q
1
n
abq q bq abc q q
n
n
2
acq q
mn
n
abq abq aq cq q
n
Recurrence relation
x P x a b c q
n
A P x a b c q A C P x a b c q C P x a b c q
n n n n n n n
where
n n n
aq cq abq
A
n
n n
abq abq
n n n
q bq abc q
n
C acq
n
n n
abq abq
q Di erence equation
n n n
q q abq x y x B x y q x B x D x y x D x y q x
where
y x P x a b c q
n
and
B x aq x bx c
D x x aq x cq
Generating functions
1
X
cq q aq x bc x
n
n
q xt q cq t P x a b c q t
n
aq bq bq q q
n
n
1
X
aq q bc x cq x
n
n
P x a b c q t q aq t q xt
n
abc q abc q q q cq
n
n
Remarks The big q Jacobi p olynomials with c and the little q Jacobi p olynomials de ned
by are related in the following way
n
x bq q
n
n
n n
2
a q b a q p P x a b q
n n
aq q aq
n
Sometimes the big q Jacobi p olynomials are de ned in terms of four parameters instead of
three In fact the p olynomials given by the de nition
n n
q abq ac q x
q q P x a b c d q
n
aq ac dq
are orthogonal on the interval d c with resp ect to the weight function
c q x d q x q
1
d x
q
ac q x bd q x q
1
These p olynomials are not really di erent from those de ned by since we have
P x a b c d q P ac q x a b ac d q
n n
and
P x a b c q P x a b aq cq q
n n
References
Sp ecial case
Big q Legendre
De nition The big q Legendre p olynomials are big q Jacobi p olynomials with a b
n n
q q x
q q P x c q
n
q cq
Orthogonality
q
Z
n
c q q q
n
n
2
cq q P x c q P x c q d x q c
mn m n q
n
q cq q
n
cq
Recurrence relation
x P x c q A P x c q A C P x c q C P x c q
n n n n n n n n
where
n n
q cq
A
n
n n
q q
n n
q c q
n
C cq
n
n n
q q
q Di erence equation
n n n
q q q x y x B x y q x B x D x y x D x y q x
where
y x P x c q
n
and
B x q x x c
D x x q x cq
Generating functions
1
X
cq q c x q x
n
n
P x c q t q cq t q xt
n
q q q q q
n
n
1
X
P x c q c x cq x
n
n
q q t q xt t
c q cq c q q
n
n
References
q Hahn
De nition
n n x
q q q
x
q q n N Q q N jq
n
N
q q
Orthogonality
N
N
X
q q q
x
x x x
q Q q N jq Q q N jq
m n
N
q q q
x
x
N n
n
q q q q q q q q
n N
N n
2
q
mn
N N n
q q q q q q q q
N n
Recurrence relation
x x x x x
q Q q A Q q A C Q q C Q q
n n n n n n n n
where
x x
Q q Q q N jq
n n
and
n N n n
q q q
A
n
n n
q q
n n n N n
q q q q q
C
n
n n
q q
q Di erence equation
n n n
q q q y x B x y x B x D x y x D x y x
where
x
y x Q q N jq
n
and
x N x
B x q q
x x N
D x q q q
Generating functions
N
N x x N
X
q q q q
n
x x n
q q t q q t Q q N jq t
n
q q q q q
n
n
N x x N
q q
x
q q t q q t
N N
q q
N
X
q q
n
x n
Q q N jq t
n
N
q q q
n
n
Remarks The q Hahn p olynomials de ned by and the dual q Hahn p olynomials given
by are related in the following way
x
Q q N jq R n N jq
n x
with
n n
n q q
or
n
R x N jq Q q N jq
n x
where
x x
x q q
For x N the generating function can also b e written as
N
x N x N
X
q q q q
n
x n x
q q t Q q N jq t q q t
n
q q q q q
n
n
References
Dual q Hahn
De nition
n x x
q q q
q q n N R x N jq
n
N
q q
where
x x
x q q
Orthogonality
N
x N
X
x
q q q q q
x
N x
2
q R x N jq R x N jq
m n
N x
q q q q q q
x
x
N
q q q q q
N n
N n
q q
mn
N
q q q q q
N n
Recurrence relation
x x
q q R x
n
A R x A C R x C R x
n n n n n n n
where
R x R x N jq
n n
and
n N n
A q q
n
n n N
C q q q
n
q Di erence equation
n n
q q y x B x y x B x D x y x D x y x
where
y x R x N jq
n
and
x x x N
q q q
B x
x x
q q
x N x x x N
q q q q
D x
x x
q q
Generating functions
N
x N x
X
q t q q q q q
1 n
x n
q q t R x N jq t
n
N x
q q t q q q
1 n
n
N
N N x x
X
q q q t q q q
n 1
x n
q q t R x N jq t
n
x
q q t q q q
1 n
n
N
N x N x
X
q q q q q q t q
n 1
x n
q q t R x N jq t
n
N x N
q q t q q q q
1 n
n
Remarks The dual q Hahn p olynomials de ned by and the q Hahn p olynomials given
by are related in the following way
x
Q q N jq R n N jq
n x
with
n n
n q q
or
n
R x N jq Q q N jq
n x
where
x x
x q q
For x N the generating function can also b e written as
N
N x x
X
q q q q
n
x n N
q q t R x N jq t q t q
n N x
q q q
n
n
For x N the generating function can also b e written as
N
N x N x
X
q q q q q
n
x n
q q t q t q R x N jq t
x n
N N
q q q q
n
n
References
Al Salam Chihara
De nition
n i i
q ae ae ab q
n
q q Q x a bjq
n
n
ab a
n i
q ae
i i in
q b q e be q e x cos
n
n i
b q e
Orthogonality When a and b are real or complex conjugates and max jaj jbj then we
have the following orthogonality relation
Z
w x
mn
p
Q x a bjq Q x a bjq dx
m n
n n
q abq q
x
1
where
1 1
i
2 2
e q h x h x h x q h x q
1
w x w x a bjq
i i
ae be q h x a h x b
1
with
1
Y
i i k k
e e q x cos xq q h x
1
k
If a jbj and jabj then we have another orthogonality relation given by
Z
w x
p
Q x a bjq Q x a bjq dx
m n
x
X
mn
w Q x a bjq Q x a bjq
k m k n k
n n
q abq q
1
k
k
aq a
where w x is as b efore
k k
aq aq
x
k
and
k
k
2
a q a q a ab q
1 k
k
w q
k
q ab a b q a q ab q q a b
1 k
Recurrence relation
xQ x A Q x a a A C Q x C Q x
n n n n n n n n
where
n
a Q x a bjq
n
Q x
n
ab q
n
and
n
A a abq
n
n
C a q
n
q Di erence equation
h i
1 1
2 2
w x aq q D bq jq D y x
q q
n n
q q w x a bjq y x y x Q x a bjq
n
where
w x a bjq
p
w x a bjq
x
and
1 f x 1
q
i i i
2 2
e f q e x cos D f x with f e f q
q q
x
q
If we de ne
n
q az az ab q
n
q q P z
n
n
a ab
then the q di erence equation can also b e written in the form
n n
q q P z A z P q z A z A z P z A z P q z
n n n n
where
az bz
A z
z q z
Generating functions
1
i i
X
ae be Q x a bjq
n
i n
q e t t x cos
i
ab e t q ab q q
1 n
n
1
X
at bt q Q x a bjq
1 n
n
t x cos
i i
e t e t q q q
1 n
n
References
q Meixner Pollaczek
De nition
n i i
q ae ae a q
n
n in
q q P x ajq a e
n
a q q
n
n i i
q ae ae q
n
i in
q q a e e x cos
n i
a q e q q
n
Orthogonality
Z
mn
w cos ajq P cos ajq P cos ajq d
m n
n
q q q a q q
n 1
where
a
and
1 1
i
2 2
h x h x h x q h x q e q
1
w x ajq
i i
i i
h x ae h x ae
ae ae q
1
with
1
Y
i i k k
h x e e q x cos xq q
1
k
Recurrence relation
n
xP x ajq q P x ajq
n n
n n
aq cos P x ajq a q P x ajq
n n
q Di erence equation
h i
1
n n
2
w x aq q D jq D y x q q w x ajq y x y x P x ajq
q q n
where
w x ajq
p
w x ajq
x
and
1 1
f x
q
i i i
2 2
D f x e f q e x cos with f e f q
q q
x
q
Generating functions
1
i
X
ae t q
1
n
P x ajq t x cos
n
i
e t q
1
n
1
i i
X
P x ajq ae ae
n
i n
q e t t x cos
i
a a q
e t q
n
1
n
References
Continuous q Jacobi
1 1 3 1 3 1 1 1
2 4 2 4 2 4 2 4
De nitions If we take a q b q c q and d q in the de nition
of the Askey Wilson p olynomials we nd after renormalizing
1 1 1 1
i i n n
2 4 2 4
e q e q q q
q q
n
q q x cos P xjq
1 1
n
2 2
q q
q q q
n
1 1 1 1
2 2 2 2
b q c q and d q to obtain after renormalizing In M Rahman takes a q
1 1
n n i i
2 2
q q q q q q e q e
n
P x q q q x cos
n
q q q q q q
n
These two q analogues of the Jacobi p olynomials are not really di erent since they are connected
by the quadratic transformation
q q
n
n
P xjq q P x q
n n
q q
n
and the orthogonality relations are resp ectively Orthogonality For
Z
w xjq
p
P xjq P xjq dx
m n
x
1 1
2 2
q q q
1
1 1
2 2
q q q q q q
1
1
2
q q q q q 1
n
n
2
q
mn
1
n
2
q q q q q
n
where
i
e q
1
w xjq w x q q jq
1 1 1 1 1 3 1 3
i i i i
2 4 2 4 2 4 2 4
q e q e q e q e q
1
1
i i
2
e e q
1
1 1 1 1 1
i i
2 4 2 4 2
q e q e q
1
1 1
2 2
h x h x h x q h x q
1 1 1 1 1 3 1 3
2 4 2 4 2 4 2 4
h x q h x q h x q h x q
with
1
Y
k k i i
xq q e e q x cos h x
1
k
and
Z
w x q
p
P x q P x q dx
m n
x
q q
1
q q q q q q q q
1
q q q q q q q
n
n
q
mn
n
q q q q q q q
n
where
i
e q
1
w x q w x q q q
1 1 1 1
i i i i
2 2 2 2
q e q e q e q e q
1
i i
h x h x e e q
1
1 1 1 1
i i
2 2 2 2
q h x q e q e q h x q
1
with
1
Y
k k i i
xq q e e q x cos h x
1
k
Recurrence relations
i h
1 1 1 1
2 4 2 4
q A C P xjq C P xjq xP xjq A P xjq q
n n n n n n n n
where
q q
n
P xjq P xjq
n
n
q q
n
and
1 1
n n n n
2 2
q q q q
A
n
1 1
n n
2 4
q q q
1 1 1 1
n n n n
2 4 2 2
q q q q q
C
n
n n
q q
i h
1 1
2 2
q A C P x q P x q C P x q q xP x q A
n n n n n n n n
where
q q q
n
P x q P x q
n
n
q q q
n
and
n n n n
q q q q
A
n
1
n n
2
q q q
1
n n n n
2
q q q q q
C
n
n n
q q
q Di erence equations
i h
1 1
2 2
q jq D y x w x q q jq y x y x P xjq q D w x q
q n q
n
where
w x q q jq
p
w x q q jq
x
n n n
q q q
n
and
1 f x 1
q
i i i
2 2
e f q e x cos D f x with f e f q
q q
x
q
h i
1 1
2 2
w x q q D q q D y x w x q q q y x y x P x q
q q n
n
where
w x q q q
p
w x q q q
x
n n n
q q q
n
and
1 1 f x
q
i i i
2 2
with f e f q e f q e x cos D f x
q q
x
q
Generating functions
1 3 1 3 1 1 1 1
i i i i
2 4 2 4 2 4 2 4
e q e e q e q q
i i
q e t q e t
q q
1 1
1
X
2 2
q xjq P q q
n
n
n
t x cos
1 1
n
q q q
2 4
n q
n
1 1 1 3 3 1 1 1
i i i i
2 4 2 4 2 4 2 4
q e q e e q e q
i i
q e t q e t
1 1
2 2
q q
1
1
X
2
q q P xjq
n
n
n
t x cos
1 1 1
n
2 2 4
q q q
n
n
1 1 1 3 3 1 1 1
i i i i
2 4 2 4 2 4 2 4
e q e e q e q q
i i
q e t q e t
1 1
2 2
q q
1
1
X
2
q q P xjq
n
n
n
t x cos
1 1 1
n
2 2 4
q q q
n
n
1 1 1 1
i i i i
2 2 2 2
q e q e e q e q
i i
q e t q e t
q q
1
X
P x q q q q
n
n
n
t x cos
1
n
q q q
2
n q
n
1 1 1 1
i i i i
2 2 2 2
q e q e e q e q
i i
q e t q e t
q q
1
X
q q q P x q
n
n
n
t x cos
1
n
q q q
2
n q
n
1 1 1 1
i i i i
2 2 2 2
q e q e e q e q
i i
q e t q e t
q q
1
X
q q P x q
n
n
n
t x cos
1
n
q q
2
q n
n
Remark The continuous q Jacobi p olynomials given by and the continuous q
ultraspherical or Rogers p olynomials given by are connected by the quadratic trans
formations
1 1
1 q q q
n
n
2 2
2
x q P q C x q jq
n
n
1 1
2 2
q q q
n
and
1 1
q q 1
n
n
2 2
2
C x q jq xP q x q
n
n
1 1
2 2
q q q
n
References
Sp ecial cases
Continuous q ultraspherical Rogers
1 1 1 1 1 1
2 2 2 2 2 2
b q c and d q in the de nition De nition If we set a
