Hypergeometric Orthogonal Polynomials and Their G-Analogues

Home , Askey scheme, Dual Hahn polynomials, Hahn polynomials, Orthogonal polynomials, Romanovski polynomials

Additions to the formula lists in “Hypergeometric orthogonal polynomials and their q-analogues” by Koekoek, Lesky and Swarttouw

Tom H. Koornwinder

September 17, 2021

Abstract This report gives a rather arbitrary choice of formulas for (q-)hypergeometric orthogonal polynomials which the author missed while consulting Chapters 9 and 14 in the book “Hypergeometric orthogonal polynomials and their q-analogues” by Koekoek, Lesky and Swarttouw. The systematics of these chapters will be followed here, in particular for the numbering of subsections and of references.

Introduction This report contains some formulas about (q-)hypergeometric orthogonal polynomials which I missed but wanted to use while consulting Chapters 9 and 14 in the book [KLS]: R. Koekoek, P. A. Lesky and R. F. Swarttouw, Hypergeometric orthogonal polynomials and their q-analogues, Springer-Verlag, 2010. These chapters form together the (slightly extended) successor of the report R. Koekoek and R. F. Swarttouw, The Askey-scheme of hypergeometric orthogonal polynomials and its q-analogue, Report 98-17, Faculty of Technical Mathematics and Informatics, Delft Uni- versity of Technology, 1998; http://aw.twi.tudelft.nl/~koekoek/askey/. Certainly these chapters give complete lists of formulas of special type, for instance orthogonality relations and three-term recurrence relations. But outside these narrow categories there are many other formulas for (q-)orthogonal polynomials which one wants to have available. Of- ten one can ﬁnd the desired formula in one of the standard references listed at the end of this report. Sometimes it is only available in a journal or a less common monograph. Just for my own comfort, I have brought together some of these formulas. This will possibly also be helpful for some other users. Usually, any type of formula I give for a special class of polynomials, will suggest a similar formula for many other classes, but I have not aimed at completeness by ﬁlling in a formula of such type at all places. The resulting choice of formulas is rather arbitrary, just depending on the formulas which I happened to need or which raised my interest. For each formula I give a suitable reference or I sketch a proof. It is my intention to gradually extend this collection of formulas.

1 Conventions The (x.y) and (x.y.z) type subsection numbers, the (x.y.z) type formula numbers, and the [x] type citation numbers refer to [KLS]. The (x) type formula numbers refer to this manuscript and the [Kx] type citation numbers refer to citations which are not in [KLS]. Some standard references like [DLMF] are given by special acronyms. N is always a positive integer. Always assume n to be a nonnegative integer or, if N is present, to be in {0, 1,...,N}. Throughout assume 0 < q < 1. For each family the coeﬃcient of the term of highest degree of the orthogonal polynomial of degree n can be found in [KLS] as the coeﬃcient of pn(x) in the formula after the main formula under the heading “Normalized Recurrence Relation”. If that main formula is numbered as (x.y.z) then I will refer to the second formula as (x.y.zb). In the notation of q-hypergeometric orthogonal polynomials we will follow the convention that the parameter list and q are separated by ‘ | ’ in the case of a q-quadratic lattice (for instance Askey–Wilson) and by ‘;’ in the case of a q-linear lattice (for instance big q-Jacobi). This convention is mostly followed in [KLS], but not everywhere, see for instance little q-Laguerre / Wall.

Acknowledgement Many thanks to Howard Cohl for having called my attention so often to typos and inconsistencies. Thanks also to Roberto Costas Santos for observing an error and to Gregory Natanson for historical information about Routh.

2 Contents Introduction Conventions Generalities 9.1 Wilson 9.2 Racah 9.3 Continuous dual Hahn 9.4 Continuous Hahn 9.5 Hahn 9.6 Dual Hahn 9.7 Meixner–Pollaczek 9.8 Jacobi 9.8.1 Gegenbauer / Ultraspherical 9.8.2 Chebyshev 9.9 Pseudo Jacobi (or Romanovski–Routh) 9.10 Meixner 9.11 Krawtchouk 9.12 Laguerre 9.13 Bessel 9.14 Charlier 9.15 Hermite 14.1 Askey–Wilson 14.2 q-Racah 14.3 Continuous dual q-Hahn 14.4 Continuous q-Hahn 14.5 Big q-Jacobi 14.7 Dual q-Hahn 14.8 Al-Salam–Chihara 14.9 q-Meixner–Pollaczek 14.10 Continuous q-Jacobi 14.10.1 Continuous q-ultraspherical / Rogers 14.11 Big q-Laguerre 14.12 Little q-Jacobi 14.14 Quantum q-Krawtchouk 14.16 Aﬃne q-Krawtchouk 14.17 Dual q-Krawtchouk 14.20 Little q-Laguerre / Wall 14.21 q-Laguerre 14.27 Stieltjes-Wigert 14.28 Discrete q-Hermite I 14.29 Discrete q-Hermite II Standard references References from [KLS]

3 Other references

4 Generalities Criteria for uniqueness of orthogonality measure According to Shohat & Tamarkin [K30, p.50] orthonormal polynomials pn have a unique orthogonality measure (up to positive constant factor) if for some z ∈ C we have ∞ X 2 |pn(z)| = ∞. (1) n=0

Also (see Shohat & Tamarkin [K30, p.59]), monic orthogonal polynomials pn with three-term recurrence relation xpn(x) = pn+1(x) + Bnpn(x) + Cnpn−1(x)(Cn necessarily positive) have a unique orthogonality measure if ∞ X −1/2 (Cn) = ∞. (2) n=1 Furthermore, if orthogonal polynomials have an orthogonality measure with bounded sup- port, then this is unique (see Chihara [146]).

Even orthogonality measure If {pn} is a system of orthogonal polynomials with respect to an even orthogonality measure which satisﬁes the three-term recurrence relation

xpn(x) = Anpn+1(x) + Cnpn−1(x) then p (0) C 2n = − 2n−1 . (3) p2n−2(0) A2n−1

Appell’s bivariate hypergeometric function F4 This is deﬁned by ∞ 0 X (a)m+n(b)m+n m n 1 1 F (a, b; c, c ; x, y) := x y (|x| 2 + |y| 2 < 1), (4) 4 (c) (c0) m! n! m,n=0 m n see [HTF1, 5.7(9), 5.7(44)] or [DLMF, (16.13.4)]. There is the reduction formula −x −y a, 1 + a − b F a, b; b, b; , = (1 − x)a(1 − y)a F ; xy , 4 (1 − x)(1 − y) (1 − x)(1 − y) 2 1 b see [HTF1, 5.10(7)]. When combined with the quadratic transformation [HTF1, 2.11(34)] (here a − b − 1 should be replaced by a − b + 1), see also [DLMF, (15.8.15)], this yields −x −y F a, b; b, b; , 4 (1 − x)(1 − y) (1 − x)(1 − y) (1 − x)(1 − y)a 1 a, 1 (a + 1) 4xy = F 2 2 ; . 1 + xy 2 1 b (1 + xy)2 This can be rewritten as 1 a, 1 (a + 1) 4xy F (a, b; b, b; x, y) = (1 − x − y)−a F 2 2 ; . (5) 4 2 1 b (1 − x − y)2 1 1 4xy 2 2 Note that, if x, y ≥ 0 and x + y < 1, then 1 − x − y > 0 and 0 ≤ (1−x−y)2 < 1.

5 q-Hypergeometric series of base q−1 By [GR, Exercise 1.4(i)]:

−1 −1 ! a1, . . . , ar −1 a1 , . . . ar , 0,..., 0 qa1 . . . arz rφs ; q , z = s+1φs −1 −1 ; q, (6) b1, . . . bs b1 , . . . , bs b1 . . . bs for r ≤ s + 1, a1, . . . , ar, b1, . . . , bs 6= 0. In the non-terminating case, for 0 < q < 1, there is convergence if |z| < b1 . . . bs/(qa1 . . . ar).

A transformation of a terminating 2φ1 By [GR, Exercise 1.15(i)] we have q−n, b q−n, c/b, 0 φ ; q, z = (bz/(cq); q−1) φ ; q, q . (7) 2 1 c n 3 2 c, cq/(bz)

Very-well-poised q-hypergeometric series The notation of [GR, (2.1.11)] will be followed:

1 1  2 2  a1, qa1 , −qa1 , a4, . . . , ar+1 r+1Wr(a1; a4, a5, . . . , ar+1; q, z) := r+1φr 1 1 ; q, z . (8) 2 2 a1 , −a1 , qa1/a4, . . . , qa1/ar+1

Theta function The notation of [GR, (11.2.1)] will be followed:

θ(x; q) := (x, q/x; q)∞, θ(x1, . . . , xm; q) := θ(x1; q) . . . θ(xm; q). (9)

9.1 Wilson

Symmetry The Wilson polynomial Wn(y; a, b, c, d) is symmetric in a, b, c, d. This follows from the orthogonality relation (9.1.2) together with the value of its coeﬃcient of yn given in (9.1.5b). Alternatively, combine (9.1.1) with [AAR, Theorem 3.1.1]. As a consequence, it is suﬃcient to give generating function (9.1.12). Then the generating functions (9.1.13), (9.1.14) will follow by symmetry in the parameters.

Hypergeometric representation In addition to (9.1.1) we have (see [513, (2.2)]):

2 (a − ix)n(b − ix)n(c − ix)n(d − ix)n Wn(x ; a, b, c, d) = (−2ix)n 1 ! 2ix − n, ix − 2 n + 1, a + ix, b + ix, c + ix, d + ix, −n × 7F6 1 ; 1 . ix − 2 n, 1 − n − a + ix, 1 − n − b + ix, 1 − n − c + ix, 1 − n − d + ix, 1 + 2ix (10)

The symmetry in a, b, c, d is clear from (10).

Special value 2 Wn(−a ; a, b, c, d) = (a + b)n(a + c)n(a + d)n , (11) 2 2 2 and similarly for arguments −b , −c and −d by symmetry of Wn in a, b, c, d.

6 Uniqueness of orthogonality measure Under the assumptions on a, b, c, d for (9.1.2) or (9.1.3) the orthogonality measure is unique up to constant factor. For the proof assume without loss of generality (by the symmetry in a, b, c, d) that Re a ≥ 0. Write the right-hand side of (9.1.2) or (9.1.3) as hnδm,n. Observe from (9.1.2) and (11) that

|W (−a2; a, b, c, d)|2 n = O(n4Re a−1) as n → ∞. hn

Therefore (1) holds, from which the uniqueness of the orthogonality measure follows. By a similar, but necessarily more complicated argument Ismail et al. [281, Section 3] proved the uniqueness of orthogonality measure for associated Wilson polynomials.