of the Askey Wilson p olynomials and change the normalization we obtain the continuous
q ultraspherical or Rogers p olynomials
1 1
i i n n
2 2
e e 1 q q
q
n
n
2
q q C x jq
1 1
n
2 2
q q
q q
n
n i
q e q
n
n in
q q e
q q
n
n
q q
n
i in
q q e x cos e
n
q q q
n
Orthogonality
Z
w x q q q
1 n
p
C x jq C x jq dx j j
m n mn
n
q q q q q
x
1 n
where
i i
e q e q
1 1
w x w x jq
1 1 1 1 1 1
i
i i i i
e q
2 2 2 2 2 2
e q e e q e q 1
1
1 1
2 2
h x q h x h x h x q
1 1 1 1 1 1
2 2 2 2 2 2
h x q h x h x q h x
with
1
Y
i i k k
e e q x cos xq q h x
1
k
Recurrence relation
n n n
q xC x jq q C x jq q C x jq
n n n
q Di erence equation
h i
1
2
w x q q D jq D y x w x jq y x y x C x jq
q q n n
where
w x jq
p
w x jq
x
n n n
q q q
n
and
f x 1 1
q
i i i
2 2
D f x with f e f q e f q e x cos
q q
x
q
Generating functions
1
i i
X
e t e t q
1
n
C x jq t x cos
n
i i
e t e t q
1
n
1
i
X
C x jq e
n
i n
q e t t x cos
i
e t q q
1 n
n
n
1
n n i
X
2
q e
i i n
q e t e t q C x jq t x cos
1 n
q
n
n
1 1 1 1 1 1
i i i i
2 2 2 2 2 2
e q e e q e
i i
q e t q e t
1 1
2 2
q q
1
1
X
2
q q
n
n
C x jq t x cos
n
1
2
q q
n
n
1 1 1 1 1 1
i i i i
2 2 2 2 2 2
e e q e q e
i i
q e t q e t
q
1 1
1
X
2 2
q q q
n
n
C x jq t x cos
n
q q
n
n
1 1 1 1 1 1
i i i i
2 2 2 2 2 2
e q e q e e
i i
q e t q e t
1 1
2 2
q q
1
1
X
2
q q
n
n
C x jq t x cos
n
1
2
q q
n
n
Remarks The continuous q ultraspherical or Rogers p olynomials can also b e written as
n
X
q q
k n k
i n k
e x cos C x jq
n
q q q q
k n k
k
They can b e obtained from the continuous q Jacobi p olynomials de ned by in the
1
2
following way Set in the de nition and change q by and we nd the
continuous q ultraspherical or Rogers p olynomials with a di erent normalization We have
1
1
+
2
2
q q 1
! q
n
n
2
C x jq xjq P
n
n
q
n
1
2
in the de nition of the q ultraspherical or Rogers p olynomials If we set q
we nd the continuous q Jacobi p olynomials given by with In fact we have
1
q q
n
2
q x q C P xjq
n
1 1
n
n
2 4
q q q
n
If we change q to q we nd
n
C x jq q C x jq
n n
The sp ecial case q of the continuous q ultraspherical or Rogers p olynomials equals the
Chebyshev p olynomials of the second kind de ned by In fact we have
sin n
U x x cos C x q jq
n n
sin
The limit case leads to the Chebyshev p olynomials of the rst kind given by in the
following way
n
q
lim C x jq cos n T x x cos n
n n
!
The continuous q Jacobi p olynomials given by and the continuous q ultraspherical or
Rogers p olynomials given by are connected by the quadratic transformations
1 1
q q q 1
n
n
2 2
2
x q C x q jq q P
n
n
1 1
2 2
q q q
n
and
1 1
q q 1
n
n
2 2
2
C x q jq xP x q q
n
n
1 1
2 2
q q q
n
Finally we remark that the continuous q ultraspherical or Rogers p olynomials are related to
the continuous q Legendre p olynomials de ned by in the following way
1
n
2
C x q jq q P x q
n n
References
Continuous q Legendre
De nition The continuous q Legendre p olynomials are continuous q Jacobi p olynomials with
If we set in the de nition of the continuous q Jacobi p olynomials
we obtain
1 1
i i n n
2 2
e q e q q q
q q x cos P x q
n
q q q
If we set in the de nition we nd
1 1
n n i i
4 4
q q q e q e
P xjq q q x cos
1
n
2
q q q
but these are not really di erent in view of the quadratic transformation
P xjq P x q
n n
Orthogonality
Z
n n
w x q q q q q
n 1
p
P x q P x q dx
mn m n
q q q q
x
1 1
where
i i i
h x h x e e q e q
1 1
w x
1 1 1 1 1 1 1 1
i i i i i i
2 2 2 2 2 2 2 2
q q h x q e q e q e q e q e q e q h x q
1 1
with
1
Y
k k i i
xq q e e q x cos h x
1
k
Recurrence relation
1 1
n n n
2 2
q xP x q q q P x q q q P x q
n n n
q Di erence equation
1
2
q y x y x P x q q D w x q q D y x w x q
n q q n
where
n n n
q q q
n
and
w x a q
p
w x a q
x
where
1 1
i
2 2
h x h x h x q h x q e q
1
w x a q
i i i i
ae ae ae ae q h x a h x a h x a h x a
1
with
1
Y
k k i i
x cos xq q e e q h x
1
k
and
f x 1 1
q
i i i
2 2
D f x e f q e x cos with f e f q
q q
x
q
Generating functions
1 1 1 1
i i i i
2 2 2 2
q e q e e q e q
i i
q e t q e t
q q
1
X
P x q
n
n
t x cos
1
n
2
q
n
1 1 1 1
i i i i
2 2 2 2
e q e e q e q q
i i
q e t q e t
q q
1
X
q q q P x q
n n
n
t x cos
1
n
q q q
2
n q
n
3 3 1 1
i i i i
2 2 2 2
e q e e q e q q
i i
q e t q e t
q q
1
n
X
q q P x q
n n
n
t x cos
1
n
q q q q
2
q n
n
1 1 3 3
i i i i
2 2 2 2
q e q e e q e q
i i
q e t q e t
q q
1
X
P x q q q
n n
n
t x cos
1
n
q q
2
n q
n
1 3 1 3
i i i i
2 2 2 2
q e q e e q e q
i i
q e t q e t
q q
1
X
P x q q q
n n
n
t x cos
1
n
q q
2
n q
n
Remarks The continuous q Legendre p olynomials can also b e written as
n
X
1
q q q q
k n k
i n k n
2
e x cos P x q q
n
q q q q
k n k
k
They are related to the continuous q ultraspherical or Rogers p olynomials given by
in the following way
1
n
2
P x q q C x q jq
n n
References
Big q Laguerre
De nition
n
q x
q q P x a b q
n
aq bq
n
x q aq x
q
n
aq b q q b
n
Orthogonality
aq
Z
a x b x q
1
P x a b q P x a b q d x
m n q
x q
1
bq
n
q a b ab q q q q
1 n
n
2
aq q abq q
mn
aq bq q aq bq q
1 n
Recurrence relation
x P x a b q A P x a b q A C P x a b q C P x a b q
n n n n n n n n
where
n n
A aq bq
n
n n
C abq q
n
q Di erence equation
n n
q q x y x B x y q x B x D x y x D x y q x
where
y x P x a b q
n
and
B x abq x
D x x aq x bq
Generating functions
1
X
bq q aq x
n
n
q xt bq t q P x a b q t
1 n
aq q q
n
n
n
1
n
X
2
q x
n
P x a b q t q t t q
n 1
q q aq bq
n
n
Remark The big q Laguerre p olynomials de ned by and the a ne q Krawtchouk
p olynomials given by are related in the following way
Af f x x N
K q p N q P q p q q
n
n
References
Little q Jacobi
De nition
n n
q abq
q q x p x a bjq
n
aq
Orthogonality
1
X
bq q
k
k k k
aq p q a bjq p q a bjq
m n
q q
k
k
n
q bq q abq aq abq q
n 1
aq b q
mn
n
aq q abq aq abq q
1 n
Recurrence relation
xp x a bjq A p x a bjq A C p x a bjq C p x a bjq
n n n n n n n n
where
n n
aq abq
n
A q
n
n n
abq abq
n n
q bq
n
C aq
n
n n
abq abq
q Di erence equation
n n n
q q abq xy x
B x y q x B x D x y x D x y q x y x p x a bjq
n
where
B x a bq x
D x x
Generating function
n
1
n
X
2
q bq x
n
p x a bjq t q t q xt
n
bq q q bq aq
n
n
Remarks The little q Jacobi p olynomials de ned by and the big q Jacobi p olynomials
given by are related in the following way
n
bq q
n
n
n n
2
b q P bq x b a q p x a bjq
n n
aq q
n
The little q Jacobi p olynomials and the q Meixner p olynomials de ned by are related
in the following way
x n n x
M q b c q p c q b b q jq
n n
References
Sp ecial case
Little q Legendre
De nition The little q Legendre p olynomials are little q Jacobi p olynomials with a b
n n
q q
q q x p xjq
n
q
Orthogonality
1
n
X
q
k k k
q p q jq p q jq
m n mn
n
q
k
Recurrence relation
xp xjq A p xjq A C p xjq C p xjq
n n n n n n n n
where
n
q
n
A q
n
n n
q q
n
q
n
C q
n
n n
q q
q Di erence equation
n n n
q q q xy x B x y q x B x D x y x D x y q x
where
y x p xjq
n
and
B x q x
D x x
Generating function
n
1
n
X
2
q q x
n
p xjq t q t q xt
n
q q q q q
n
n
References
q Meixner
De nition
n n x
q q q
x
q M q b c q
n
bq c
Orthogonality
1
X
x
bq q
x
x x x
2
c q M q b c q M q b c q
m n
q bcq q
x
x
c q q c q q
1 n
n
q bq c
mn
bcq q bq q
1 n
Recurrence relation
n x x n x
q q M q c bq M q
n n
n n n x n n x
c bq q q c q M q q q c q M q
n n
where
x x
M q M q b c q
n n
q Di erence equation
n
q y x B x y x B x D x y x D x y x
where
x
y x M q b c q
n
and
x x
B x cq bq
x x
D x q bcq
Generating functions
1
x x
X
M q b c q q t q
n
n
q t
bq t q c q q
1 n
n
1
x
X
bq q b c q
n
x n
q bq t M q b c q t
n
c q t q c q q q
1 n
n
Remarks The q Meixner p olynomials de ned by and the little q Jacobi p olynomials
given by are related in the following way
x n n x
M q b c q p c q b b q jq
n n
The q Meixner p olynomials and the quantum q Krawtchouk p olynomials de ned by
are related in the following way
x x N q tm
q p N q M q q p q K
n
n
References
Quantum q Krawtchouk
De nition
n x
q q
n q tm x
q pq n N K q p N q
n
N
q
Orthogonality
N
X
x
pq q
N x
N x q tm x q tm x
2
q K q p N q K q p N q
m n
q q q q
x N x
x
n N
N +1 n+1
p q q q pq q
N n n
N n
2 2
q
mn
q q q
N
Recurrence relation
q tm
n x q tm x n N x
pq q K q q K q
n
n
q tm
n N n n q tm x n n x
q K q q pq q q q pq K q
n
n
where
q tm x q tm x
K q K q p N q
n n
q Di erence equation
n
p q y x B x y x B x D x y x D x y x
where
q tm x
y x K q p N q
n
and
x x N
B x q q
x x N
D x q p q
Generating function
N
N x N x
X
q q q q t q
n 1
x q tm x n
q q t K q p N q t
n
pq t q pq q q
1 n
n
Remarks The quantum q Krawtchouk p olynomials de ned by and the q Meixner
p olynomials given by are related in the following way
q tm x x N
K q p N q M q q p q
n
n
The quantum q Krawtchouk p olynomials are related to the a ne q Krawtchouk p olynomials
de ned by by the transformation q q in the following way
n
n
p
x N Af f x q tm
2
q q p N q K q p N q p q q K
n
n n
q
For x N the generating function can also b e written as
N
N x N
X
q q q
n
x x q tm x n
q q t q t q K q p N q t
x
n
pq pq q q
n
n
References
q Krawtchouk
De nition
n x n
q q pq
x
q q K q p N q
n
N
q
n x x N
q q q q
n
n N
q pq n N
N nx N x n
q q q q
n
Orthogonality
N
N
X
q q
x
x x x
p K q p N q K q p N q
m n
q q
x
x
N
p q pq q
n
N n
p q q pq
n
N +1
2
n
N N n
2
pq q p q pq q
N mn
Recurrence relation
x x x x x
K q A K q A C K q C K q q
n n n n n n n n
where
x x
K q K q p N q
n n
and
n N n
q pq
A
n
n n
pq pq
n N n
pq q
n N
C pq
n
n n
pq pq
q Di erence equation
n n n x N
q q pq y x q y x
x N x x
q p q y x p q y x
where
x
y x K q p N q
n
Generating functions
N
x N x N
X
n
q q q q
n
x x n
2
q pq t q q t q K q p N q t
n
q q
n
n
N
x x N
X
K q p N q q
n
N x x n
q pq t q q t t
N N N
pq q pq q q
n
n
Remarks The q Krawtchouk p olynomials de ned by and the dual q Krawtchouk
p olynomials given by are related in the following way
x N
K q p N q K n pq N jq
n x
with
n n
n q pq
or
n N
K x c N jq K q cq N q
n x
with
x x N
x q cq
The generating function must b e seen as an equality in terms of formal p ower series
For x N this generating function can also b e written as
N
N x x N
X
n
q q q q
n
x x n
2
q pq t q q t q K q p N q t
n
q q
n
n
References
A ne q Krawtchouk
De nition
n x
q q
Af f x
q q K q p N q
n
N
pq q
n
n x n x N
2
pq q q q q
q n N
N
q pq q p
n
Orthogonality
N
X
pq q q q
x N
x Af f x Af f x
pq K q p N q K q p N q
m n
q q q q
x N x
x
q q q q
n N n
n N
pq pq
mn
pq q q q
n N
Recurrence relation
Af f
x Af f x n N n x
q K q q pq K q
n
n
Af f
n N n n N n Af f x n N n x
q pq pq q K q pq q K q
n
n
where
Af f x Af f x
K q K q p N q
n n
q Di erence equation
n n
q q y x B x y x B x D x y x D x y x
where
Af f x
y x K q p N q
n
and
x N x
B x q pq
x x N
D x p q q
Generating functions
N
N N x
X
q q q t q q
n 1
Af f x n
q pq t K q p N q t
n
x
pq q t q q q
1 n
n
N
x N
X
pq q q
n
Af f x n x
K q p N q t q q t pq t q
1
n
N
q q q
n
n
Remarks The a ne q Krawtchouk p olynomials de ned by and the big q Laguerre
p olynomials given by are related in the following way
Af f x x N
K q p N q P q p q q
n
n
The a ne q Krawtchouk p olynomials are related to the quantum q Krawtchouk p olynomials
de ned by by the transformation q q in the following way
q tm x N Af f x
K q p N q K q p N q
n n
p q q
n
For x N the generating function can also b e written as
N
N x
X
q q q
n
Af f x n N
q pq t K q p N q t q t q
N x
n
q q pq
n
n
References
Dual q Krawtchouk
De nition
n x x N
q q cq
q q K x c N jq
n
N
q
x N n x
q q q q
n
x
q cq n N
N x n N nx
q q q q
n
where
x x N
x q cq
Orthogonality
N
N N x N
X
cq q q cq
x
x x N x
c q K x K x
m n
N
q cq q cq
x
x
q q
n
N n
q c cq
N mn
N
q q
n
where
K x K x c N jq
n n
Recurrence relation
x x N n N
q cq K x q K x
n n
n N N n N n
q cq q K x cq q K x
n n
where
K x K x c N jq
n n
q Di erence equation
n n
q q y x B x y x B x D x y x D x y x
where
y x K x c N jq
n
and
x N x N
q cq
B x
x N x N
cq cq
x x
q cq
x N
D x cq
x N x N
cq cq
Generating functions
N
N N N
X
q t cq t q q q
1 n
n
K x c N jq t
n
x x N
q t cq t q q q
1 n
n
N
x x
X
K x c N jq q c q
n
x N n
q cq t t
N x
q q t q q q
1 n
n
Remark The dual q Krawtchouk p olynomials de ned by and the q Krawtchouk
p olynomials given by are related in the following way
x N
K q p N q K n pq N jq
n x
with
n n
n q pq
or
n N
K x c N jq K q cq N q
n x
with
x x N
x q cq
For x N the generating function can also b e written as
N
N
X
q q
n
n N N
K x c N jq t q t q cq t q
n N x x
q q
n
n
References
Continuous big q Hermite
De nition
n i i
q ae ae
n
q q H x ajq a
n
n i
q ae
n i in
x cos q q e e
Orthogonality When a is real and jaj then we have the following orthogonality relation
Z
w x
mn
p
H x ajq H x ajq dx
m n
n
q q
x
1
where
1 1
i
2 2
h x h x h x q e q h x q
1
w x w x ajq
i
ae q h x a
1
with
1
Y
k k i i
xq q e e q x cos h x
1
k
If a then we have another orthogonality relation given by
Z
w x
p
H x ajq H x ajq dx
m n
x
X
mn
w H x ajq H x ajq
k m k n k
n
q q
1
k
k
aq a
where w x is as b efore
k k
aq aq
x
k
and
k
k
2
3 1
a q a q a q
k 1
k k
2 2
q w
k
q q a q q a
1 k
Recurrence relation
n n
xH x ajq H x ajq aq H x ajq q H x ajq
n n n n
q Di erence equations
h i
1
n n
2
w x aq q D jq D y x q q w x ajq y x y x H x ajq
q q n
where
w x ajq
p
w x ajq
x
and
1 1
f x
q
i i i
2 2
with f e f q e f q e x cos D f x
q q
x
q
If we de ne
n
q az az
n
q q P z a
n
then the q di erence equation can also b e written in the form
n n
q q P z A z P q z A z A z P z A z P q z
n n n n
where
az
A z
z q z
Generating function
1
X
H x ajq at q
n 1
n
t x cos
i i
e t e t q q q
1 n
n
References
Continuous q Laguerre
De nitions We have two kinds of continuous q Laguerre p olynomials coming from the con
tinuous q Jacobi p olynomials de ned by and
1 1 1 1
n i i
2 4 2 4
q q q q e q e
n
P xjq q q
n
q q q
n
1 1
3 1
n i
i
2 4
2 4
q q e
1 e q 1 1 1 q
n
n in i
2 4 2 4
q x cos e e q q
1 1
n
i
2 4
q q
e q
n
and
1 1
n i i
2 2
q q q q e q e
n
P x q q q x cos
n
q q q q
n
These two q analogues of the Laguerre p olynomials are connected by the following quadratic
transformation
n
P xjq q P x q
n n
the orthogonality relations are resp ectively Orthogonality For
Z
w xjq 1 q q
n
n
2
p
P xjq P xjq dx q
mn
m n
q q q q q
x
1 n
where
1
i i i
2
e q e e q
1 1
w xjq w x q jq
1 1 1 3 1 1 1
i i i
2 4 2 4 2 4 2
q q e q e q e q
1 1
1 1
2 2
h x q h x h x h x q
1 1 3 1
2 4 2 4
h x q h x q
with
1
Y
i i k k
e e q x cos xq q h x
1
k
and
Z
w x q
p
P x q P x q dx
m n
x
q q q
n
n
q
mn
q q q q q q q q
1 n
where
i i i
h x h x e e q e q
1 1
w x q w x q q
1 1 1 1 1
i i i i
2 2 2 2 2
q e q e q e q q e q h x q
1 1
with
1
Y
k k i i
h x xq q e e q x cos
1
k
Recurrence relations
1 1
n
2 4
q P xjq xP xjq q
n
n
1 1 1 1 1
n n
2 4 2 2 4
q P q P xjq xjq q q
n
n
1
n
2
xP x q q q P x q
n
n
1 1
n n
2 2
q P x q q q P x q q
n n
q Di erence equations
h i
1
n n
2
w x q q D jq D y x q q w x q jq y x y x P xjq
q q
n
where
w x q jq
p
w x q jq
x
and
f x 1 1
q
i i i
2 2
D f x with f e f q e f q e x cos
q q
x
q
i h
1
n n
2
q D y x q q w x q q y x y x P x q w x q q D
q q
n
where
w x q q
p
w x q q
x
and
f x 1 1
q
i i i
2 2
D f x with f e f q e f q e x cos
q q
x
q
Generating functions
1
1
X
2
q t q t q
1
n
P xjq t x cos
1 1 1 1
n
i i
2 4 2 4
q e t q e t q
1
n
1 1 1 3
1
n i i
X
2 4 2 4
q xjq t P e q e
n
i
x cos q e t
1 1
i
n
e t q q
2 4
1 q q q
n
n
1 1
1
i i
X
2 2
1
q t q e q e q
1
n i
2
P x q t x cos e t q q
1
n
i
q
2
q e t q
1
n
1 1 1
1
i i
X
2 2 2
q q q t q e q e q P x q
n
n 1
i n
q e t t x cos
1
i
n
q e t q q q
2
q 1 n
n
1 1 1
i i
2 2 2
q t q e q e q
1
i
q e t
i
e t q q
1
1
X
q q x q P
n
n
n
t x cos
1
n
q q
2
q n
n
References
Little q Laguerre Wall
De nition
n
q
q q x p x ajq
n
aq
n
x q x
q
n
a q q a
n
Orthogonality
1
n k
X
q q aq aq
n
k k
p q ajq p q ajq aq
m n mn
q q aq q aq q
k 1 n
k
Recurrence relation
xp x ajq A p x ajq A C p x ajq C p x ajq
n n n n n n n n
where
n n
A q aq
n
n n
C aq q
n
q Di erence equation
n n
q q xy x ay q x x a y x x y q x y x p x ajq
n
Generating function
n
1
n
X
2
q
n
p x ajq t q xt t q
n 1
q q aq
n
n
Remark If we set a q and change q to q we nd the q Laguerre p olynomials de ned
by in the following way
q q
n
L x q p x q jq
n
n
q q
n
References
q Laguerre
De nition
n
q q q
n
n
L x q q xq
n
q q q
n
n
q x
n
q q
q q
n
Orthogonality The q Laguerre p olynomials satisfy two kinds of orthogonality relations
an absolutely continuous one and a discrete one These orthogonality relations are given by
resp ectively
1
Z
x q q q q
1 n
L x q L x q dx
mn
m n
n
x q q q q q q
1 1 n
and
1
k k
X
q
k k
L cq q L cq q
m n
k
cq q
1
k 1
q q q cq c q q
n 1
c
mn
n
q c c q q q q q
1 n
Recurrence relation
n n
q xL x q q L x q
n
n
n n n
q q q L x q q q L x q
n
n
q Di erence equation
n
q q xy x q x y q x q x y x y q x
where
y x L x q
n
Generating functions
1
X
L x q xt q
n
1
n
q xt t
q t q q q
1 n
n
1
X
x
n
L x q t q q t
n
t q
1
n
Remarks The q Laguerre p olynomials are sometimes called the generalized Stieltjes Wigert
p olynomials
If we change q to q we obtain the little q Laguerre or Wall p olynomials given by
in the following way
q q
n
p x q jq L x q
n
n
n
q q q
n
The q Laguerre p olynomials de ned by and the alternative q Charlier p olynomials
given by are related in the following way
x
K q a q
n
x n n
L aq q
n
q q
n
The q Laguerre p olynomials de ned by and the q Charlier p olynomials given by
are related in the following way
C x q q
n
x q L
n
q q
n
Since the Stieltjes and Hamburger moment problems corresp onding to the q Laguerre p olyno
mials are indeterminate there exist many di erent weight functions
References
Alternative q Charlier
De nition
n n
q aq
q q x K x a q
n
n
q
n n
q axq xq q
n
n
xq
n n
q q x
n
n
q axq
a
Orthogonality
n+1
1
n k
X
2
k+1
a q a
k k n
2
q K q a q K q a q q q aq q a
m n n 1 mn
n
q q aq
k
k
Recurrence relation
xK x a q A K x a q A C K x a q C K x a q
n n n n n n n n
where
n
aq
n
A q
n
n n
aq aq
n
q
n
C aq
n
n n
aq aq
q Di erence equation
n n n
q q aq xy x axy q x ax x y x x y q x
where
y x K x a q
n
Generating functions
1
X
K x a q x
n
n
q xt q aq xt t
q q
n
n
n
1
n
X
2
xt q t q
1
n
q aq xt K x a q t
n
t xt q q q
1 n
n
Remarks The alternative q Charlier p olynomials de ned by and the q Laguerre
p olynomials given by related in the following way
x
K q a q
n
n x n
aq q L
n
q q
n
The generating function must b e seen as an equality in terms of formal p ower series
For x N this generating function can also b e written as
1
x x
X
K q a q q
n
x x n
q aq t q q t t
q q
n
n
References
q Charlier
De nition
n n x
q q q
x
q C q a q
n
a
n x n
q q
q a q q
n
a q a
Orthogonality
1
x
X
x
a
x x n
2
q C q a q C q a q q a q a q q q a
m n 1 n mn
q q
x
x
Recurrence relation
n x x x
q q C q aC q
n n
n n x n n x
a q q a q C q q q a q C q
n n
where
x x
C q C q a q
n n
q Di erence equation
n x x x x
q y x aq y x q a y x q y x y x C q a q
n
Generating functions
1
x x
X
q C q a q
n
n
q a q t t
t q q q
1 n
n
1
x x
X
q t q C q a q
1 n
x n
q q t t
a q t q a q q q
1 n
n
Remark The q Charlier p olynomials de ned by and the q Laguerre p olynomials