9.2 Racah Racah in terms of Wilson In the Remark on p.196 Racah polynomials are expressed in terms of Wilson polynomials. This can be equivalently written as

Rn x(x − N + δ); α, β, −N − 1, δ 1 2 1 1 1 1 Wn − (x + (δ − N)) ; (δ − N), α + 1 − (δ − N), β + (δ + N) + 1, − (δ + N) = 2 2 2 2 2 . (12) (α + 1)n(β + δ + 1)n(−N)n

9.3 Continuous dual Hahn

Symmetry The continuous dual Hahn polynomial Sn(y; a, b, c) is symmetric in a, b, c. This follows from the orthogonality relation (9.3.2) together with the value of its coeﬃcient of yn given in (9.3.5b). Alternatively, combine (9.3.1) with [AAR, Corollary 3.3.5]. As a consequence, it is suﬃcient to give generating function (9.3.12). Then the generating functions (9.3.13), (9.3.14) will follow by symmetry in the parameters.

Special value 2 Sn(−a ; a, b, c) = (a + b)n(a + c)n , (13)

2 2 and similarly for arguments −b and −c by symmetry of Sn in a, b, c.

Uniqueness of orthogonality measure Under the assumptions on a, b, c for (9.3.2) or (9.3.3) the orthogonality measure is unique up to constant factor. For the proof assume without loss of generality (by the symmetry in a, b, c) that Re a ≥ 0. Write the right-hand side of (9.3.2) or (9.3.3) as hnδm,n. Observe from (9.3.2) and (13) that

|S (−a2; a, b, c)|2 n = O(n2Re a−1) as n → ∞. hn

Therefore (1) holds, from which the uniqueness of the orthogonality measure follows.

7 Special continuous dual Hahn in terms of Wilson

2n 1 2 1 1 1 1 1 Sn x; a, b, 2 = Wn 4 x; 2 a, 2 (a + 1), 2 b, 2 (b + 1) . (14) (a + b + n)n

For the proof compare the weight functions and the values for x = −a2.

Generating functions By (9.3.17) the generating function (9.3.16) has the generating function (9.7.13) for Meixner–Pollaczek polynomials as a limit case.

9.4 Continuous Hahn

Orthogonality relation and parameter symmetry The orthogonality relation (9.4.2) holds under the more general assumption that Re (a, b, c, d) > 0 and (c, d) = (a, b) or (b, a). Thus, under these assumptions, the continuous Hahn polynomial pn(x; a, b, c, d) is symmetric in a, b and in c, d. This follows from the orthogonality relation (9.4.2) together with the value of its coeﬃcient of xn given in (9.4.4b). As a consequence, it is suﬃcient to give generating function (9.4.11). Then the generating function (9.4.12) will follow by symmetry in the parameters.

Symmetry n pn(−x; a, b, a, b) = (−1) pn(x; a, b, a, b). (15)

Special value in(a + a) (a + b) p (ia; a, b, a, b) = n n . (16) n n!

Similarly, pn(x; a, b, a, b) has special values for x = −ia, ib and −ib.

Quadratic transformation For a, b ∈ R or b = a we have [K20, (2.29), (2.30)]

2 1 2 1 p (x; a, b, a, b) Wn(x ; a, b, , 0) p (x; a, b, a, b) xWn(x ; a, b, , 1) 2n = 2 , 2n+1 = 2 . (17) 2 1 2 1 p2n(ia; a, b, a, b) Wn(−a ; a, b, 2 , 0) p2n+1(ia; a, b, a, b) iaWn(−a ; a, b, 2 , 1)

Explicit expression For a, b ∈ R or b = a we have by (17), (9.1.1) and reversion of direction of summation that

(n + a + b + a + b − 1)n 1 n−2[ 2 n] 1 1 pn(x; a, b, a, b) = x (− 2 n + ix + 1)[ 1 n](− 2 n − ix + 1)[ 1 n] n! 2 2 1 1 1 1 1 ! − 2 n, − 2 n + 2 , − 2 n − a + 1, − 2 n − b + 1 × 4F3 3 1 1 ; 1 . (18) −n − a − b + 2 , − 2 n + ix + 1, − 2 n − ix + 1

8 Special cases In the following special case there is a reduction to Meixner–Pollaczek:

1 1 1 (2a)n(2a + 2 )n (2a) 1 pn(x; a, a + 2 , a, a + 2 ) = Pn (2x; 2 π). (19) (4a)n

See [342, (2.6)] (note that in [342, (2.3)] the Meixner–Pollaczek polyonmials are deﬁned diﬀerent from (9.7.1), without a constant factor in front). For 0 < a < 1 the continuous Hahn polynomials pn(x; a, 1 − a, a, 1 − a) are orthogonal on (−∞, ∞) with respect to the weight function cosh(2πx) − cos(2πa)−1 (by straightforward 1 computation from (9.4.2)). For a = 4 the two special cases coincide: Meixner–Pollaczek with weight function cosh(2πx)−1.

Uniqueness of orthogonality measure The coeﬃcient of pn−1(x) in (9.4.4) behaves as O(n2) as n → ∞. Hence (2) holds, by which the orthogonality measure is unique.

9.5 Hahn Special values

n (−1) (β + 1)n Qn(0; α, β, N) = 1,Qn(N; α, β, N) = . (20) (α + 1)n

Use (9.5.1) and compare with (9.8.1) and (43). From (9.5.3) and (3) it follows that

1 ( 2 )n(N + α + 1)n Q2n(N; α, α, 2N) = 1 . (21) (−N + 2 )n(α + 1)n From (9.5.1) and [DLMF, (15.4.24)] it follows that

(−N − β)x QN (x; α, β, N) = (x = 0, 1,...,N). (22) (α + 1)x

Symmetries By the orthogonality relation (9.5.2):

Qn(N − x; α, β, N) = Qn(x; β, α, N), (23) Qn(N; α, β, N)

It follows from (30) and (25) that

QN−n(x; α, β, N) = Qn(x; −N − β − 1, −N − α − 1,N)(x = 0, 1,...,N). (24) QN (x; α, β, N)

Duality The Remark on p.208 gives the duality between Hahn and dual Hahn polynomials:

Qn(x; α, β, N) = Rx(n(n + α + β + 1); α, β, N)(n, x ∈ {0, 1,...N}). (25)

9 9.6 Dual Hahn Special values By (22) and (25) we have

(−N − δ)n Rn(N(N + γ + δ + 1); γ, δ, N) = . (26) (γ + 1)n It follows from (20) and (25) that x (−1) (δ + 1)x RN (x(x + γ + δ + 1); γ, δ, N) = (x = 0, 1,...,N). (27) (γ + 1)x

Symmetries Write the weight in (9.6.2) as 2x + γ + δ + 1 (γ + 1)x N wx(α, β, N) := N! . (28) (x + γ + δ + 1)N+1 (δ + 1)x x Then (δ + 1)N wN−x(γ, δ, N) = (−γ − N)N wx(−δ − N − 1, −γ − N − 1,N). (29) Hence, by (9.6.2),

Rn((N − x)(N − x + γ + δ + 1); γ, δ, N) = Rn(x(x−2N −γ −δ−1); −N −δ−1, −N −γ −1,N). Rn(N(N + γ + δ + 1); γ, δ, N) (30) Alternatively, (30) follows from (9.6.1) and [DLMF, (16.4.11)]. It follows from (23) and (25) that

RN−n(x(x + γ + δ + 1); γ, δ, N) = Rn(x(x + γ + δ + 1); δ, γ, N)(x = 0, 1,...,N). (31) RN (x(x + γ + δ + 1); γ, δ, N)

Re: (9.6.11). The generating function (9.6.11) can be written in a more conceptual way as

N x − N, x + γ + 1 N! X (1 − t)x F ; t = ω R (λ(x); γ, δ, N) tn, (32) 2 1 −δ − N (δ + 1) n n N n=0 where γ + nδ + N − n ω := , (33) n n N − n i.e., the denominator on the right-hand side of (9.6.2). By the duality between Hahn polynomials and dual Hahn polynomials (see (25)) the above generating function can be rewritten in terms of Hahn polynomials:

N n − N, n + α + 1 N! X (1 − t)n F ; t = w Q (x; α, β, N) tx, (34) 2 1 −β − N (β + 1) x n N x=0 where α + xβ + N − x w := , (35) x x N − x i.e., the weight occurring in the orthogonality relation (9.5.2) for Hahn polynomials.

10 Re: (9.6.15). There should be a closing bracket before the equality sign.

9.7 Meixner–Pollaczek

Re: (9.7.1) In addition to the hypergeometric representation (9.7.1) we have, by the Pfaﬀ transformation [HTF1, 2.9(3)], that

(2λ) −n, λ − ix P (λ)(x; φ) = n e−inφ F ; 1 − e2iφ . (36) n n! 2 1 2λ

Special values By (9.7.1) and (36) we have:

(2λ) (2λ) P (λ)(iλ; φ) = n einφ,P (λ)(−iλ; φ) = n e−inφ. (37) n n! n n!

Symmetry (λ) n (λ) Pn (x; φ) = (−1) Pn (−x; π − φ). (38)

Quadratic transformations [K20, (2.33), (2.34)]

(a) 1 2 1 (a) 1 2 1 P (x; π) Sn(x ; a, , 0) P (x; π) xSn(x ; a, , 1) 2n 2 = 2 , 2n+1 2 = 2 . (39) (a) 2 1 (a) 2 1 1 Sn(−a ; a, , 0) 1 iaSn(−a ; a, , 1) P2n (ia; 2 π) 2 P2n+1(ia; 2 π) 2

These are limit cases of (17) by the limits (9.1.16), (9.4.14).

Uniqueness of orthogonality measure The coeﬃcient of pn−1(x) in (9.7.4) behaves as O(n2) as n → ∞. Hence (2) holds, by which the orthogonality measure is unique.

Generating functions By (9.3.17) the generating function (9.3.16) for continuous dual Hahn polynomials has the generating function (9.7.13) as a limit case. By (9.7.14) formula (9.7.13) has the generating function (9.12.12) for Laguerre polynomials as a limit case.

9.8 Jacobi

Orthogonality relation Write the right-hand side of (9.8.2) as hn δm,n. Then

α+β+1 hn n + α + β + 1 (α + 1)n(β + 1)n 2 Γ(α + 1)Γ(β + 1) = , h0 = , h0 2n + α + β + 1 (α + β + 2)n n! Γ(α + β + 2) (40) h n + α + β + 1 (β + 1) n! n = n . (α,β) 2 2n + α + β + 1 (α + 1) (α + β + 2) h0 (Pn (1)) n n

11 In (9.8.3) the numerator factor Γ(n + α + β + 1) in the last line should be Γ(β + 1). When thus corrected, (9.8.3) can be rewritten as:

Z ∞ (α,β) (α,β) α β Pm (x) Pn (x)(x − 1) (x + 1) dx = hn δm,n , 1 1 − 1 − β > α > −1, m, n < − 2 (α + β + 1), (41) α+β+1 hn n + α + β + 1 (α + 1)n(β + 1)n 2 Γ(α + 1)Γ(−α − β − 1) = , h0 = . h0 2n + α + β + 1 (α + β + 2)n n! Γ(−β)

Following Lesky [382] the Jacobi polynomials in case of orthogonality relation (41) may be called Romanovski–Jacobi polynomials.

Symmetry (α,β) n (β,α) Pn (−x) = (−1) Pn (x). (42) Use (9.8.2) and (9.8.5b) or see [DLMF, Table 18.6.1].

Special values

(α + 1) (−1)n(β + 1) P (α,β)(−1) (−1)n(β + 1) P (α,β)(1) = n ,P (α,β)(−1) = n , n = n . (43) n n (α,β) n! n! Pn (1) (α + 1)n

Use (9.8.1) and (42) or see [DLMF, Table 18.6.1].

Generating functions Formula (9.8.15) was ﬁrst obtained by Brafman [109, (12)]. Alterna- tively (see [109, (9)] or use [DLMF, (16.16.6)]), the left-hand side of (9.8.15) can be written as Appell’s hypergeometric function F4:

∞ 1 1 X (γ)k(α + β + 1 − γ)k (α,β) k F4 γ, α+β +1−γ; α+1, β +1; 2 t(x−1), 2 t(x+1) = Pk (x)t (44) (α + 1)k(β + 1)k k=0

The generating function (9.12.12) for Laguerre polynomials is a limit case of (44) by (9.8.16). 1 1+xy Formula (9.8.15) with t, x replaced by 2 (x + y), x+y , respectively, takes the form

γ, α + β + 1 − γ γ, α + β + 1 − γ F ; 1 (1 − x) F ; 1 (1 + y) 2 1 α + 1 2 2 1 β + 1 2 ∞ X (γ)k(α + β + 1 − γ)k k (α,β) 1 + xy = (x + y) Pk . (45) (α + 1)k(β + 1)k x + y k=0

In [109, (14)] the case γ nonpositive integer of (9.8.15) is given. When we di this for (45) with γ = −n ∈ Z≤0 this yields the inverse of Bateman’s bilinear sum, as is given in [331, (2.19), (2.20)], [DLMF, (18.18.25), (18.18.26)].

12 Bilinear generating functions For 0 ≤ r < 1 and x, y ∈ [−1, 1] we have in terms of F4 (see (4)): ∞ X (α + β + 1)n n! 1 rn P (α,β)(x) P (α,β)(y) = (α + 1) (β + 1) n n (1 + r)α+β+1 n=0 n n r(1 − x)(1 − y) r(1 + x)(1 + y) × F 1 (α + β + 1), 1 (α + β + 2); α + 1, β + 1; , , (46) 4 2 2 (1 + r)2 (1 + r)2 ∞ X 2n + α + β + 1 (α + β + 2)n n! 1 − r rn P (α,β)(x) P (α,β)(y) = n + α + β + 1 (α + 1) (β + 1) n n (1 + r)α+β+2 n=0 n n r(1 − x)(1 − y) r(1 + x)(1 + y) × F 1 (α + β + 2), 1 (α + β + 3); α + 1, β + 1; , . (47) 4 2 2 (1 + r)2 (1 + r)2 Formulas (46) and (47) were ﬁrst given by Bailey [91, (2.1), (2.3)]. See Stanton [485] for a shorter proof. (However, in the second line of [485, (1)] z and Z should be interchanged.) As observed 1 1 − 2 (α+β−1) d 2 (α+β+1) in Bailey [91, p.10], (47) follows from (46) by applying the operator r dr ◦ r to both sides of (46). In view of (40), formula (47) is the Poisson kernel for Jacobi polynomials. The right-hand side of (47) makes clear that this kernel is positive. See also the discussion in Askey [46, following (2.32)].

Quadratic transformations

(α+ 1 ) (α,− 1 ) C 2 (x) P (α,α)(x) P 2 (2x2 − 1) 2n = 2n = n , (48) (α+ 1 ) (α,α) (α,− 1 ) 2 P (1) 2 C2n (1) 2n Pn (1) (α+ 1 ) (α,α) (α, 1 ) C 2 (x) P (x) x P 2 (2x2 − 1) 2n+1 = 2n+1 = n . (49) (α+ 1 ) (α,α) (α, 1 ) 2 P (1) 2 C2n+1 (1) 2n+1 Pn (1) See p.221, Remarks, last two formulas together with (43) and (60). Or see [DLMF, (18.7.13), (18.7.14)].

Diﬀerentiation formulas Each diﬀerentiation formula is given in two equivalent forms. d (1 − x)αP (α,β)(x) = −(n + α) (1 − x)α−1P (α−1,β+1)(x), dx n n (50) d (1 − x) − α P (α,β)(x) = −(n + α) P (α−1,β+1)(x). dx n n

d (1 + x)βP (α,β)(x) = (n + β) (1 + x)β−1P (α+1,β−1)(x), dx n n (51) d (1 + x) + β P (α,β)(x) = (n + β) P (α+1,β−1)(x). dx n n Formulas (50) and (51) follow from [DLMF, (15.5.4), (15.5.6)] together with (9.8.1). They also follow from each other by (42).

13 Generalized Gegenbauer polynomials These are deﬁned by

(α,β) (α,β) 2 (α,β) (α,β+1) 2 S2m (x) := const.Pm (2x − 1),S2m+1(x) := const. x Pm (2x − 1) (52)

(λ,µ) in the notation of [146, p.156] (see also [K5]), while [K9, Section 1.5.2] has Cn (x) = const. 1 1 (λ− 2 ,µ− 2 ) × Sn (x). For α, β > −1 we have the orthogonality relation

Z 1 (α,β) (α,β) 2β+1 2 α Sm (x) Sn (x) |x| (1 − x ) dx = 0 (m 6= n). (53) −1 For β = α − 1 generalized Gegenbauer polynomials are limit cases of continuous q-ultraspherical polynomials, see (175). If we deﬁne the Dunkl operator Tµ by

f(x) − f(−x) (T f)(x) := f 0(x) + µ (54) µ x and if we choose the constants in (52) as

(α,β) (α + β + 1)m (α,β) 2 (α,β) (α + β + 1)m+1 (α,β+1) 2 S2m (x) = Pm (2x − 1),S2m+1(x) = x Pm (2x − 1) (β + 1)m (β + 1)m+1 (55) then (see [K6, (1.6)]) (α,β) (α+1,β) T 1 Sn = 2(α + β + 1) S . (56) β+ 2 n−1 Formula (56) with (55) substituted gives rise to two diﬀerentiation formulas involving Jacobi polynomials which are equivalent to (9.8.7) and (51). Composition of (56) with itself gives

2 (α,β) (α+2,β) T 1 Sn = 4(α + β + 1)(α + β + 2) Sn−2 , β+ 2 which is equivalent to the composition of (9.8.7) and (51):

d2 2β + 1 d + P (α,β)(2x2 − 1) = 4(n + α + β + 1)(n + β) P (α+2,β)(2x2 − 1). (57) dx2 x dx n n−1

Formula (57) was also given in [332, (2.4)].

9.8.1 Gegenbauer / Ultraspherical

λ (λ) Notation Here the Gegenbauer polynomial is denoted by Cn instead of Cn .

Orthogonality relation Write the right-hand side of (9.8.20) as hn δm,n. Then

1 2 1 hn λ (2λ)n π Γ(λ + 2 ) hn λ n! = , h0 = , λ 2 = . (58) h0 λ + n n! Γ(λ + 1) h0 (Cn (1)) λ + n (2λ)n

14 Hypergeometric representation Beside (9.8.19) we have also

bn/2c ` 1 1 1 X (−1) (λ)n−` (λ)n − n, − n + 1 Cλ(x) = (2x)n−2` = (2x)n F 2 2 2 ; . (59) n `!(n − 2`)! n! 2 1 1 − λ − n x2 `=0

See [DLMF, (18.5.10)].

Special value (2λ) Cλ(1) = n . (60) n n! Use (9.8.19) or see [DLMF, Table 18.6.1].

Expression in terms of Jacobi

(λ− 1 ,λ− 1 ) λ 2 2 1 1 Cn (x) Pn (x) λ (2λ)n (λ− 2 ,λ− 2 ) = ,C (x) = Pn (x). (61) λ (λ− 1 ,λ− 1 ) n 1 Cn (1) 2 2 (λ + )n Pn (1) 2

Re: (9.8.21) By iteration of recurrence relation (9.8.21):

(n + 1)(n + 2) n2 + 2nλ + λ − 1 x2Cλ(x) = Cλ (x) + Cλ(x) n 4(n + λ)(n + λ + 1) n+2 2(n + λ − 1)(n + λ + 1) n (n + 2λ − 1)(n + 2λ − 2) + Cλ (x). (62) 4(n + λ)(n + λ − 1) n−2

Bilinear generating functions

∞ 1 1 2 2 2 ! X n! n λ λ 1 2 λ, 2 (λ + 1) 4r (1 − x )(1 − y ) r C (x) C (y) = 2F1 ; (2λ) n n (1 − 2rxy + r2)λ λ + 1 (1 − 2rxy + r2)2 n=0 n 2 (r ∈ (−1, 1), x, y ∈ [−1, 1]). (63)

For the proof put β := α in (46), then use (5) and (61). The Poisson kernel for Gegenbauer polynomials can be derived in a similar way from (47), or alternatively by applying the operator −λ+1 d λ r dr ◦ r to both sides of (63):

∞ X λ + n n! 1 − r2 rn Cλ(x) Cλ(y) = λ (2λ) n n (1 − 2rxy + r2)λ+1 n=0 n 1 1 2 2 2 ! 2 (λ + 1), 2 (λ + 2) 4r (1 − x )(1 − y ) × 2F1 1 ; 2 2 (r ∈ (−1, 1), x, y ∈ [−1, 1]). (64) λ + 2 (1 − 2rxy + r )

Formula (64) was obtained by Gasper & Rahman [234, (4.4)] as a limit case of their formula for the Poisson kernel for continuous q-ultraspherical polynomials.

15 Trigonometric expansions By [DLMF, (18.5.11), (15.8.1)]: n X (λ)k(λ)n−k (λ)n −n, λ Cλ(cos θ) = ei(n−2k)θ = einθ F ; e−2iθ (65) n k!(n − k)! n! 2 1 1 − λ − n k=0 −iθ (λ)n − 1 iλπ i(n+λ)θ −λ λ, 1 − λ ie = e 2 e (sin θ) F ; (66) 2λn! 2 1 1 − λ − n 2 sin θ ∞ 1 (λ)n X (λ)k(1 − λ)k cos((n − k + λ)θ + 2 (k − λ)π) = k+λ . (67) n! (1 − λ − n)kk! (2 sin θ) k=0 1 5 1 In (66) and (67) we require that 6 π < θ < 6 π. Then the convergence is absolute for λ > 2 and 1 conditional for 0 < λ ≤ 2 . By [DLMF, (14.13.1), (14.3.21), (15.8.1)]]:

1 ∞ λ 2Γ(λ + 2 ) (2λ)n 1−2λ X (1 − λ)k(n + 1)k Cn (cos θ) = 1 (sin θ) sin (2k + n + 1)θ (68) 2 (λ + 1)n (n + λ + 1)kk! π Γ(λ + 1) k=0 1 2Γ(λ + 2 ) (2λ)n 1−2λ i(n+1)θ 1 − λ, n + 1 2iθ = 1 (sin θ) Im e 2F1 ; e π 2 Γ(λ + 1) (λ + 1)n n + λ + 1 λ 1 iθ 2 Γ(λ + ) (2λ)n 1 λ, 1 − λ e 2 −λ − 2 iλπ i(n+λ)θ = 1 (sin θ) Re e e 2F1 ; π 2 Γ(λ + 1) (λ + 1)n 1 + λ + n 2i sin θ 2λ 1 ∞ 1 2 Γ(λ + 2 ) (2λ)n X (λ)k(1 − λ)k cos((n + k + λ)θ − 2 (k + λ)π) = 1 k+λ . (69) 2 (λ + 1)n (1 + λ + n)kk! (2 sin θ) π Γ(λ + 1) k=0 1 5 We require that 0 < θ < π in (68) and 6 π < θ < 6 π in (69) The convergence is absolute for 1 1 λ > 2 and conditional for 0 < λ ≤ 2 . For λ ∈ Z>0 the above series terminate after the term with k = λ − 1. Formulas (68) and (69) are also given in [Sz, (4.9.22), (4.9.25)].