given by are related in the following way
C x q q
n
L x q
n
q q
n
References
Al Salam Carlitz I
De nition
n
n
q x q x
a n
2
q U x q a q
n
a
Orthogonality
Z
a a
q x a q x q U x q U x q d x
1 q
m n
a
n
n
2
a q q q q a a q q q a
n 1 mn
Recurrence relation
a a
a n a n n
xU x q U x q a q U x q aq q U x q
n n
n n
q Di erence equation
n n n n
q x y x aq y q x aq q x a x y x
n a
q x a x y q x y x U x q
n
Generating function
1
a
X
x q U t q at q
n
1 1
n
t
xt q q q
1 n
n
Remark The Al Salam Carlitz I p olynomials are related to the Al Salam Carlitz I I p olyno
mials de ned by in the following way
a a
U x q V x q
n n
References
Al Salam Carlitz II
De nition
n n
n
q q x
a n
2
q V x q a q
n
a
Orthogonality
2
1
k k n
X
q a q q a
n
a k a k
a V q q V q q
mn
2
m n
n
q q aq q
aq q q
k k
1
k
Recurrence relation
a a
a n a n n
xV x q V x q a q V x q aq q V x q
n n
n n
q Di erence equation
n
q x y x x a x y q x x a x aq y x
a
aq y q x y x V x q
n
Generating functions
n
1
n
X
2
xt q q
1
a n
V x q t
n
t q at q q q
1 1 n
n
n
1
X
2
x q
a n
V x q t at q q t
1
n
q q at
n
n
Remark The Al Salam Carlitz I I p olynomials are related to the Al Salam Carlitz I p olyno
mials de ned by in the following way
a a
V x q U x q
n n
References
Continuous q Hermite
De nition
n
q
n i in
x cos q q e H xjq e
n
Orthogonality
Z
w x
mn
p
H xjq H xjq dx
m n
n
q q
x
1
where
1 1
i
2 2
h x h x h x q w x e q h x q
1
with
1
Y
k k i i
xq q e e q x cos h x
1
k
Recurrence relation
n
xH xjq H xjq q H xjq
n n n
q Di erence equation
n n
q D w x D y x q q w x y x y x H xjq
q q n
where
w x
p
w x
x
and
1 f x 1
q
i i i
2 2
e f q e x cos D f x with f e f q
q q
x
q
Generating functions
1
X
H xjq
n
n
t x cos
i
q q
n
j e t q j
1
n
n
1
n
X
2
q
i n i
q e t H xjq t x cos e t q
n 1
i
q q e t
n
n
Remark The continuous q Hermite p olynomials can also b e written as
n
X
q q
n
i n k
H xjq e x cos
n
q q q q
k n k
k
References
Stieltjes Wigert
De nition
n
q
n
q xq S x q
n
q q
n
Orthogonality
1
Z
q q S x q S x q ln q
1 m n
dx
mn
n
x q q x q q q q
1 1 n
Recurrence relation
n n n
q xS x q q S x q q q S x q q S x q
n n n n
q Di erence equation
n
x q y x xy q x x y x y q x y x S x q
n
Generating functions
1
X
n
S x q t q q xt
n
t q
1
n
1
X
n
n n
2
q S x q t q q xt t q
n 1
t
n
Remark Since the Stieltjes and Hamburger moment problems corresp onding to the Stieltjes
Wigert p olynomials are indeterminate there exist many di erent weight functions For instance
they are also orthogonal with resp ect to the weight function
p
w x exp ln x x with
ln q
References
Discrete q Hermite I
De nition The discrete q Hermite I p olynomials are Al Salam Carlitz I p olynomials with
a
n
n
q x
2
q q x h x q U x q q
n
n
n n n
q q q
n
q x
x
Orthogonality
Z
n
2
q x q x q h x q h x q d x q q q q q q q
1 m n q n 1 mn
Recurrence relation
n n
xh x q h x q q q h x q
n n n
q Di erence equation
n
q x y x y q x q y x q x y q x y x h x q
n
Generating function
1
X
h x q t q t q
n 1 1
n
t
xt q q q
1 n
n
Remark The discrete q Hermite I p olynomials are related to the discrete q Hermite I I p oly
nomials de ned by in the following way
n
h x q h ix q i
n n
References
Discrete q Hermite II
De nition The discrete q Hermite I I p olynomials are Al Salam Carlitz I I p olynomials with
a
n
n
q ix
n n n
2
q q h x q i V ix q i q
n
n
n n
q q q
n
q x
x
Orthogonality
1
h i
X
k k k k k k
h cq q h cq q h cq q h cq q w cq q
m n m n
k 1
q c q c q q q q
1 n
c
mn
2
n
q c c q q
q
1
where
w x
ix q ix q x q
1 1 1
Recurrence relation
n n
xh x q h x q q q h x q
n n n
q Di erence equation
n
h x q x h q x q x q h x q q h q x q q x
n n n n
Generating functions
n
1
X
2
q xt q
1
n
h x q t
n
t q q q
1 n
n
n
1
n
X
2
q ix
n
h x q t q it it q
n 1
q q it
n
n
Remark The discrete q Hermite I I p olynomials are related to the discrete q Hermite I p oly
nomials de ned by in the following way
n
h x q i h ix q
n n
References
Chapter
Limit relations b etween basic
hyp ergeometric orthogonal
p olynomials
Askey Wilson ! Continuous dual q Hahn
The continuous dual q Hahn p olynomials de ned by simply follow from the Askey Wilson
p olynomials given by by setting d in
p x a b c jq p x a b cjq
n n
Askey Wilson ! Continuous q Hahn
The continuous q Hahn p olynomials de ned by can b e obtained from the Askey Wilson
i i i
p olynomials given by by the substitutions a ae b be c ce and
i
d de
i i i i
p cos ae be ce de jq p cos a b c d q
n n
Askey Wilson ! Big q Jacobi
The big q Jacobi p olynomials de ned by can b e obtained from the Askey Wilson p olyno
a x b a q c a q and d a in mials by setting x
n
a p x a b c djq
n
p x a b c djq
n
ab ac ad q
n
de ned by and then taking the limit a
x q q a
q P x q lim p a
n n
a!
a a a
Askey Wilson ! Continuous q Jacobi
1 1 3 1 3 1 1 1
2 4 2 4 2 4 2 4
If we take a q b q c q and d q in the de nition
of the Askey Wilson p olynomials and change the normalization we nd the continuous q Jacobi
p olynomials given by
1 1 1 1 3 1 1 1 3 1
n
2 4 2 4 2 4 2 4 2 4
x q p q q q q q
n
P xjq
1 1
n
2 2
q q q q
n
1 1 1 1
2 2 2 2
b q c q and d q to obtain after a change of In M Rahman takes a q
normalization the continuous q Jacobi p olynomials de ned by
1 1 1 1 1
n
2 2 2 2 2
q x q p q q q q
n
P x q
n
q q q q
n
As was p ointed out in section these two q analogues of the Jacobi p olynomials are not really
di erent since they are connected by the quadratic transformation
q q
n
n
P xjq q P x q
n n
q q
n
Askey Wilson ! Continuous q ultraspherical Rogers
1 1 1 1 1 1
2 2 2 2 2 2
b q c and d q in the de nition of the Askey Wilson If we set a
p olynomials and change the normalization we obtain the continuous q ultraspherical or Rogers
p olynomials de ned by In fact we have
1 1 1 1 1 1
2 2 2 2 2 2
q q q x q p
n n
C x jq
n
1 1
2 2
q q q q
n
q Racah ! Big q Jacobi
The big q Jacobi p olynomials de ned by can b e obtained from the q Racah p olynomials
by setting in the de nition
x
R x a b c jq P q a b c q
n n
q Racah ! q Hahn
The q Hahn p olynomials follow from the q Racah p olynomials by the substitution and
N
q q in the de nition of the q Racah p olynomials
N x
R x q jq Q q N jq
n n
Another way to obtain the q Hahn p olynomials from the q Racah p olynomials is by setting
N
and q in the de nition
N x
R x q jq Q q N jq
n n
N N
And if we take q q q and in the de nition of the q Racah
p olynomials we nd the q Hahn p olynomials given by in the following way
N N x
R x q q jq Q q N jq
n n
x
Note that x q in each case
q Racah ! Dual q Hahn
To obtain the dual q Hahn p olynomials from the q Racah p olynomials we have to take and
N
q q in
N
R x q jq R x N jq
n n
with
x x
x q q
N
We may also take and q in to obtain the dual q Hahn p olynomials
from the q Racah p olynomials
N
R x q jq R x N jq
n n
with
x x
x q q
N N
And if we take q q q and in the de nition of the q Racah
p olynomials we nd the dual q Hahn p olynomials given by in the following way
N N
R x q q jq R x N jq
n n
with
x x
x q q
q Racah ! q Krawtchouk
The q Krawtchouk p olynomials de ned by can b e obtained from the q Racah p olyno
N N
mials by setting q q pq and in the de nition of the q Racah
p olynomials
x N N x
R q q pq jq K q p N q
n n
x
Note that x q in this case
q Racah ! Dual q Krawtchouk
The dual q Krawtchouk p olynomials de ned by easily follow from the q Racah p olynomials
N
given by by using the substitutions q q and c
N
R x q cjq K x c N jq
n n
Note that
x x N
x x q cq
Continuous dual q Hahn ! Al Salam Chihara
The Al Salam Chihara p olynomials de ned by simply follow from the continuous dual
q Hahn p olynomials by taking c in the de nition of the continuous dual q Hahn
p olynomials
p x a b jq Q x a bjq
n n
Continuous q Hahn ! q Meixner Pollaczek
The q Meixner Pollaczek p olynomials de ned by simply follow from the continuous q Hahn
p olynomials if we set d a and b c in the de nition of the continuous q Hahn
p olynomials
p cos a a q
n
P cos ajq
n
q q
n
Big q Jacobi ! Big q Laguerre
If we set b in the de nition of the big q Jacobi p olynomials we obtain the big q Laguerre
p olynomials given by
P x a c q P x a c q
n n
Big q Jacobi ! Little q Jacobi
The little q Jacobi p olynomials de ned by can b e obtained from the big q Jacobi p olyno
mials by the substitution x cq x in the de nition and then by the limit c
lim P cq x a b c q p x a bjq
n n
c!1
Big q Jacobi ! q Meixner
If we take the limit a in the de nition of the big q Jacobi p olynomials we simply
obtain the q Meixner p olynomials de ned by
x x
lim P q a b c q M q c b q
n n
a!1
q Hahn ! Little q Jacobi
If we set x N x in the de nition of the q Hahn p olynomials and take the limit N
we nd the little q Jacobi p olynomials
x N x
lim Q q N jq p q jq
n n
N !1
x
where p q jq is de ned by
n
q Hahn ! q Meixner
The q Meixner p olynomials de ned by can b e obtained from the q Hahn p olynomials by
N
setting b and b c q in the de nition of the q Hahn p olynomials and
letting N
x N x
lim Q q b b c q N jq M q b c q
n n
N !1
q Hahn ! Quantum q Krawtchouk
The quantum q Krawtchouk p olynomials de ned by simply follow from the q Hahn p oly
nomials by setting p in the de nition of the q Hahn p olynomials and taking the limit
x q tm x
lim Q q p N jq K q p N q
n
n
!1
q Hahn ! q Krawtchouk
If we set q p in the de nition of the q Hahn p olynomials and then let
we obtain the q Krawtchouk p olynomials de ned by
p
x x
N q K q p N q lim Q q
n n
!