Fourier transform 1 λ Γ(λ + 1) Z C (y) 1 n 2 λ− 2 ixy n λ −λ 1 1 λ (1 − y ) e dy = i 2 Γ(λ + 1) x Jλ+n(x). (70) Γ(λ + 2 ) Γ( 2 ) −1 Cn (1) See [DLMF, (18.17.17) and (18.17.18)].

Laplace transforms Z ∞ 2 n+2λ−1 −t2 λ Hn(tx) t e dt = Cn (x). (71) n! Γ(λ) 0 See Nielsen [K26, p.48, (4) with p.47, (1) and p.28, (10)] (1918) or Feldheim [K10, (28)] (1942).

1 λ 2 Z C (t) 1 2 2 2 n 2 λ− 2 −1 n+2λ+1 −x /t −n −x 1 1 λ (1 − t ) t (x/t) e dt = 2 Hn(x) e (λ > − 2 ). (72) Γ(λ + 2 ) 0 Cn (1) 1 Use Askey & Fitch [K2, (3.29)] for α = ± 2 together with (42), (48), (49), (97) and (98).

16 Addition formula (see [AAR, (9.8.50)]])

n k 1 1 X (−1) (−n)k (n + 2α + 1)k (α,α) 2 2 2 2 Rn xy + (1 − x ) (1 − y ) t = 2k 2 2 ((α + 1)k) k=0 1 1 1 1 2 k/2 (α+k,α+k) 2 k/2 (α+k,α+k) (α− 2 ,α− 2 ) (α− 2 ,α− 2 ) × (1 − x ) Rn−k (x) (1 − y ) Rn−k (y) ωk Rk (t), (73) where R 1 α β (α,β) (α,β) (α,β) (α,β) −1(1 − x) (1 + x) dx Rn (x) := Pn (x)/Pn (1), ωn := . R 1 (α,β) 2 α β −1(Rn (x)) (1 − x) (1 + x) dx

9.8.2 Chebyshev

In addition to the Chebyshev polynomials Tn of the ﬁrst kind (9.8.35) and Un of the second kind (9.8.36),

1 1 (− 2 ,− 2 ) Pn (x) Tn(x) := 1 1 = cos(nθ), x = cos θ, (74) (− 2 ,− 2 ) Pn (1) 1 1 ( 2 , 2 ) Pn (x) sin((n + 1)θ) Un(x) := (n + 1) 1 1 = , x = cos θ, (75) ( 2 , 2 ) sin θ Pn (1) we have Chebyshev polynomials Vn of the third kind and Wn of the fourth kind,

1 1 (− 2 , 2 ) 1 Pn (x) cos((n + 2 )θ) Vn(x) := 1 1 = 1 , x = cos θ, (76) (− 2 , 2 ) cos( θ) Pn (1) 2 1 1 ( 2 ,− 2 ) 1 Pn (x) sin((n + 2 )θ) Wn(x) := (2n + 1) 1 1 = 1 , x = cos θ, (77) ( 2 ,− 2 ) sin( θ) Pn (1) 2 see [K23, Section 1.2.3]. Then there is the symmetry

n Vn(−x) = (−1) Wn(x). (78)

The names of Chebyshev polynomials of the third and fourth kind and the notation Vn(x) are due to Gautschi [K11]. The notation Wn(x) was ﬁrst used by Mason [K22]. Names and notations for Chebyshev polynomials of the third and fourth kind are interchanged in [AAR, Remark 2.5.3] and [DLMF, Table 18.3.1].

9.9 Pseudo Jacobi (or Romanovski-Routh)

In this section in [KLS] the pseudo Jacobi polynomial Pn(x; ν, N) in (9.9.1) is considered for 1 N ∈ Z≥0 and n = 0, 1, . . . , n. However, we can more generally take − 2 < N ∈ R (so here I overrule my convention formulated in the beginning of this paper), N0 integer such that 1 1 N − 2 ≤ N0 < N + 2 , and n = 0, 1,...,N0 (see [382, §5, case A.4]). The orthogonality relation (9.9.2) is valid for m, n = 0, 1,...,N0.

17 History These polynomials were ﬁrst observed by Routh [K29] in 1885, but not as orthogonal polynomials (see Natanson [K25] about the history). Romanovski [463] (see also Lesky [382]) independently obtained them in 1929 as orthogonal polynomials.

Limit relation: Pseudo big q-Jacobi −→ Pseudo Jacobi See also (158).

References See also [Ism, §20.1], [51], [384], [K17], [K21], [K27].

9.10 Meixner History In 1934 Meixner [406] (see (1.1) and case IV on pp. 10, 11 and 12) gave the orthogonality measure for the polynomials Pn given by the generating function

∞ X tn exu(t) f(t) = P (x) , n n! n=0 where

1 k2 α−β β(α−β) u(t) 1 − βt (1 − βt) e = , f(t) = (k2 < 0; α > β > 0 or α < β < 0). 1 − αt k2 (1 − αt) α(α−β)

Then Pn can be expressed as a Meixner polynomial:

x + k α−1 P (x) = (−k (αβ)−1) βn M − 2 , −k (αβ)−1, βα−1 . n 2 n n α − β 2

In 1938 Gottlieb [K15, §2] introduces polynomials ln “of Laguerre type” which turn out to −nλ −λ be special Meixner polynomials: ln(x) = e Mn(x; 1, e ).

Uniqueness of orthogonality measure The coeﬃcient of pn−1(x) in (9.10.4) behaves as O(n2) as n → ∞. Hence (2) holds, by which the orthogonality measure is unique.

9.11 Krawtchouk Special values By (9.11.1) and the binomial formula:

−1 n Kn(0; p, N) = 1,Kn(N; p, N) = (1 − p ) . (79)

The self-duality (p.240, Remarks, ﬁrst formula)

Kn(x; p, N) = Kx(n; p, N)(n, x ∈ {0, 1,...,N}) (80) combined with (79) yields:

−1 x KN (x; p, N) = (1 − p ) (x ∈ {0, 1,...,N}). (81)

18 Symmetry By the orthogonality relation (9.11.2):

Kn(N − x; p, N) = Kn(x; 1 − p, N). (82) Kn(N; p, N) By (82) and (80) we have also

KN−n(x; p, N) = Kn(x; 1 − p, N)(n, x ∈ {0, 1,...,N}), (83) KN (x; p, N) and, by (83), (82) and (79), p n+x−N K (N − x; p, N) = K (x; p, N)(n, x ∈ {0, 1,...,N}). (84) N−n p − 1 n A particular case of (82) is: 1 n 1 Kn(N − x; 2 ,N) = (−1) Kn(x; 2 ,N). (85) Hence 1 K2m+1(N; 2 , 2N) = 0. (86) From (9.11.11): 1 1 ( 2 )m K2m(N; 2 , 2N) = 1 . (87) (−N + 2 )m

Quadratic transformations 1 1 ( 2 )m 2 1 1 K2m(x + N; 2 , 2N) = 1 Rm(x ; − 2 , − 2 ,N), (88) (−N + 2 )m 3 1 ( 2 )m 2 1 1 K2m+1(x + N; 2 , 2N) = − 1 x Rm(x − 1; 2 , 2 ,N − 1), (89) N (−N + 2 )m 1 1 ( 2 )m 1 1 K2m(x + N + 1; 2 , 2N + 1) = 1 Rm(x(x + 1); − 2 , 2 ,N), (90) (−N − 2 )m 3 1 ( 2 )m 1 1 1 K2m+1(x + N + 1; 2 , 2N + 1) = 1 (x + 2 ) Rm(x(x + 1); 2 , − 2 ,N), (91) (−N − 2 )m+1 where Rm is a dual Hahn polynomial (9.6.1). For the proofs use (9.6.2), (9.11.2), (9.6.4) and (9.11.4).

Generating functions

N X N K (x; p, N)K (x; q, N)zx x m n x=0 p − z + pz m q − z + qz n (p − z + pz)(q − z + qz) = (1 + z)N−m−nK n; − ,N . p q m z (92) This follows immediately from Rosengren [K28, (3.5)], which goes back to Meixner [K24].

19 9.12 Laguerre

α (α) Notation Here the Laguerre polynomial is denoted by Ln instead of Ln .

Hypergeometric representation (α + 1) −n Lα(x) = n F ; x (93) n n! 1 1 α + 1 (−x)n −n, −n − α 1 = F ; − (94) n! 2 0 − x (−x)n = C (n + α; x), (95) n! n where Cn in (95) is a Charlier polynomial. Formula (93) is (9.12.1). Then (94) follows by reversal of summation. Finally (95) follows by (94) and (107). It is also the remark on top of p.244 in [KLS], and it is essentially [416, (2.7.10)].

Uniqueness of orthogonality measure The coeﬃcient of pn−1(x) in (9.12.4) behaves as O(n2) as n → ∞. Hence (2) holds, by which the orthogonality measure is unique.

Special value (α + 1) Lα(0) = n . (96) n n! Use (9.12.1) or see [DLMF, 18.6.1)].

Quadratic transformations

n 2n −1/2 2 H2n(x) = (−1) 2 n! Ln (x ), (97) n 2n+1 1/2 2 H2n+1(x) = (−1) 2 n! x Ln (x ). (98) See p.244, Remarks, last two formulas. Or see [DLMF, (18.7.19), (18.7.20)].

Fourier transform Z ∞ α n 1 Ln(y) −y α ixy n y α e y e dy = i n+α+1 , (99) Γ(α + 1) 0 Ln(0) (iy + 1) see [DLMF, (18.17.34)].

Diﬀerentiation formulas Each diﬀerentiation formula is given in two equivalent forms. d d (xαLα(x)) = (n + α) xα−1Lα−1(x), x + α Lα(x) = (n + α) Lα−1(x). (100) dx n n dx n n

d d e−xLα(x) = −e−xLα+1(x), − 1 Lα(x) = −Lα+1(x). (101) dx n n dx n n Formulas (100) and (101) follow from [DLMF, (13.3.18), (13.3.20)] together with (9.12.1).