q
q Hahn ! A ne q Krawtchouk
The a ne q Krawtchouk p olynomials de ned by can b e obtained from the q Hahn p oly
nomials by the substitution p and in
x Af f x
Q q p N jq K q p N q
n
n
Dual q Hahn ! A ne q Krawtchouk
The a ne q Krawtchouk p olynomials de ned by can b e obtained from the dual q Hahn
p olynomials by the substitution p and in
Af f x
R x p N jq K q p N q
n
n
x
Note that x q in this case
Dual q Hahn ! Dual q Krawtchouk
The dual q Krawtchouk p olynomials de ned by can b e obtained from the dual q Hahn
N
p olynomials by setting c q in and then letting
c
N
lim R x q q K x c N jq
n n
!
Al Salam Chihara ! Continuous big q Hermite
If we take the limit b in the de nition of the Al Salam Chihara p olynomials we simply
obtain the continuous big q Hermite p olynomials given by
lim Q x a bjq H x ajq
n n
b!
Al Salam Chihara ! Continuous q Laguerre
The continuous q Laguerre p olynomials de ned by can b e obtained from the Al Salam
1 1 1 3
2 4 2 4
Chihara p olynomials given by by taking a q and b q
1 1 3 q q 1
n
2 4 2 4
xjq P q q x q Q
n
1 1
n
n
2 4
q
q Meixner Pollaczek ! Continuous q ultraspherical
Rogers
If we take and a in the de nition of the q Meixner Pollaczek p olynomials we
obtain the continuous q ultraspherical or Rogers p olynomials given by
P cos jq C cos jq
n n
Continuous q Jacobi ! Continuous q Laguerre
The continuous q Laguerre p olynomials given by and follow simply from the
continuous q Jacobi p olynomials de ned by and resp ectively by taking the limit
lim P xjq P xjq
n n
!1
and
P x q
n
lim P x q
n
!1
q q
n
Continuous q ultraspherical Rogers ! Continuous
q Hermite
The continuous q Hermite p olynomials de ned by can b e obtained from the continuous
q ultraspherical or Rogers p olynomials given by by taking the limit In fact we
have
H xjq
n
lim C x jq
n
!
q q
n
Big q Laguerre ! Little q Laguerre Wall
The little q Laguerre or Wall p olynomials de ned by can b e obtained from the big
q Laguerre p olynomials by taking x bq x in and then letting b
lim P bq x a b q p x ajq
n n
b!1
Big q Laguerre ! Al Salam Carlitz I
If we set x aq x and b ab in the de nition of the big q Laguerre p olynomials and take
the limit a we obtain the Al Salam Carlitz I p olynomials given by
P aq x a ab q
n
b
lim U x q
n
n
a!
a
Little q Jacobi ! Little q Laguerre Wall
The little q Laguerre or Wall p olynomials de ned by are little q Jacobi p olynomials with
b So if we set b in the de nition of the little q Jacobi p olynomials we obtain the
little q Laguerre or Wall p olynomials
p x a jq p x ajq
n n
Little q Jacobi ! q Laguerre
If we substitute a q and x b q x in the de nition of the little q Jacobi p olyno
mials and then let b tend to in nity we nd the q Laguerre p olynomials given by
q q x
n
L x q q b q lim p
n
n
b!1
bq q q
n
Little q Jacobi ! Alternative q Charlier
If we set b a q b in the de nition of the little q Jacobi p olynomials and then take
the limit a we obtain the alternative q Charlier p olynomials given by
b
lim p x a q K x b q
n n
a!
aq
q Meixner ! q Laguerre
The q Laguerre p olynomials de ned by can b e obtained from the q Meixner p olynomials
x
given by by setting b q and q cq x in the de nition of the q Meixner
p olynomials and then taking the limit c
q q
n
L x q lim M cq x q c q
n
n
c!1
q q
n
q Meixner ! q Charlier
The q Meixner p olynomials and the q Charlier p olynomials de ned by and resp ec
tively are simply related by the limit b in the de nition of the q Meixner p olynomials
In fact we have
M x a q C x a q
n n
q Meixner ! Al Salam Carlitz II
The Al Salam Carlitz II p olynomials de ned by can b e obtained from the q Meixner
p olynomials de ned by by setting b c a in the de nition of the q Meixner
p olynomials and then taking the limit c
n
n
a
a
2
q V x q c q x lim M
n
n
c#
c a
Quantum q Krawtchouk ! Al Salam Carlitz II
N
If we set p a q in the de nition of the quantum q Krawtchouk p olynomials and
let N we obtain the Al Salam Carlitz I I p olynomials given by In fact we have
n
n
q tm N a
2
lim K x a q N q q V x q
n n
N !1
a
q Krawtchouk ! Alternative q Charlier
If we set x N x in the de nition of the q Krawtchouk p olynomials and then take the
limit N we obtain the alternative q Charlier p olynomials de ned by
x N x
lim K q p N q K q p q
n n
N !1
q Krawtchouk ! q Charlier
The q Charlier p olynomials given by can b e obtained from the q Krawtchouk p olyno
N
mials de ned by by setting p a q in the de nition of the q Krawtchouk
p olynomials and then taking the limit N
x N x
q a q N q C q a q lim K
n n
N !1
A ne q Krawtchouk ! Little q Laguerre Wall
If we set x N x in the de nition of the a ne q Krawtchouk p olynomials and take the
limit N we simply obtain the little q Laguerre or Wall p olynomials de ned by
Af f x N x
lim K q p N q p q p q
n
n
N !1
Dual q Krawtchouk ! Al Salam Carlitz I
If we set c a in the de nition of the dual q Krawtchouk p olynomials and take the
limit N we simply obtain the Al Salam Carlitz I p olynomials given by
n
n
a x
2
q U q q N lim K q x
n
n
N !1
a a
x x N
Note that x q a q
Continuous big q Hermite ! Continuous q Hermite
The continuous q Hermite p olynomials de ned by can easily b e obtained from the con
tinuous big q Hermite p olynomials given by by taking a
H x jq H xjq
n n
Continuous q Laguerre ! Continuous q Hermite
The continuous q Hermite p olynomials given by can b e obtained from the continuous
q Laguerre p olynomials de ned by by taking the limit in the following way
P xjq H xjq
n
n
lim
1 1
n
!1
q q
2 4
q n
q Laguerre ! Stieltjes Wigert
If we set x xq in the de nition of the q Laguerre p olynomials and take the limit
we simply obtain the Stieltjes Wigert p olynomials given by
xq q S x q lim L
n
n
!1
Alternative q Charlier ! Stieltjes Wigert
The Stieltjes Wigert p olynomials de ned by can b e obtained from the alternative q
Charlier p olynomials by setting x a x in the de nition of the alternative q Charlier
p olynomials and then taking the limit a In fact we have
x
lim K a q q q S x q
n n n
a!1
a
q Charlier ! Stieltjes Wigert
x
If we set q ax in the de nition of the q Charlier p olynomials and take the limit
a we obtain the Stieltjes Wigert p olynomials given by in the following way
lim C ax a q q q S x q
n n n
a!1
Al Salam Carlitz I ! Discrete q Hermite I
The discrete q Hermite I p olynomials de ned by can easily b e obtained from the Al
Salam Carlitz I p olynomials given by by the substitution a
U x q h x q
n
n
Al Salam Carlitz II ! Discrete q Hermite II
The discrete q Hermite II p olynomials de ned by follow from the Al Salam Carlitz I I
p olynomials given by by the substitution a in the following way
n
i V ix q h x q
n
n
Chapter
From basic to classical
hyp ergeometric orthogonal
p olynomials
Askey Wilson ! Wilson
To nd the Wilson p olynomials de ned by from the Askey Wilson p olynomials we set
a b c d i ix x
a q b q c q d q and e q or ln q in the de nition and take the
limit q
ix ix a b c d
q q q q q q jq p
n
W x a b c d lim
n
n
q "
q
q Racah ! Racah
If we set q q q q in the de nition of the q Racah p olynomials
and let q we easily obtain the Racah p olynomials de ned by
lim R x q q q q jq R x
n n
q "
where
x x
x q q
x x x
Continuous dual q Hahn ! Continuous dual Hahn
To nd the continuous dual Hahn p olynomials de ned by from the continuous dual q Hahn
a b c i ix x
p olynomials we set a q b q c q and e q or ln q in the de nition
and take the limit q
ix ix a b c
p q q q q q jq
n
lim S x a b c
n
n
q "
q
Continuous q Hahn ! Continuous Hahn
a b c d i ix x
If we set a q b q c q d q and e q or ln q in the de nition
of the continuous q Hahn p olynomials and take the limit q we nd the continuous Hahn
p olynomials given by in the following way
x a b c d
p cos ln q q q q q q
n
n
lim sin p x a b c d
n
n
q "
q q q
n
Big q Jacobi ! Jacobi
If we set c a q and b q in the de nition of the big q Jacobi p olynomials and let
q we nd the Jacobi p olynomials given by
P x
n
lim P x q q q
n
q "
P
n
If we take c q for arbitrary real instead of c we nd
x P
n
lim P x q q q q
n
q "
P
n
Big q Legendre ! Legendre Spherical
If we set c in the de nition of the big q Legendre p olynomials and let q we simply
obtain the Legendre or spherical p olynomials de ned by
lim P x q P x
n n
q "
If we take c q for arbitrary real instead of c we nd
lim P x q q P x
n n
q "
q Hahn ! Hahn
The Hahn p olynomials de ned by simply follow from the q Hahn p olynomials given by
after setting q and q in the following way
x
lim Q q q q N jq Q x N
n n
q "
Dual q Hahn ! Dual Hahn
The dual Hahn p olynomials given by follow from the dual q Hahn p olynomials by simply
taking the limit q in the de nition of the dual q Hahn p olynomials after applying the
substitution q and q
lim R x q q N jq R x N
n n
q "
where
x x
x q q
x x x
Al Salam Chihara ! Meixner Pollaczek
i i i ix i
If we set a q e b q e and e q e in the de nition of the Al Salam Chihara
p olynomials and take the limit q we obtain the Meixner Pollaczek p olynomials given by
in the following way
x i i
Q cos ln q q e q e jq
n
lim P x
n
q "
q q
n
q Meixner Pollaczek ! Meixner Pollaczek
To nd the Meixner Pollaczek p olynomials de ned by from the q Meixner Pollaczek p oly
i ix x
nomials we substitute a q and e q or ln q in the de nition of the
q Meixner Pollaczek p olynomials and take the limit q to nd
x
lim P cos ln q q jq P x
n
n
q "
Continuous q Jacobi ! Jacobi
If we take the limit q in the de nitions and of the continuous q Jacobi
p olynomials we simply nd the Jacobi p olynomials de ned by
lim P xjq P x
n n
q "
and
lim P x q P x
n n
q "
Continuous q ultraspherical Rogers ! Gegenbauer Ultra
spherical
If we set q in the de nition of the continuous q ultraspherical or Rogers p olyno
mials and let q tend to one we obtain the Gegenbauer or ultraspherical p olynomials given by
lim C x q jq C x
n
n
q "
Continuous q Legendre ! Legendre Spherical
The Legendre or spherical p olynomials de ned by easily follow from the continuous
q Legendre p olynomials given by by taking the limit q
lim P x q P x
n n
q "
Of course we also have
lim P xjq P x
n n
q "
Big q Laguerre ! Laguerre
The Laguerre p olynomials de ned by can b e obtained from the big q Laguerre p olynomials
by the substitution a q and b q q in the de nition of the big q Laguerre
p olynomials and the limit q
L x
n
lim P x q q q q
n
q "
L
n
Little q Jacobi ! Jacobi
The Jacobi p olynomials de ned by simply follow from the little q Jacobi p olynomials
de ned by in the following way
P x
n
lim p x q q jq
n
q "
P
n
Little q Legendre ! Legendre Spherical
If we take the limit q in the de nition of the little q Legendre p olynomials we simply
nd the Legendre or spherical p olynomials given by
lim p xjq P x
n n
q "
Little q Jacobi ! Laguerre
If we take a q b q for arbitrary real and x q x in the de nition of
the little q Jacobi p olynomials and then take the limit q we obtain the Laguerre p olynomials
given by
L x
n
q x q q q lim p
n
q "
L
n
q Meixner ! Meixner
To nd the Meixner p olynomials de ned by from the q Meixner p olynomials given by
we set b q and c c c and let q
c
x
q q lim M q M x c
n n
q "
c
Quantum q Krawtchouk ! Krawtchouk
The Krawtchouk p olynomials given by easily follow from the quantum q Krawtchouk
p olynomials de ned by in the following way
q tm x
lim K q p N q K x p N
n
n
q "
q Krawtchouk ! Krawtchouk
If we take the limit q in the de nition of the q Krawtchouk p olynomials we simply
nd the Krawtchouk p olynomials given by in the following way
x
lim K q p N q K x N
n n
q "
p
A ne q Krawtchouk ! Krawtchouk
If we let q in the de nition of the a ne q Krawtchouk p olynomials we obtain
Af f x
lim K q p N jq K x p N
n
n
q "
where K x p N is the Krawtchouk p olynomial de ned by
n
Dual q Krawtchouk ! Krawtchouk
If we set c p in the de nition of the dual q Krawtchouk p olynomials and take the
limit q we simply nd the Krawtchouk p olynomials given by
N jq K x p N lim K x
n n
q "
p
Continuous big q Hermite ! Hermite
q
If we set a and x x q in the de nition of the continuous big q Hermite
p olynomials and let q tend to one we obtain the Hermite p olynomials given by in the
following way
1
q
2
q x H
n
lim H x
n
n
q
2
q "
q
p
q in the de nition of the continuous q and x x If we take a a
big q Hermite p olynomials and take the limit q we nd the Hermite p olynomials de ned by
with shifted argument
1
p
q
2
x H q q a
n
lim H x a
n
n
q
2
q "
Continuous q Laguerre ! Laguerre
x
If we set x q in the de nitions and of the continuous q Laguerre p olynomials
and take the limit q we nd the Laguerre p olynomials de ned by In fact we have
x
lim P q jq L x
n n
q "
and
x
lim P q q L x
n n
q "
Little q Laguerre Wall ! Laguerre
If we set a q and x q x in the de nition of the little q Laguerre or Wall
p olynomials and let q tend to one we obtain the Laguerre p olynomials given by
L x
n
lim p q x q jq
n
q "
L
n
Little q Laguerre Wall ! Charlier
x
If we set a q a and x q in the de nition of the little q Laguerre or Wall
p olynomials and take the limit q we obtain the Charlier p olynomials given by in the
following way
x
p q q ajq C x a
n n
lim
n n
q "
q a
q Laguerre ! Laguerre
If we set x q x in the de nition of the q Laguerre p olynomials and take the limit
q we obtain the Laguerre p olynomials given by
lim L q x q L x
n n
q "
q Laguerre ! Charlier
x
If we set x q and q a q or ln q ln q a in the de nition
of the q Laguerre p olynomials multiply by q q and take the limit q we obtain the Charlier
n
p olynomials given by
ln q a
x
or lim q q L q q C x a q
n n
n
q "
a q ln q
Alternative q Charlier ! Charlier
x
If we set x q and a a q in the de nition of the alternative q Charlier p olynomials
and take the limit q we nd the Charlier p olynomials given by
x
K q a q q
n
n
lim a C x a
n
n
q "
q
q Charlier ! Charlier
If we set a a q in the de nition of the q Charlier p olynomials and take the limit
q we obtain the Charlier p olynomials de ned by
x
lim C q a q q C x a
n n
q "
Al Salam Carlitz I ! Charlier
x
If we set a a q and x q in the de nition of the Al Salam Carlitz I p olynomials
n n
and take the limit q after dividing by a q we obtain the Charlier p olynomials de ned
by
a q
x
U q q
n
n
lim a C x a
n
n
q "
q
Al Salam Carlitz I ! Hermite
p p
q and a a q in the de nition of the Al Salam Carlitz I If we set x x
n
2
and let q tend to one we obtain the Hermite p olynomials given p olynomials divide by q
by with shifted argument In fact we have
p
p
2
a q
U x q q H x a
n
n
lim
n
n
2
q "
q
Al Salam Carlitz II ! Charlier
x
If we set a a q and x q in the de nition of the Al Salam Carlitz I I p olynomials
and taking the limit q we nd
a q
x
V q q
n
n
a C x a lim
n
n
q "
q
Al Salam Carlitz I I ! Hermite
p p
q and a a q in the de nition of the Al Salam Carlitz I I If we set x x
n
2
p olynomials divide by q and let q tend to one we obtain the Hermite p olynomials given
by with shifted argument In fact we have
p
p
2
q a
x q q H x V
n
n
lim
n
n
2
q "
q
Continuous q Hermite ! Hermite
The Hermite p olynomials de ned by can b e obtained from the continuous q Hermite
q
q In fact we have p olynomials given by by setting x x
1
q
2
q H x
n
H x lim
n
n
q
2
q "
Stieltjes Wigert ! Hermite
The Hermite p olynomials de ned by can b e obtained from the Stieltjes Wigert p olynomials
p
given by by setting x q x q and taking the limit q in the following
way
p
q q S q x q q
n n
n
H x lim
n
n
q
2
q "
Discrete q Hermite I ! Hermite
The Hermite p olynomials de ned by can b e found from the discrete q Hermite I p olyno
mials given by in the following way
p
x h q q
n
H x
n
lim
n
n
2
q "
q
Discrete q Hermite II ! Hermite
The Hermite p olynomials de ned by can also b e found from the discrete q Hermite II
p olynomials given by in a similar way
p
q x h q
n
H x
n
lim
n
n
2
q "
q
Biblio graphy
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D M Bressoud Linearization and related formulas for q ultraspherical polynomials
SIAM Journal on Mathematical Analysis
J W Brown New generating functions for classical polynomials Pro ceedings of the Amer
ican Mathematical So ciety
J Bustoz M E H Ismail The associated ultraspherical polynomials and their q
analogues Canadian Journal of Mathematics
J Bustoz M E H Ismail Tur an inequalities for ultraspherical and continuous q
ultraspherical polynomials SIAM Journal on Mathematical Analysis
J Bustoz M E H Ismail A positive trigonometric sum SIAM Journal on Mathe
matical Analysis
J Bustoz N Savage Inequalities for ultraspherical and Laguerre polynomials SIAM
Journal on Mathematical Analysis
J Bustoz N Savage Inequalities for ultraspherical and Laguerre polynomials II
SIAM Journal on Mathematical Analysis
L Carlitz A note on the Laguerre polynomials The Michigan Mathematical Journal
L Carlitz Some generating functions of Weisner Duke Mathematical Journal
L Carlitz The generating function for the Jacobi polynomial Rendiconti del Seminario
Matematico della Universit a di Padova
L Carlitz Some generating functions for Laguerre polynomials Duke Mathematical
Journal
L Carlitz H M Srivastava Some new generating functions for the Hermite poly
nomials Journal of Mathematical Analysis and Applications
J A Charris M E H Ismail On sieved orthogonal polynomials V sieved Pol laczek
polynomials SIAM Journal on Mathematical Analysis
L Chihara On the zeros of the Askey Wilson polynomials with applications to coding
theory SIAM Journal on Mathematical Analysis
T S Chihara On indeterminate Hamburger moment problems Paci c Journal of Math
ematics
T S Chihara Orthogonal polynomials with Brenke type generating functions Duke Math
ematical Journal
T S Chihara A characterization and a class of distribution functions for the Stieltjes
Wigert polynomials Canadian Mathematical Bulletin
T S Chihara An introduction to orthogonal polynomials Gordon and Breach New York
T S Chihara On generalized Stieltjes Wigert and related orthogonal polynomials Journal
of Computational and Applied Mathematics
T S Chihara M E H Ismail Extremal measures for a system of orthogonal polyno
mials Constructive Approximation
Z Ciesielski Explicit formula relating the Jacobi Hahn and Bernstein polynomials SIAM
Journal on Mathematical Analysis
M E Cohen Some classes of generating functions for the Laguerre and Hermite polyno
mials Mathematics of Computation
R D Cooper M R Hoare M Rahman Stochastic processes and special functions
On the probabilistic origin of some positive kernels associated with classical orthogonal poly
nomials Journal of Mathematical Analysis and Applications
A E Danese Some identities and inequalities involving ultraspherical polynomials Duke
Mathematical Journal
J S Dehesa On a general system of orthogonal q polynomials Journal of Computational
and Applied Mathematics
P Delsarte Association schemes and t designs in regular semilattices Journal of Com
binatorial Theory A
P Delsarte J M Goethals Alternating bilinear forms over GF q Journal of
Combinatorial Theory A
C F Dunkl A Krawtchouk polynomial addition theorem and wreath products of symmetric
groups Indiana University Mathematics Journal
C F Dunkl An addition theorem for some q Hahn polynomials Monatshefte fur Mathe
matik
C F Dunkl An addition theorem for Hahn polynomials the spherical functions SIAM
Journal on Mathematical Analysis
C F Dunkl Orthogonal polynomials with symmetry of order three Canadian Journal of
Mathematics
C F Dunkl D E Ramirez Krawtchouk polynomials and the symmetrization of hy
pergroups SIAM Journal on Mathematical Analysis
A Elbert A Laforgia A note on the paper of Ahmed Muldoon and Spigler SIAM
Journal on Mathematical Analysis
A Elbert A Laforgia New properties of the zeros of a Jacobi polynomial in relation
to their centroid SIAM Journal on Mathematical Analysis
A Erdelyi et al Higher transcendental functions Volume I I of the Bateman Manuscript
Pro ject McGraw Hill Bo ok Company New York
A Erdelyi Asymptotic forms for Laguerre polynomials Journal of the Indian Mathe
matical So ciety
H Exton Basic Laguerre polynomials Pure and Applied Mathematika Sciencses
D Foata J Labelle Mod eles combinatoires pour les polyn omes de Meixner Europ ean
Journal of Combinatorics
D Foata P Leroux Polyn omes de Jacobi interpr etation combinatoire et fonction
g en eratrice Pro ceedings of the American Mathematical So ciety
G Gasper Linearization of the product of Jacobi polynomials I Canadian Journal of
Mathematics
G Gasper Linearization of the product of Jacobi polynomials II Canadian Journal of
Mathematics
G Gasper On the extension of Turan s inequality to Jacobi polynomials Duke Mathe
matical Journal
G Gasper Positivity and the convolution structure for Jacobi series Annals of Mathe
matics
G Gasper Banach algebras for Jacobi series and positivity of a kernel Annals of Math
ematics
G Gasper Nonnegativity of a discrete Poisson kernel for the Hahn polynomials Journal
of Mathematical Analysis and Applications
G Gasper Projection formulas for orthogonal polynomials of a discrete variable Journal
of Mathematical Analysis and Applications
G Gasper Positive sums of the classical orthogonal polynomials SIAM Journal on Math
ematical Analysis
G Gasper Orthogonality of certain functions with respect to complex valued weights
Canadian Journal of Mathematics
G Gasper Roger s linearization formula for the continuous q ultraspherical polynomials
and quadratic transformation formulas SIAM Journal on Mathematical Analysis
G Gasper q Extensions of Clausen s formula and of the inequalities used by De Branges