20 Generating functions The generating function (9.12.12) is a limit case of the generating function (44) for Jacobi polynomials by (9.8.16). By (9.7.14) the generating function (9.12.12) is also a limit case of the generating function (9.7.13) for Meixner–Pollaczek polynomials.

Generalized Hermite polynomials See [146, p.156], [K9, Section 1.5.1]. These are deﬁned by 1 1 µ µ− 2 2 µ µ+ 2 2 H2m(x) := const.Lm (x ),H2m+1(x) := const. x Lm (x ). (102) 1 Then for µ > − 2 we have orthogonality relation Z ∞ µ µ 2µ −x2 Hm(x) Hn (x) |x| e dx = 0 (m 6= n). (103) −∞

Let the Dunkl operator Tµ be deﬁned by (54). If we choose the constants in (102) as

m 1 m 1 µ (−1) (2m)! µ− 2 2 µ (−1) (2m + 1)! µ+ 2 2 H2m(x) = 1 Lm (x ),H2m+1(x) = 1 x Lm (x ) (104) (µ + 2 )m (µ + 2 )m+1 then (see [K6, (1.6)]) µ µ TµHn = 2n Hn−1. (105) Formula (105) with (104) substituted gives rise to two diﬀerentiation formulas involving Laguerre polynomials which are equivalent to (9.12.6) and (100). Composition of (105) with itself gives

2 µ µ Tµ Hn = 4n(n − 1) Hn−2, which is equivalent to the composition of (9.12.6) and (100):

d2 2α + 1 d + Lα(x2) = −4(n + α) Lα (x2). (106) dx2 x dx n n−1

9.13 Bessel Hypergeometric representation The constraint n = 0, 1, 2,...,N can be omitted. All formulas in §9.13 except (9.13.2) remain valid for all integer n ≥ 0. These more general values of n are even needed in the generating function (9.13.10).

Notation In the notation of Grosswald [255] the left-hand side of (9.13.1) has to be replaced by yn(x; a + 2).

Orthogonality relation Replace the constraint a < −2N − 1 in (9.13.2) by m, n = 0, 1,...,N = d−(3 + a)/2e. Following Lesky [382] the Bessel polynomials in case of orthogonality relation (9.13.2) may be called Romanovski–Bessel polynomials.

21 9.14 Charlier Hypergeometric representation

−n, −x 1 C (x; a) = F ; − (107) n 2 0 − a (−x) −n = n F ; a (108) an 1 1 x − n + 1 n! = Lx−n(a), (109) (−a)n n

α where Ln(x) is a Laguerre polynomial. Formula (107) is (9.14.1). Then (108) follows by reversal of the summation. Finally (109) follows by (108) and (9.12.1). It is also the Remark on p.249 of [KLS], and it was earlier given in [416, (2.7.10)].

Uniqueness of orthogonality measure The coeﬃcient of pn−1(x) in (9.14.4) behaves as O(n) as n → ∞. Hence (2) holds, by which the orthogonality measure is unique.

9.15 Hermite

Uniqueness of orthogonality measure The coeﬃcient of pn−1(x) in (9.15.4) behaves as O(n) as n → ∞. Hence (2) holds, by which the orthogonality measure is unique.

Fourier transforms Z ∞ 1 − 1 y2 ixy n − 1 x2 √ Hn(y) e 2 e dy = i Hn(x) e 2 , (110) 2π −∞ see [AAR, (6.1.15) and Exercise 6.11]. Z ∞ 1 −y2 ixy n n − 1 x2 √ Hn(y) e e dy = i x e 4 , (111) π −∞ see [DLMF, (18.17.35)].

n Z ∞ i n − 1 y2 −ixy −x2 √ y e 4 e dy = Hn(x) e , (112) 2 π −∞ see [AAR, (6.1.4)].

14.1 Askey–Wilson

Symmetry The Askey–Wilson polynomials pn(x; a, b, c, d | q) are symmetric in a, b, c, d. This follows from the orthogonality relation (14.1.2) together with the value of its coeﬃcient of xn given in (14.1.5b). Alternatively, combine (14.1.1) with [GR, (III.15)]. As a consequence, it is suﬃcient to give generating function (14.1.13). Then the generating functions (14.1.14), (14.1.15) will follow by symmetry in the parameters.

22 Basic hypergeometric representation In addition to (14.1.1) we have (in notation (8)):

−iθ −iθ −iθ −iθ (ae , be , ce , de ; q)n inθ pn(cos θ; a, b, c, d | q) = −2iθ e (e ; q)n −n 2iθ iθ iθ iθ iθ −n 2−n × 8W7 q e ; ae , be , ce , de , q ; q, q /(abcd) . (113)

This follows from (14.1.1) by combining (III.15) and (III.19) in [GR]. It is also given in [513, (4.2)], but be aware for some slight errors. The symmetry in a, b, c, d is evident from (113).

Special value and diﬀerent notation

1 −1 −n pn 2 (a + a ); a, b, c, d | q = a (ab, ac, ad; q)n , (114)

1 −1 1 −1 1 −1 and similarly for arguments 2 (b + b ), 2 (c + c ) and 2 (d + d ) by symmetry of pn in a, b, c, d. Formula (114) is an immediate consequence of (14.1.1). We will also write

1 −1 −n n−1 −1 pn( (z + z ); a, b, c, d | q) q , q abcd, az, az R (z; a, b, c, d | q) := 2 = φ ; q, q . (115) n 1 −1 4 3 pn( 2 (a + a ); a, b, c, d | q) ab, ac, ad Here there is no longer full symmetry in a, b, c, d, only in b, c, d.

Trivial symmetry From (14.1.1) we see [72, (1.34)]

n pn(x; a, b, c, d | q) = (−1) pn(−x; −a, −b, −c, −d | q), (116) Rn(z; a, b, c, d | q) = Rn(−z; −a, −b, −c, −d | q).

Duality Deﬁne parametersa, ˜ ˜b, c,˜ d˜ in terms of a, b, c, d by

−1 1 a˜ = (q abcd) 2 , ˜b = ab/a,˜ c˜ = ac/a,˜ d˜= ad/a.˜ (117)

Jumping from one branch to the other branch in the square root in the formula fora ˜ implies thata, ˜ ˜b, c,˜ d˜ move to −a,˜ −˜b, −c,˜ −d˜. Repetition of the parameter transformation recovers the original parameters up to a possible common multiplication of a, b, c, d by −1, while the branch choice fora ˜ is irrelevant:

1 a = q−1a˜˜bc˜d˜ 2 , b =a ˜˜b/a, c =a ˜c/a,˜ d =a ˜d/a.˜ (118)

From (115) we have the duality relation

m n ˜ ˜ Rn aq ; a, b, c, d | q = Rm aq˜ ;a, ˜ b, c,˜ d | q (m, n ∈ Z≥0). (119)

By (116) both sides of (119) are invariant under common multiplication by −1 of a, b, c, d, respectivelya, ˜ ˜b, c,˜ d˜.

23 Orthogonality relation The conditions on the parameters in (14.1.2) can be slightly relaxed: Let |a|, |b|, |c|, |d| ≤ 1 such that pairwise products of a, b, c, d are not equal to 1 and such that non-real parameters occur in complex conjugate pairs. In fact, the only possible cases which then oﬀend the condition |a|, |b|, |c|, |d| < 1 are that either precisely one parameter has absolute value 1 and equals 1 or −1, or precisely two parameter values have absolute value 1, one equal to 1 and the other equal to −1. Then the weight fucntion will not cause a singularity by its factors 1 ± eiθ and 1 ± e−iθ in the denominator, since these are compensated by the factors 1 − e2iθ and 1 − e−2iθ in the numerator. The orthogonality (14.1.3) involving discrete terms can be given for more general parameter values as in [72, Theorem 2.5]. There a, b, c, d are real or occur in complex conjugate pairs if non-real, and pairwise products have absolute value ≤ 1 but are not equal to 1.

Re: (14.1.5) Let

pn(x; a, b, c, d | q) n n−1 pn(x) := n n−1 = x + eknx + ··· . (120) 2 (abcdq ; q)n Then (1 − qn)(a + b + c + d − (abc + abd + acd + bcd)qn−1) ekn = − . (121) 2(1 − q)(1 − abcdq2n−2) ˜ ˜ 1 −1 This follows because kn −kn+1 equals the coefficient 2 a+a −(An +Cn) of pn(x) in (14.1.5). q-Difference equation The q-difference operator acting on Pn(z) on the right-hand side of −1 (14.1.7), gives, when acting on Qn(z) := (az, az ; q)∞, the result −n n n−1 −n n−1 n−1 n−1 n q (1 − q )(1 − abcdq )Qn(z) − q (1 − abq )(1 − acq )(1 − adq )(1 − q )Qn−1(z) −1 −1 −1 = A(z)Qn(qz) − A(z) + A(z ) Qn(z) + A(z )Qn(q z). (122) This formula is implicit in [K32]. Use there (3.1) with the Askey–Wilson parameters (7.15) and (7.8), and combine it with (14.1.7).

Generating functions Rahman [449, (4.1), (4.9)] gives: ∞ −1 n X (abcdq ; q)na tn p (cos θ; a, b, c, d | q) (ab, ac, ad, q; q) n n=0 n −1 −1 1 −1 1 1 1 iθ −iθ ! (abcdtq ; q)∞ (abcdq ) 2 , −(abcdq ) 2 , (abcd) 2 , −(abcd) 2 , ae , ae = 6φ5 −1 −1 ; q, q (t; q)∞ ab, ac, ad, abcdtq , qt −1 iθ −iθ (abcdq , abt, act, adt, ae , ae ; q)∞ + −1 iθ −iθ (ab, ac, ad, t , ate , ate ; q)∞ −1 1 −1 1 1 1 iθ −iθ ! t(abcdq ) 2 , −t(abcdq ) 2 , t(abcd) 2 , −t(abcd) 2 , ate , ate × φ ; q, q (|t| < 1). (123) 6 5 abt, act, adt, abcdt2q−1, qt In the limit (124) the ﬁrst term on the right-hand side of (123) tends to the left-hand side of (9.1.15), while the second term tends formally to 0. The special case ad = bc of (123) was earlier given in [236, (4.1), (4.6)].

24 Limit relations Askey–Wilson −→ Wilson Instead of (14.1.21) we can keep a polynomial of degree n while the limit is approached:

1 2 a b c d pn(1 − 2 x(1 − q) ; q , q , q , q | q) lim = Wn(x; a, b, c, d). (124) q→1 (1 − q)3n

For the proof ﬁrst derive the corresponding limit for the monic polynomials by comparing (14.1.5) with (9.4.4).

Askey–Wilson −→ Continuous Hahn Instead of (14.4.15) we can keep a polynomial of degree n while the limit is approached:

a iφ b iφ a −iφ b −iφ pn cos φ − x(1 − q) sin φ; q e , q e , q e , q e | q lim q↑1 (1 − q)2n n = (−2 sin φ) n! pn(x; a, b, a, b) (0 < φ < π). (125)

Here the right-hand side has a continuous Hahn polynomial (9.4.1). For the proof ﬁrst derive the corresponding limit for the monic polynomials by comparing (14.1.5) with (9.1.5). In fact, deﬁne the monic polynomial

a iφ b iφ a −iφ b −iφ pn cos φ − x(1 − q) sin φ; q e , q e , q e , q e | q p (x) := . en n n−1 (−2(1 − q) sin φ) (abcdq ; q)n

Then it follows from (14.1.5) that

a iφ −a −iφ (1 − q )e + (1 − q )e + Aen + Cen Aen−1Cen x pn(x) = pn+1(x) + pn(x) + pn−1(x), e e 2(1 − q) sin φ e (1 − q)2 sin2 φ e

a iφ b iφ a −iφ b −iφ where Aen and Cen are as given after (14.1.3) with a, b, c, d replaced by q e , q e , q e , q e . Then the recurrence equation for pen(x) tends for q ↑ 1 to the recurrence equation (9.4.4) with c = a, d = b.

Askey–Wilson −→ Meixner–Pollaczek Instead of (14.9.15) we can keep a polynomial of degree n while the limit is approached:

λ iφ λ −iφ pn cos φ − x(1 − q) sin φ; q e , 0, q e , 0 | q (λ) lim = n! Pn (x; π − φ) (0 < φ < π). (126) q↑1 (1 − q)n

Here the right-hand side has a Meixner–Pollaczek polynomial (9.7.1). For the proof ﬁrst derive the corresponding limit for the monic polynomials by comparing (14.1.5) with (9.7.4). In fact, deﬁne the monic polynomial

λ iφ λ −iφ pn cos φ − x(1 − q) sin φ; q e , 0, q e , 0 | q p (x) := . en (−2(1 − q) sin φ)n

25 Then it follows from (14.1.5) that

λ iφ −λ −iφ (1 − q )e + (1 − q )e + Aen + Cen Aen−1Cen x pn(x) = pn+1(x) + pn(x) + pn−1(x), e e 2(1 − q) sin φ e (1 − q)2 sin2 φ e

λ iφ λ −iφ where Aen and Cen are as given after (14.1.3) with a, b, c, d replaced by q e , 0, q e , 0. Then the recurrence equation for pen(x) tends for q ↑ 1 to the recurrence equation (9.7.4).

References See also Koornwinder [K18].

14.2 q-Racah Symmetry

−1 −N−1 (βq, αδ q; q)n n −1 −N−1 −1 Rn(x; α, β, q , δ | q) = δ Rn(δ x; β, α, q , δ | q). (127) (αq, βδq; q)n

This follows from (14.2.1) combined with [GR, (III.15)]. In particular,

−N−1 (βq, −αq; q)n n −N−1 Rn(x; α, β, q , −1 | q) = (−1) Rn(−x; β, α, q , −1 | q), (128) (αq, −βq; q)n and −N−1 n −N−1 Rn(x; α, α, q , −1 | q) = (−1) Rn(−x; α, α, q , −1 | q), (129)

Trivial symmetry Clearly from (14.2.1):

−1 −1 −1 Rn(x; α, β, γ, δ | q) = Rn(x; βδ, αδ , γ, δ | q) = Rn(x; γ, αβγ , α, γδα | q). (130)

For α = q−N−1 this shows that the three cases αq = q−N or βδq = q−N or γq = q−N of (14.2.1) are not essentially diﬀerent.

Duality It follows from (14.2.1) that

−y y+1 −N−1 −n n−N −N−1 Rn(q + γδq ; q , β, γ, δ | q) = Ry(q + βq ; γ, δ, q , β | q)(n, y = 0, 1,...,N). (131)

14.3 Continuous dual q-Hahn The continuous dual q-Hahn polynomials are the special case d = 0 of the Askey–Wilson polynomials: pn(x; a, b, c | q) := pn(x; a, b, c, 0 | q). Hence all formulas in §14.3 are specializations for d = 0 of formulas in §14.1.

26 14.4 Continuous q-Hahn The continuous q-Hahn polynomials are the special case of Askey–Wilson polynomials with parameters aeiφ, beiφ, ae−iφ, be−iφ: iφ iφ −iφ −iφ pn(x; a, b, φ | q) := pn(x; ae , be , ae , be | q).

In [72, (4.29)] and [GR, (7.5.43)] (who write pn(x; a, b | q), x = cos(θ + φ)) and in [KLS, §14.4] (who writes pn(x; a, b, c, d; q), x = cos(θ + φ)) the parameter dependence on φ is incorrectly omitted. Since all formulas in §14.4 are specializations of formulas in §14.1, there is no real need to give these specializations explicitly. In particular, the limit (14.4.15) is in fact a limit from Askey–Wilson to continuous Hahn. See also (125).

14.5 Big q-Jacobi Diﬀerent notation See p.442, Remarks: q−n, qn+1ab, qac−1x P (x; a, b, c, d; q) := P (qac−1x; a, b, −ac−1d; q) = φ ; q, q . (132) n n 3 2 qa, −qac−1d Furthermore, Pn(x; a, b, c, d; q) = Pn(λx; a, b, λc, λd; q), (133)

−1 −1 −1 Pn(x; a, b, c; q) = Pn(−q c x; a, b, −ac , 1; q) (134)

Orthogonality relation (equivalent to (14.5.2), see also [K19, (2.42), (2.41), (2.36), (2.35)]). Let c, d > 0 and either a ∈ (−c/(qd), 1/q), b ∈ (−d/(cq), 1/q) or a/c = −b/d∈ / R. Then Z c (qx/c, −qx/d; q)∞ Pm(x; a, b, c, d; q)Pn(x; a, b, c, d; q) dqx = hn δm,n , (135) −d (qax/c, −qbx/d; q)∞ where 2 2 n hn 1 q a d 1 − qab (q, qb, −qbc/d; q)n 2 n(n−1) = q 2n+1 (136) h0 c 1 − q ab (qa, qab, −qad/c; q)n and 2 (q, −d/c, −qc/d, q ab; q)∞ h0 = (1 − q)c . (137) (qa, qb, −qbc/d, −qad/c; q)∞

Other hypergeometric representation and asymptotics −1 −n −n −1 −1 (−qbd x; q)n q , q b , cx Pn(x; a, b, c, d; q) = −n −1 −1 3φ2 −n −1 −1 ; q, q (138) (−q a cd ; q)n qa, −q b dx −1 −n −n −1 −1 −1 n (qb, cx ; q)n q , q a , −qbd x n+1 −1 = (qac x) −1 3φ2 1−n −1 ; q, −q ac d (139) (qa, −qac d; q)n qb, q c x n −1 −1 (qb, q; q)n X (cx ; q)n−k (−qbd x; q)k 1 −1 n k 2 k(k−1) −1 k = (qac x) −1 (−1) q (−dx ) . (−qac d; q)n (q, qa; q)n−k (qb, q; q)k k=0 (140)

27 Formula (138) follows from (132) by [GR, (III.11)] and next (139) follows by series inversion [GR, Exercise 1.4(ii)]. Formulas (138) and (140) are also given in [Ism, (18.4.28), (18.4.29)]. It follows from (139) or (140) that (see [298, (1.17)] or [Ism, (18.4.31)])

−1 −1 −1 −n (cx , −dx ; q)∞ lim (qac x) Pn(x; a, b, c, d; q) = −1 , (141) n→∞ (−qac d, qa; q)∞

m m uniformly for x in compact subsets of C\{0}. (Exclusion of the spectral points x = cq , dq (m = 0, 1, 2,...), as was done in [298] and [Ism], is not necessary. However, while (141) yields 0 at these points, a more reﬁned asymptotics at these points is given in [298] and [Ism].) For the proof of (141) use that

−1 −1 −1 −n (qb, cx ; q)n −qbd x −1 lim (qac x) Pn(x; a, b, c, d; q) = −1 1φ1 ; q, −dx , (142) n→∞ (qa, −qac d; q)n qb which can be evaluated by [GR, (II.5)]. Formula (142) follows formally from (139), and it follows rigorously, by dominated convergence, from (140).

Symmetry (see [K19, §2.5]).

Pn(−x; a, b, c, d; q) = Pn(x; b, a, d, c; q). (143) Pn(−d/(qb); a, b, c, d; q)

Special values

Pn(c/(qa); a, b, c, d; q) = 1, (144) n ad (qb, −qbc/d; q)n Pn(−d/(qb); a, b, c, d; q) = − , (145) bc (qa, −qad/c; q)n n 1 n(n+1) ad (−qbc/d; q)n Pn(c; a, b, c, d; q) = q 2 , (146) c (−qad/c; q)n

1 n(n+1) n (qb; q)n Pn(−d; a, b, c, d; q) = q 2 (−a) . (147) (qa; q)n

Quadratic transformations (see [K19, (2.48), (2.49)] and (178)). These express big q-Jacobi polynomials Pm(x; a, a, 1, 1; q) in terms of little q-Jacobi polynomials (see §14.12).

2 −1 2 2 pn(x ; q , a ; q ) P2n(x; a, a, 1, 1; q) = −2 −1 2 2 , (148) pn((qa) ; q , a ; q ) 2 2 2 qax pn(x ; q, a ; q ) P2n+1(x; a, a, 1, 1; q) = −2 2 2 . (149) pn((qa) ; q, a ; q )

28 Hence, by (14.12.1), [GR, Exercise 1.4(ii)] and (178),

2 2 −n −n+1 (qa ; q )n n q , q 2 −2 Pn(x; a, a, 1, 1; q) = 2 (qax) 2φ1 −2n+1 −2 ; q , (ax) (150) (qa ; q)n q a [ 1 n] 2 2 2 (q; q)n n X k k(k−1) (qa ; q )n−k n−2k = 2 (qa) (−1) q 2 2 x . (151) (qa ; q)n (q ; q )k (q; q)n−2k k=0

− 1 1 q-Chebyshev polynomials In (132), with c = d = 1, the cases a = b = q 2 and a = b = q 2 can be considered as q-analogues of the Chebyshev polynomials of the ﬁrst and second kind, respectively (§9.8.2) because of the limit (14.5.17). The quadratic relations (148), (149) can also be specialized to these cases. The deﬁnition of the q-Chebyshev polynomials may vary by normalization and by dilation of argument. They were considered in [K4]. By [24, p.279] and (148), (149), the Al-Salam-Ismail polynomials Un(x; a, b)(q-dependence suppressed) in the case a = q can be expressed as q-Chebyshev polynomials of the second kind:

n+1 −3 1 n 1 − q − 1 1 1 U (x, q, b) = (q b) 2 P (b 2 x; q 2 , q 2 , 1, 1; q). n 1 − q n

Similarly, by [K7, (5.4), (5.1), (5.3)] and (148), (149), Cigler’s q-Chebyshev polynomials Tn(x, s, q) and Un(x, s, q) can be expressed in terms of the q-Chebyshev cases of (132):

1 n − 1 − 1 − 1 Tn(x, s, q) = (−s) 2 Pn((−qs) 2 x; q 2 , q 2 , 1, 1; q), n+1 −2 1 n 1 − q − 1 1 1 U (x, s, q) = (−q s) 2 P ((−qs) 2 x; q 2 , q 2 , 1, 1; q). n 1 − q n

Limit to Discrete q-Hermite I

−n n lim a Pn(x; a, a, 1, 1; q) = q hn(x; q). (152) a→0

Here hn(x; q) is given by (14.28.1). For the proof of (152) use (138).