in his proof of the Bieberbach Robertson and Milin conjectures SIAM Journal on Mathe
matical Analysis
G Gasper M Rahman Positivity of the Poisson kernel for the continuous q
ultraspherical polynomials SIAM Journal on Mathematical Analysis
G Gasper M Rahman Product formulas of Watson Bailey and Bateman types and
positivity of the Poisson kernel for q Racah polynomials SIAM Journal on Mathematical
Analysis
G Gasper M Rahman Positivity of the Poisson kernel for the continuous q Jacobi
polynomials and some quadratic transformation formulas for basic hypergeometric series
SIAM Journal on Mathematical Analysis
G Gasper M Rahman A nonterminating q Clausen formula and some related product
formulas SIAM Journal on Mathematical Analysis
G Gasper M Rahman Basic Hypergeometric Series Encyclop edia of Mathematics
and Its Applications Cambridge University Press Cambridge
L Gatteschi New inequalities for the zeros of Jacobi polynomials SIAM Journal on
Mathematical Analysis
J Gillis Integrals of products of Laguerre polynomials SIAM Journal on Mathematical
Analysis
J Gillis M E H Ismail T Offer An asymptotic problem in derangement theory
SIAM Journal on Mathematical Analysis
J Gillis J Jedwab D Zeilberger A combinatorial interpretation of the integral
of the products of Legendre polynomials SIAM Journal on Mathematical Analysis
I S Gradshteyn I M Ryzhik Table of integrals series and products Corrected
and enlarged edition Academic Press Orlando Florida
D P Gupta M E H Ismail D R Masson Associated continuous Hahn polynomials
Canadian Journal of Mathematics
W Hahn Uber Orthogonalpolynome die q Di erenzengleichungen genugen Mathematis
che Nachrichten
R Horton Jacobi polynomials IV A family of variation diminishing kernels SIAM
Journal on Mathematical Analysis
M E H Ismail On obtaining generating functions of Boas and Buck type for orthogonal
polynomials SIAM Journal on Mathematical Analysis
M E H Ismail Connection relations and bilinear formulas for the classical orthogonal
polynomials Journal of Mathematical Analysis and Applications
M E H Ismail A queueing model and a set of orthogonal polynomials Journal of Math
ematical Analysis and Applications
M E H Ismail On sieved orthogonal polynomials I Symmetric Pol laczek analogues
SIAM Journal on Mathematical Analysis
M E H Ismail Asymptotics of the Askey Wilson and q Jacobi polynomials SIAM Journal
on Mathematical Analysis
M E H Ismail On sieved orthogonal polynomials IV Generating functions Journal of
Approximation Theory
M E H Ismail J Letessier G Valent Linear birth and death models and associated
Laguerre and Meixner polynomials Journal of Approximation Theory
M E H Ismail J Letessier G Valent Quadratic birth and death processes and
associated continuous dual Hahn polynomials SIAM Journal on Mathematical Analysis
M E H Ismail J Letessier D R Masson G Valent Birth and death processes
and orthogonal polynomials In Orthogonal Polynomials Theory and Practice ed P
Nevai Kluwer Academic Publishers Dordrecht
M E H Ismail J Letessier G Valent J Wimp Two families of associated Wilson
polynomials Canadian Journal of Mathematics
M E H Ismail J Letessier G Valent J Wimp Some results on associated Wilson
polynomials In Orthogonal Polynomials and their Applications eds C Brezinski L Gori
A Ronveaux Volume of IMACS Annals on Computing and Applied Mathematics J C
Baltzer Scienti c Publishing Company Basel
M E H Ismail D R Masson M Rahman Complex weight functions for classical
orthogonal polynomials Canadian Journal of Mathematics
M E H Ismail M E Muldoon A discrete approach to monotonicity of zeros of
orthogonal polynomials Transactions of the American Mathematical So ciety
M E H Ismail M Rahman The associated Askey Wilson polynomials Transactions
of the American Mathematical So ciety
M E H Ismail D Stanton On the Askey Wilson and Rogers polynomials Canadian
Journal of Mathematics
M E H Ismail D Stanton G Viennot The combinatorics of q Hermite polyno
mials and the Askey Wilson integral Europ ean Journal of Combinatorics
M E H Ismail M V Tamhankar A combinatorial approach to some positivity prob
lems SIAM Journal on Mathematical Analysis
M E H Ismail J A Wilson Asymptotic and generating relations for the q Jacobi and
polynomials Journal of Approximation Theory
D M Jackson Laguerre polynomials and derangements Mathematical Pro ceedings of
the Cambridge Philosophical So ciety
E G Kalnins W Miller Jr q Series and orthogonal polynomials associated with
Barnes rst lemma SIAM Journal on Mathematical Analysis
S Karlin J McGregor Linear growth birth and death processes Journal of Math
ematics and Mechanics
S Karlin J L McGregor The Hahn polynomials formulas and an application
Scripta Mathematicae
H T Koelink T H Koornwinder The Clebsch Gordan coe cients for the quantum
group S U and q Hahn polynomials Indagationes Mathematicae
T H Koornwinder The addition formula for Jacobi polynomials I Summary of results
Indagationes Mathematicae
T H Koornwinder The addition formula for Jacobi polynomials and spherical harmon
ics SIAM Journal on Applied Mathematics
T Koornwinder Jacobi polynomials II An analytic proof of the product formula SIAM
Journal on Mathematical Analysis
T Koornwinder Jacobi polynomials III An analytic proof of the addition formula
SIAM Journal on Mathematical Analysis
T Koornwinder The addition formula for Laguerre polynomials SIAM Journal on
Mathematical Analysis
T H Koornwinder Yet another proof of the addition formula for Jacobi polynomials
Journal of Mathematical Analysis and Applications
T H Koornwinder Positivity proofs for linearization and connection coe cients of
orthogonal polynomials satisfying an addition formula Journal of the London Mathematical
So ciety
T H Koornwinder Clebsch Gordan coe cients for SU and Hahn polynomials Nieuw
Archief vo or Wiskunde
T H Koornwinder Krawtchouk polynomials a uni cation of two di erent group theo
retic interpretations SIAM Journal on Mathematical Analysis
T H Koornwinder Special orthogonal polynomial systems mapped onto each other by the
Fourier Jacobi transform In Polyn omes Orthogonaux et Applications eds C Brezinski
et al Lecture Notes in Mathematics Springer Verlag New York
T H Koornwinder Group theoretic interpretations of Askey s scheme of hypergeometric
orthogonal polynomials In Orthogonal Polynomials and Their Applications eds M Alfaro
et al Lecture Notes in Mathematics Springer Verlag New York
T H Koornwinder Continuous q Legendre polynomials as spherical matrix elements of
irreducible representations of the quantum SU group C W I Quarterly
T H Koornwinder Representations of the twisted SU quantum group and some q
hypergeometric orthogonal polynomials Indagationes Mathematicae
T H Koornwinder Jacobi functions as limit cases of q ultraspherical polynomials Jour
nal of Mathematical Analysis and Applications
T H Koornwinder Orthogonal polynomials in connection with quantum groups In
Orthogonal Polynomials Theory and Practice ed P Nevai Kluwer Dordrecht
T H Koornwinder The addition formula for little q Legendre polynomials and the SU
quantum group SIAM Journal on Mathematical Analysis
T H Koornwinder Askey Wilson polynomials for root systems of type BC Contemp o
rary Mathematics
T H Koornwinder Askey Wilson polynomials as zonal spherical functions on the SU
quantum group SIAM Journal on Mathematical Analysis
A Laforgia A monotonic property for the zeros of ultraspherical polynomials Pro ceed
ings of the American Mathematical So ciety
T P Laine Projection formulas and a new proof of the addition formula for the Jacobi
polynomials SIAM Journal on Mathematical Analysis
D A Leonard Orthogonal polynomials duality and association schemes SIAM Journal
on Mathematical Analysis
P Lesky Uber Polynomsysteme die Sturm Liouvil leschen Di erenzengleichungen genu
gen Mathematische Zeitschrift
J Letessier G Valent The generating function method for quadratic asymptotical ly
symmetric birth and death processes SIAM Journal on Applied Mathematics
J Letessier G Valent Dual birth and death processes and orthogonal polynomials
SIAM Journal on Applied Mathematics
Y L Luke The special functions and their approximations Volume Mathematics in
Science and Engineering Volume Academic Press New York
D R Masson Associated Wilson polynomials Constructive Approximation
T Masuda K Mimachi Y Nakagami M Noumi K Ueno Representations of the
quantum group SU and the little q Jacobi polynomials Journal of Functional Analysis
q
H G Meijer Asymptotic expansion of Jacobi polynomials In Polyn omes Orthogonaux et
Applications eds C Brezinski et al Lecture Notes in Mathematics Springer
Verlag New York
J Meixner Orthogonale Polynomsysteme mit einer besonderen Gestalt der erzeugenden
Funktion Journal of the London Mathematical So ciety
W Miller Jr A note on Wilson polynomials SIAM Journal on Mathematical Analysis
W Miller Jr Symmetry techniques and orthogonality for q series In q Series and
Partitions ed D Stanton The IMA Volumes in Mathematics and Its Applications Volume
Springer Verlag New York
D S Moak The q analogue of the Laguerre polynomials Journal of Mathematical Analysis
and Applications
n n
D S Moak E B Saff R S Varga On the zeros of Jacobi polynomials P x
n
Transactions of the American Mathematical So ciety
B Nassrallah M Rahman Projection formulas a reproducing kernel and a gener
ating function for q Wilson polynomials SIAM Journal on Mathematical Analysis
A F Nikiforov S K Suslov V B Uvarov Classical orthogonal polynomials of
a discrete variable Springer Series in Computational Physics Springer Verlag New York
M Noumi K Mimachi Roger s q ultraspherical polynomials on a quantum sphere
Duke Mathematical Journal
F W J Olver Asymptotics and special functions Computer Science and Applied Math
ematics Academic Press New York
M Perlstadt A property of orthogonal polynomial families with polynomial duals SIAM
Journal on Mathematical Analysis
J Prasad H Hayashi On the uniform approximation of smooth functions by Jacobi
polynomials Canadian Journal of Mathematics
M Rahman Construction of a family of positive kernels from Jacobi polynomials SIAM
Journal on Mathematical Analysis
M Rahman A ve parameter family of positive kernels from Jacobi polynomials SIAM
Journal on Mathematical Analysis
M Rahman Some positive kernels and bilinear sums for Hahn polynomials SIAM Journal
on Mathematical Analysis
M Rahman On a generalization of the Poisson kernel for Jacobi polynomials SIAM
Journal on Mathematical Analysis
M Rahman A generalization of Gasper s kernel for Hahn polynomials application to
Pol laczek polynomials Canadian Journal of Mathematics