Pseudo big q-Jacobi polynomials Let a, b, c, d ∈ , z > 0, z < 0 such that (ax,bx;q)∞ > 0 C + − (cx,dx;q)∞ for x ∈ z−qZ ∪ z+qZ. Then (ab)/(qcd) > 0. Assume that (ab)/(qcd) < 1. Let N be the largest nonnegative integer such that q2N > (ab)/(qcd). Then

Z (ax, bx; q)∞ Pm(cx; c/b, d/a, c/a; q) Pn(cx; c/b, d/a, c/a; q) dqx = hnδm,n z−qZ∪z+qZ (cx, dx; q)∞ (m, n = 0, 1,...,N), (153) where

2 n hn c 1 (q, qd/a, qd/b; q)n 1 − qcd/(ab) n 2 n(n−1) 2n = (−1) q q 2n+1 (154) h0 ab (qcd/(ab), qc/a, qc/b; q)n 1 − q cd/(ab)

29 and Z (ax, bx; q)∞ (q, a/c, a/d, b/c, b/d; q)∞ θ(z−/z+, cdz−z+; q) h0 = dqx = (1 − q)z+ . z−qZ∪z+qZ (cx, dx; q)∞ (ab/(qcd); q)∞ θ(cz−, dz−, cz+, dz+; q) (155) See Groenevelt & Koelink [K16, Prop. 2.2]. Formula (155) was ﬁrst given by Slater [K31, (5)] as an evaluation of a sum of two 2ψ2 series. The same formula is given in Slater [471, (7.2.6)] and in [GR, Exercise 5.10], but in both cases with the same slight error, see [K16, 2nd paragraph after Lemma 2.1] for correction. The theta function is given by (9). Note that

−1 Pn(cx; c/b, d/a, c/a; q) = Pn(−q ax; c/b, d/a, −a/b, 1; q). (156)

In [K14] the weights of the pseudo big q-Jacobi polynomials occur in certain measures on the space of N-point conﬁgurations on the so-called extended Gelfand-Tsetlin graph.

Limit relations Pseudo big q-Jacobi −→ Discrete Hermite II

n 1 n(n−1) −1 −1 lim i q 2 Pn(q a ix; a, a, 1, 1; q) = ehn(x; q). (157) a→∞

−1 −1 For the proof use (151) and (213). Note that Pn(q a ix; a, a, 1, 1; q) is obtained from the right-hand side of (156) by replacing a, b, c, d by −ia−1, ia−1, i, −i.

Pseudo big q-Jacobi −→ Pseudo Jacobi

1 (−N−1+iν) −N−1 −N−1 −N+iν−1 Pn(x; ν, N) lim Pn(iq 2 x; −q , −q , q ; q) = . (158) q↑1 Pn(−i; ν, N)

Here the big q-Jacobi polynomial on the left-hand side equals Pn(cx; c/b, d/a, c/a; q) with 1 (N+1−iν) 1 (N+1+iν) 1 (−N−1+iν) 1 (−N−1−iν) a = iq 2 , b = −iq 2 , c = iq 2 , d = −iq 2 .

14.7 Dual q-Hahn Orthogonality relation More generally we have (14.7.2) with positive weights in any of the following cases: (i) 0 < γq < 1, 0 < δq < 1; (ii) 0 < γq < 1, δ < 0; (iii) γ < 0, δ > q−N ; (iv) γ > q−N , δ > q−N ; (v) 0 < qγ < 1, δ = 0. This also follows by inspection of the positivity of the coeﬃcient of pn−1(x) in (14.7.4). Case (v) yields Aﬃne q-Krawtchouk in view of (14.7.13).

Symmetry

−1 −N (δ q ; q)n N+1n −1 −1 −1−N −1 −N−1 −1 −N−1 Rn(x; γ, δ, N | q) = γδq Rn(γ δ q x; δ q , γ q ,N | q). (γq; q)n (159) This follows from (14.7.1) combined with [GR, (III.11)].

30 14.8 Al-Salam–Chihara Standardization and notation The deﬁnition (14.8.1) by q-hypergeometric representation follows the convention of [72, p.25] that Qn(x; a, b | q) = pn(x; a, b, 0, 0 | q), where pn(x; a, b, c, d | q) is the Askey–Wilson polynomial (14.1.1). In [Ism, (15.1.6)] these polynomials are notated n pn(x; a, b | q), equal to a /(ab; q)n times Qn(x; a, b | q) as in (14.8.1).

Symmetry The Al-Salam–Chihara polynomials Qn(x; a, b | q) are symmetric in a, b. This follows from the orthogonality relation (14.8.2) together with the value of its coeﬃcient of xn given in (14.8.5b).

Orthogonality relation Just as in Section 14.1 the condition |a|, |b| < 1 on the parameters in (14.8.2) can be slightly relaxed into |a|, |b| ≤ 1, ab 6= 1. q−1-Al-Salam–Chihara

Re: (14.8.1) For x ∈ Z≥0: 1 1 −x −1 x −1 n n − 2 n(n−1) −1 Qn( 2 (aq + a q );a, b | q ) = (−1) b q (ab) ; q n q−n, q−x, a−2qx × φ ; q, qnab−1 (160) 3 1 (ab)−1 −1 −x −2 x 1 (qba ; q)x q , a q −1 x − 2 x(x+1) n+1 = (−ab ) q −1 −1 2φ1 −1 ; q, q (161) (a b ; q)x qba −1 1 (qba ; q)x −1 x − 2 x(x+1) n −1 −1 = (−ab ) q −1 −1 px(q ; ba , (qab) ; q). (162) (a b ; q)x Formula (160) follows from the ﬁrst identity in (14.8.1). Next (161) follows from [GR, (III.8)]. Finally (162) gives the little q-Jacobi polynomials (14.12.1). See also [79, §3].

Orthogonality

∞ 2x −2 −2 −1 X (1 − q a )(a , (ab) ; q)x 2 (ba−1)xqx (Q Q )( 1 (aq−x + a−1qx); a, b | q−1) (1 − a−2)(q, bqa−1; q) m n 2 x=0 x −2 (qa ; q)∞ −1 n −n2 = −1 (q, (ab) ; q)n (ab) q δm,n (ab > 1, qb < a). (163) (ba q; q)∞ This follows from (162) together with (14.12.2) and the completeness of the orthogonal system of the little q-Jacobi polynomials, See also [79, §3]. An alternative proof is given in [64]. There combine (3.82) with (3.81), (3.67), (3.40).

Normalized recurrence relation 1 −n 1 −n −n+1 xpn(x) = pn+1(x) + 2 (a + b)q pn(x) + 4 (q − 1)(abq − 1)pn−1(x), (164) where −1 n Qn(x; a, b | q ) = 2 pn(x).

31 14.9 q-Meixner–Pollaczek The q-Meixner–Pollaczek polynomials are the special case of Askey–Wilson polynomials with parameters aeiφ, 0, ae−iφ, 0:

1 iφ −iφ Pn(x; a, φ | q) := pn(x; ae , 0, ae , 0 | q)(x = cos(θ + φ)). (q; q)n

In [KLS, §14.9] the parameter dependence on φ is incorrectly omitted. Since all formulas in §14.9 are specializations of formulas in §14.1, there is no real need to give these specializations explicitly. See also (126). There is an error in [KLS, (14.9.6), (14.9.8)]. Read x = cos(θ + φ) instead of x = cos θ.

14.10 Continuous q-Jacobi Symmetry (α,β) n 1 (α−β)n (β,α) Pn (−x | q) = (−1) q 2 Pn (x | q). (165) This follows from (116) and (14.1.19).

14.10.1 Continuous q-ultraspherical / Rogers Re: (14.10.17)

1 1 1 1 ! 2 − 2 n 2 n 2 iθ 2 −iθ (β ; q)n − 1 n q , βq , β e , β e 1 1 Cn(cos θ; β | q) = β 2 4φ3 1 1 1 1 ; q 2 , q 2 , (166) (q; q)n −β, β 2 q 4 , −β 2 q 4 see [GR, (7.4.13), (7.4.14)].

Special value (see [63, (3.23)])

2 1 1 (β ; q)n 1 1 2 − 2 − 2 n Cn 2 (β + β ); β | q = β . (167) (q; q)n

iθ Re: (14.10.21) (another q-diﬀerence equation). Let Cn[e ; β | q] := Cn(cos θ; β | q).

2 −2 1 − βz 1 1 − βz − 1 − 1 n 1 n C [q 2 z; β | q] + C [q 2 z; β | q] = (q 2 + q 2 β) C [z; β | q], (168) 1 − z2 n 1 − z−2 n n see [351, (6.10)].

Re: (14.10.23) This can also be written as

1 − 1 − 1 n −1 Cn[q 2 z; β | q] − Cn[q 2 z; β | q] = q 2 (β − 1)(z − z )Cn−1[z; qβ | q]. (169)

32 Two other shift relations follow from the previous two equations:

Trigonometric representation (see p.473, Remarks, ﬁrst formula)

n X (β; q)k(β; q)n−k i(n−2k)θ Cn(cos θ; β | q) = e . (172) (q; q)k(q; q)n−k k=0

Limit for q ↓ −1 (see [63, pp. 74–75]). By (172) and (65) we obtain

1 1 λ 2 (λ+1) 2 2 (λ+1) 2 lim C2m(x; −q | − q) = Cm (2x − 1) + Cm−1 (2x − 1), q↑1 1 λ 2 (λ+1) 2 lim C2m+1(x; −q | − q) = 2x Cm (2x − 1). q↑1

By (61) and [HTF2, 10.6(36)] this can be rewritten as

1 1 λ (λ)m ( 2 λ, 2 λ−1) 2 lim C2m(x; −q | − q) = Pm (2x − 1), (173) q↑1 1 ( 2 λ)m 1 1 λ (λ + 1)m ( 2 λ, 2 λ) 2 lim C2m+1(x; −q | − q) = 2 x Pm (2x − 1). (174) q↑1 1 ( 2 λ + 1)m By (52) the limits (173), (174) imply that

1 1 λ ( 2 λ, 2 λ−1) lim Cn(x; −q | − q) = const.Sn (x), (175) q↑1 where the right-hand side gives a one-parameter subclass of the generalized Gegenbauer polynomial. Note that in [K13, Section 7.1] the generalized Gegenbauer polynomials are also observed as ﬁtting in the q = −1 Askey scheme, but the limit (175) is not observed there.

14.11 Big q-Laguerre

Symmetry The big q-Laguerre polynomials Pn(x; a, b; q) are symmetric in a, b. This follows from (14.11.1). As a consequence, it is suﬃcient to give generating function (14.11.11). Then the generating function (14.1.12) will follow by symmetry in the parameters.

14.12 Little q-Jacobi

Notation Here the little q-Jacobi polynomial is denoted by pn(x; a, b; q) instead of pn(x; a, b | q).

33 Basic Hypergeometric Representation In addition to (14.12.1) we have (see [K19, (2.46)])

−n n+1 −n − 1 n(n−1) (qb; q)n q , q ab, qbx pn(x; a, b; q) = (−qb) q 2 3φ2 ; q, q . (176) (qa; q)n qb, 0

Special values (see [K19, §2.4]).

pn(0; a, b; q) = 1, (177)

−1 −1 −n − 1 n(n−1) (qb; q)n pn(q b ; a, b; q) = (−qb) q 2 , (178) (qa; q)n

n 1 n(n+1) (qb; q)n pn(1; a, b; q) = (−a) q 2 . (179) (qa; q)n

14.14 Quantum q-Krawtchouk q-Hypergeometric representation For n = 0, 1,...,N (see (14.14.1) and use (7)):

q−n, y Kqtm(y; p, N; q) = φ ; q, pqn+1 (180) n 2 1 q−N q−n, q−N /y, 0 = (pyqN+1; q) φ ; q, q . (181) n 3 2 q−N , q−N−n/(py)

Special values By (180) and [GR, (II.4)]:

qtm qtm −N Kn (1; p, N; q) = 1,Kn (q ; p, N; q) = (pq; q)n. (182)

By (181) and (182) we have the self-duality

Kqtm(qx−N ; p, N; q) Kqtm(qn−N ; p, N; q) n = x (n, x ∈ {0, 1,...,N}). (183) qtm −N qtm −N Kn (q ; p, N; q) Kx (q ; p, N; q)

By (182) and (183) we have also

qtm −x N −1 KN (q ; p, N; q) = (pq ; q )x (x ∈ {0, 1,...,N}). (184)

Limit for q → 1 to Krawtchouk (see (14.14.14) and Section 9.11):

qtm −1 lim Kn (1 + (1 − q)x; p, N; q) = Kn(x; p ,N), (185) q→1 qtm −x −1 lim Kn (q ; p, N; q) = Kn(x; p ,N). (186) q→1

34 Quantum q−1-Krawtchouk By (180), (182), (6) and (189) (see also p.496, second formula):

qtm −1 −n −1 Kn (y; p, N; q ) 1 q , y −N = 2φ1 ; q, pyq (187) qtm N −1 −1 −1 −N Kn (q ; p, N; q ) (pq ; q )n q Aﬀ −N −1 = Kn (q y; p ,N; q). (188) Rewrite (188) as

qtm −1 −1 −1 −1 −1 Aﬀ −N 1−qN Km (1 + (1 − q )qx; p ,N; q ) = ((pq) ; q )n Kn 1 + (1 − q)q 1−q − x ; p, N; q . In view of (185) and (194) this tends to (82) as q → 1. The orthogonality relation (14.14.2) holds with positive weights for q > 1 if p > q−1.

History The origin of the name of the quantum q-Krawtchouk polynomials is by their interpretation as matrix elements of irreducible corepresentations of (the quantized function algebra of) the quantum group SUq(2) considered with respect to its quantum subgroup U(1). The orthogonality relation and dual orthogonality relation of these polynomials are an expression of the unitarity of these corepresentations. See for instance [343, Section 6].

14.16 Aﬃne q-Krawtchouk q-Hypergeometric representation For n = 0, 1,...,N (see (14.16.1)):

−n −N −1 Aﬀ 1 q , q y −1 Kn (y; p, N; q) = −1 −1 −1 2φ1 −N ; q, p y (189) (p q ; q )n q q−n, y, 0 = φ ; q, q . (190) 3 2 q−N , pq

Self-duality By (190):

Aﬀ −x Aﬀ −n Kn (q ; p, N; q) = Kx (q ; p, N; q)(n, x ∈ {0, 1,...,N}). (191)

Special values By (189) and [GR, (II.4)]:

Aﬀ Aﬀ −N 1 Kn (1; p, N; q) = 1,Kn (q ; p, N; q) = −1 −1 . (192) ((pq) ; q )n By (192) and (191) we have also

Aﬀ −x 1 KN (q ; p, N; q) = −1 −1 . (193) ((pq) ; q )x

Limit for q → 1 to Krawtchouk (see (14.16.14) and Section 9.11):

Aﬀ lim Kn (1 + (1 − q)x; p, N; q) = Kn(x; 1 − p, N), (194) q→1 Aﬀ −x lim Kn (q ; p, N; q) = Kn(x; 1 − p, N). (195) q→1

35 A relation between quantum and aﬃne q-Krawtchouk By (180), (189), (192) and (191) we have for x ∈ {0, 1,...,N}:

Aff −n qtm −x −1 −N−1 Kx (q ; p, N; q) KN−n(q ; p q ,N; q) = Aff −N (196) Kx (q ; p, N; q) KAff (q−x; p, N; q) = n . (197) Aff −x KN (q ; p, N; q) Formula (196) is given in [K3, formula after (12)] and [K12, (59)]. In view of (186) and (195) formula (197) has (83) as a limit case for q → 1.

Aﬃne q−1-Krawtchouk By (189), (192), (6) and (180) (see also p.505, ﬁrst formula):

Aﬀ −1 −n −N Kn (y; p, N; q ) q , q y −1 n+1 Aﬀ N −1 = 2φ1 −N ; q, p q (198) Kn (q ; p, N; q ) q qtm −N −1 = Kn (q y; p ,N; q). (199) Formula (199) is equivalent to (188). Just as for (188), it tends after suitable substitutions to (82) as q → 1. The orthogonality relation (14.16.2) holds with positive weights for q > 1 if 0 < p < q−N .

History The affine q-Krawtchouk polynomials were considered by Delsarte [161, Theorem 11], [K8, (16)] in connection with certain association schemes. He called these polynomials generalized Krawtchouk polynomials. (Note that the 2φ2 in [K8, (16)] is in fact a 3φ2 with one upper parameter equal to 0.) Next Dunkl [186, Definition 2.6, Section 5.1] reformulated this as an interpretation as spherical functions on certain Chevalley groups. He called these polynomials q-Kratchouk polynomials. The current name affine q-Krawtchouk polynomials was introduced by Stanton [488, (4.13)]. He chose this name because, in [488, pp. 115–116] the polynomials arise in connection with an affine action of a group G on a space X. Here X is the set of (v − n) × n A 0 matrices over GF(q). Let G be the group of block matrices , where A ∈ GL (q), SAB n A 0 B ∈ GL (q) and S ∈ X. Then G acts on X by · T = BTA−1 + S. v−n SAB

14.17 Dual q-Krawtchouk Symmetry n −1 −1 Kn(x; c, N | q) = c Kn(c x; c ,N | q). (200) This follows from (14.17.1) combined with [GR, (III.11)]. In particular, n Kn(x; −1,N | q) = (−1) Kn(−x; −1,N | q). (201)

14.20 Little q-Laguerre / Wall

Notation Here the little q-Laguerre polynomial is denoted by pn(x; a; q) instead of pn(x; a | q).

36 Re: (14.20.11) The right-hand side of this generating function converges for |xt| < 1. We can rewrite the left-hand side by use of the transformation 0, 0 1 − 2φ1 ; q, z = 0φ1 ; q, cz . c (z; q)∞ c Then we obtain:

∞ n 1 n(n−1) 0, 0 X (−1) q 2 (t; q) φ ; q, xt = p (x; a; q) tn (|xt| < 1). (202) ∞ 2 1 aq (q; q) n n=0 n

Expansion of xn

Divide both sides of (202) by (t; q)∞. Then coeﬃcients of the same power of t on both sides must be equal. We obtain:

n −n n X (q ; q)k nk x = (a; q)n q pk(x; a; q). (203) (q; q)k k=0

Quadratic transformations ± 1 Little q-Laguerre polynomials pn(x; a; q) with a = q 2 are related to discrete q-Hermite I polynomials hn(x; q):

n −n(n−1) 2 −1 2 (−1) q pn(x ; q ; q ) = 2 h2n(x; q), (204) (q; q )n n −n(n−1) 2 2 (−1) q xpn(x ; q; q ) = 3 2 h2n+1(x; q). (205) (q ; q )n

14.21 q-Laguerre

α (α) Notation Here the q-Laguerre polynomial is denoted by Ln(x; q) instead of Ln (x; q).

Orthogonality relation (14.21.2) can be rewritten with simpliﬁed right-hand side: Z ∞ α α α x Lm(x; q) Ln(x; q) dx = hn δm,n (α > −1) (206) 0 (−x; q)∞ with α+1 −α hn (q ; q)n (q ; q)∞ π = n , h0 = − . (207) h0 (q; q)nq (q; q)∞ sin(πα)

The expression for h0 (which is Askey’s q-gamma evaluation [K1, (4.2)]) should be interpreted by continuity in α for α ∈ Z≥0. Explicitly we can write

− 1 α(α+1) −1 hn = q 2 (q; q)α log(q )(α ∈ Z≥0). (208)

37 Expansion of xn

n −n 1 X (q ; q)k n − 2 n(n+2α+1) α+1 k α x = q (q ; q)n α+1 q Lk (x; q). (209) (q ; q)k k=0 This follows from (203) by the equality given in the Remark at the end of §14.20. Alternatively, it can be derived in the same way as (203) from the generating function (14.21.14).

Quadratic transformations

α 1 q-Laguerre polynomials Ln(x; q) with α = ± 2 are related to discrete q-Hermite II polynomials ehn(x; q): n 2n2−n −1/2 2 2 (−1) q Ln (x ; q ) = 2 2 eh2n(x; q), (210) (q ; q )n n 2n2+n 1/2 2 2 (−1) q xLn (x ; q ) = 2 2 eh2n+1(x; q). (211) (q ; q )n These follows from (204) and (205), respectively, by applying the equalities given in the Remarks at the end of §14.20 and §14.28.

14.27 Stieltjes-Wigert An alternative weight function The formula on top of p.547 should be corrected as

γ − 1 2 2 2 1 w(x) = √ x 2 exp(−γ ln x), x > 0, with γ = − . (212) π 2 ln q

− 1 For w the weight function given in [Sz, §2.7] the right-hand side of (212) equals const. w(q 2 x). See also [DLMF, §18.27(vi)].

14.28 Discrete q-Hermite I History Discrete q Hermite I polynomials (not yet with this name) ﬁrst occurred in Hahn [261], see there p.29, case V and the q-weight π(x) given by the second expression on line 4 of p.30. However note that on the line on p.29 dealing with case V, one should read k2 = q−n instead of k2 = −qn. Then, with the indicated substitutions, [261, (4.11), (4.12)] yield constant −1 −1 multiples of h2n(q x; q) and h2n+1(q x; q), respectively, due to the quadratic transformations (204), (205) together with (4.20.1).

14.29 Discrete q-Hermite II Basic hypergeometric representation (see (14.29.1)) −n −n+1 n q , q 2 2 −2 ehn(x; q) = x 2φ1 ; q , −q x . (213) 0

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T. H. Koornwinder, Korteweg-de Vries Institute, University of Amsterdam, P.O. Box 94248, 1090 GE Amsterdam, The Netherlands; email: [email protected]