M Rahman A positive kernel for Hahn Eberlein polynomials SIAM Journal on Mathe
matical Analysis
M Rahman An elementary proof of Dunkl s addition theorem for Krawtchouk polynomials
SIAM Journal on Mathematical Analysis
M Rahman A product formula and a non negative Poisson kernel for Racah Wilson
polynomials Canadian Journal of Mathematics
M Rahman A non negative representation of the linearization coe cients of the product
of Jacobi polynomials Canadian Journal of Mathematics
M Rahman A stochastic matrix and bilinear sums for Racah Wilson polynomials SIAM
Journal on Mathematical Analysis
M Rahman Families of biorthogonal rational functions in a discrete variable SIAM
Journal on Mathematical Analysis
M Rahman The linearization of the product of continuous q Jacobi polynomials Canadian
Journal of Mathematics
M Rahman Reproducing kernels and bilinear sums for q Racah and q Wilson polynomials
Transactions of the American Mathematical So ciety
M Rahman A q extension of Feldheim s bilinear sum for Jacobi polynomials and some
applications Canadian Journal of Mathematics
M Rahman A product formula for the continuous q Jacobi polynomials Journal of Math
ematical Analysis and Applications
M Rahman q Wilson functions of the second kind SIAM Journal on Mathematical
Analysis
M Rahman A simple proof of Koornwinder s addition formula for the little q Legendre
polynomials Pro ceedings of the American Mathematical So ciety
M Rahman Askey Wilson functions of the rst and second kinds series and integral rep
resentations of C x jq D x jq Journal of Mathematical Analysis and Applications
n n
M Rahman M J Shah An in nite series with products of Jacobi polynomials and
Jacobi functions of the second kind SIAM Journal on Mathematical Analysis
M Rahman A Verma Product and addition formulas for the continuous q ultra
spherical polynomials SIAM Journal on Mathematical Analysis
E D Rainville Special functions The Macmillan Company New York
T J Rivlin The Chebyshev polynomials Pure and Applied Mathematics A Wiley Inter
science series of Texts Monographs and Tracts John Wiley and Sons New York
L J Rogers On the exansion of some in nite products Pro ceedings of the London
Mathematical So ciety
L J Rogers Second memoir on the expansion of certain in nite products Pro ceedings of
the London Mathematical So ciety
L J Rogers Third memoir on the expansion of certain in nite products Pro ceedings of
the London Mathematical So ciety
M de Sainte Catherine G Viennot Combinatorial interpretation of integrals of
products of Hermite Laguerre and Tchebyche polynomials In Polyn omes Orthogonaux et
Applications eds C Brezinski et al Lecture Notes in Mathematics Springer
Verlag New York
H M Srivastava Generating functions for Jacobi and Laguerre polynomials Pro ceedings
of the American Mathematical So ciety
H M Srivastava V K Jain Some formulas involving q Jacobi and related polynomi
als Annali di Matematica Pura ed Applicata
H M Srivastava J P Singhal New generating functions for Jacobi and related
polynomials Journal of Mathematical Analysis and Applications
D Stanton A short proof of a generating function for Jacobi polynomials Pro ceedings
of the American Mathematical So ciety
D Stanton Product formulas for q Hahn polynomials SIAM Journal on Mathematical
Analysis
D Stanton Some q Krawtchouk polynomials on Cheval ley groups American Journal of
Mathematics
D Stanton Orthogonal polynomials and Cheval ley groups In Sp ecial Functions
Group Theoretical Asp ects and Applications eds R A Askey et al D Reidel Publishing
Company Dordrecht
D Stanton An introduction to group representations and orthogonal polynomials In
Orthogonal Polynomials Theory and Practice ed P Nevai Kluwer Academic Publishers
Dordrecht
T J Stieltjes Recherches sur les fractions continues Annales de la Facult e des Sciences
de Toulouse J A uvres Compl etes Volume
G Szego Orthogonal polynomials American Mathematical So ciety Collo quium Publica
tions Providence Rho de Island Fourth edition
W Van Assche T H Koornwinder Asymptotic behaviour for Wal l polynomials
and the addition formula for little q Legendre polynomials SIAM Journal on Mathematical
Analysis
G Viennot A combinatorial theory for general orthogonal polynomials with extensions
and applications In Polyn omes Orthogonaux et Applications eds C Brezinski et al
Lecture Notes in Mathematics Springer Verlag New York
B Viswanathan Generating functions for ultraspherical functions Canadian Journal of
Mathematics
H S Wall A continued fraction related to some partition formulas of Euler The American
Mathematical Monthly
L Weisner Generating functions for Hermite functions Canadian Journal of Mathemat
ics
S Wigert Sur les polynomes orthogonaux et l approximation des fonctions continues
Arkiv f or Matematik Astronomi o ch Fysik
J A Wilson Some hypergeometric orthogonal polynomials SIAM Journal on Mathemat
ical Analysis
J A Wilson Asymptotics for the F polynomials Journal of Approximation Theory
J Wimp Explicit formulas for the associated Jacobi polynomials and some applications
Canadian Journal of Mathematics
J Wimp Pol laczek polynomials and Pad e approximants some closed form expressions
Journal of Computational and Applied Mathematics
M Wyman The asymptotic behaviour of the Hermite polynomials Canadian Journal of
Mathematics
A I Zayed Jacobi polynomials as generalized Faber polynomials Transactions of the
American Mathematical So ciety
J Zeng Lin earisation de produits de polyn omes de Meixner Krawtchouk et Charlier
SIAM Journal on Mathematical Analysis
Index
A ne q Krawtchouk p olynomials Dual Hahn p olynomials
Gegenbauer p olynomials
Al Salam Carlitz I p olynomials
Hahn p olynomials
Al Salam Carlitz I I p olynomials
Hermite p olynomials
Al Salam Chihara p olynomials
Jacobi p olynomials
Alternative q Charlier p olynomials
Krawtchouk p olynomials
Askey scheme
Laguerre p olynomials
Askey Wilson p olynomials
Legendre p olynomials
Little q Jacobi p olynomials
Big q Jacobi p olynomials
Big q Laguerre p olynomials
Little q Laguerre p olynomials
Big q Legendre p olynomials
Little q Legendre p olynomials
Charlier p olynomials
Meixner p olynomials
Chebyshev p olynomials
Meixner Pollaczek p olynomials
Continuous q Hahn p olynomials
Continuous q Hermite p olynomials
q Charlier p olynomials
q Hahn p olynomials
Continuous q Jacobi p olynomials
q Krawtchouk p olynomials
Continuous q Laguerre p olynomials
q Laguerre p olynomials
q Meixner p olynomials
Continuous q Legendre p olynomials
q Meixner Pollaczek p olynomials
Continuous q ultraspherical p olynomials
q Racah p olynomials
Continuous big q Hermite p olynomials
q Scheme
Quantum q Krawtchouk p olynomials
Continuous dual q Hahn p olynomials
Continuous dual Hahn p olynomials
Racah p olynomials
Rogers p olynomials
Continuous Hahn p olynomials
Spherical p olynomials
Stieltjes Wigert p olynomials
Discrete q Hermite I p olynomials
Ultraspherical p olynomials
Discrete q Hermite II p olynomials
Wall p olynomials
Dual q Hahn p olynomials
Wilson p olynomials
Dual q Krawtchouk p olynomials
The Askey scheme of hyp ergeometric orthogonal p olynomials
and its q analogue
Ro elof Ko eko ek Ren e F Swarttouw
February
List of errata in rep ort no
Page line Replace all sets by all known sets
Page line Replace p ositive de nite by p ositive
Page line This should read
a a
r 1
a a q q
1 r
1+s r
z lim q q z F
r s r s
b b
s 1
q "1
b b q q
1 s
Page formula The right hand side should read
n a c n a d n b c n b d
mn
n a b c d n a b c d n
Page formula This should read
i h
1 1 1 1
2
2 2 2 2
bq cq dq jq D y x w x aq q D
q q
w x a b c djq y x y x p x a b c djq
n n
where
w x a b c djq
p
w x a b c djq
2
x
Page formula This should read
h i
1 1 1
2
2 2 2
w x aq q D bq cq jq D y x
q q
n+1 n
q q w x a b cjq y x y x p x a b cjq
n
where
w x a b cjq
p
w x a b cjq
2
x
Page formula This should read
i h
1 1 1 1
2
2 2 2 2
bq cq dq q D y x w x aq q D
q q
w x a b c d q y x y x p x a b c d q
n n
where
w x a b c d q
p
w x a b c d q
2
x
n 2 n
Page formula Replace ac by acq
Page line The weight function should read
1 1
c q x d q x q
1
d x
q
1 1
ac q x bd q x q
1
n 2 n
Page formula Replace c by cq
Page formula This should read
h i
1 1
2
2 2
w x aq q D bq jq D y x
q q
n+1 n
q q w x a bjq y x y x Q x a bjq
n
where
w x a bjq
p
w x a bjq
2
x
Page formula This should read
i h
1
n+1 n 2
2
jq D y x q q w x ajq y x y x P x ajq w x aq q D
q n q
where
w x ajq
p
w x ajq
2
x
Page formula This should read
h i
1 1
+ 2 + ( )
2 2
w x q q D q jq D y x w x q q jq y x y x P xjq
q q n
n
where
w x q q jq
p
w x q q jq
2
x
Page fomula This should read
h i
1 1
+ 2 + ( )
2 2
w x q q D q q D y x w x q q q y x y x P x q
q q n
n
where
w x q q q
p
w x q q q
2
x
Page formula This should read
i h
1
2
2
jq D y x w x jq y x y x C x jq w x q q D
q n n q
where
w x jq
p
w x jq
2
x
Page formula This can b e written as
1 1
2n+1 2n+2 2n
2 2
q xP x q q q P x q q q P x q
n n+1 n 1
Page formula This should read
1
2
2
q D w x q q D y x w x q q y x y x P x q
q q n n
where
w x a q
p
w x a q
2
x
n 2 n
Page formula Replace ab by abq
Page formula This should read
h i
1
2 n+1 n
2
w x aq q D jq D y x q q w x ajq y x y x H x ajq
q q n
where
w x ajq
p
w x ajq
2
x
Page formula This can also b e written as
1 1
( )
n+1 ( )
2 4
xjq q P xP xjq q
n
n+1
1 1 1 1 1
( )
n+ ( ) n+ + +
2 4 2 2 4
q q P xjq q q P xjq
n
n 1
Page formula This can also b e written as
1
( )
( ) 2n+2
2
xP x q q q P x q
n
n+1
1 1
( )
( ) 2n+2 2n+ +
2 2
q P x q q q P x q q
n n 1
Page formula This should read
i h
1
2 n+1 n ( ) +
2
q D jq D y x q q w x q jq y x y x P xjq w x q
q q
n
where
w x q jq
p
w x q jq
2
x
Page formula This should read
h i
1
+ 2 n+1 n ( )
2
w x q q D q D y x q q w x q q y x y x P x q
q q
n
where
w x q q
p
w x q q
2
x
Page formula This should read
n 2 n 1 n 1 n
q x y x aq y q x aq q x a x y x
n 1 (a)
x q q x a x y q x y x U
n
Page formula This should read
n 2
q x y x x a x y q x x a x aq y x
1 (a)
aq y q x y x V x q
n
Page formula Replace d by dx
Page formula This should read
2 n+1 n
q D w x D y x q q w x y x y x H xjq
q q n
where
w x
p
w x
2
x
Page formula This should read
n+1 2 2 1
q x y x y q x q y x q x y q x y x h x q
n
Page formula This should read
n
n
q ix
n n ( 1) n
2
q q h x q i V ix q i q
n 2 0
n
2 n n+1
q q q
2 n
q x
2 1
2
x
Page formula This can b e written as
1
h i
X
k k k k k k
h cq q h cq q h cq q h cq q w cq q
m n m n
k = 1
2 2 2 2
q c q c q q q q
1 n
c
mn
2
2 2 2 2 n
q c c q q q
1
where
w x
2 2
ix q ix q x q
1 1 1
Page formula This should read
n 2 2 2 1
q x y x x y q x x q y x q y q x y x h x q
n
Page Reference bf A should read A
Acknowledgement
We thank G Gasp er J Ko eko ek H T Ko elink and T H Ko ornwinder for p ointing us to
some of these errata