Advances in Pure Mathematics, 2019, 9, 15-42 http://www.scirp.org/journal/apm ISSN Online: 2160-0384 ISSN Print: 2160-0368
Associated Hermite Polynomials Related to Parabolic Cylinder Functions
Alfred Wünsche
Max-Planck-Group “Nonclassical Radiation” at Institut für Physik, Humboldt-Universität Berlin, Berlin, Germany
How to cite this paper: Wünsche, A. Abstract (2019) Associated Hermite Polynomials ν Related to Parabolic Cylinder Functions. In analogy to the role of Lommel polynomials R n ( z) in relation to Bessel Advances in Pure Mathematics, 9, 15-42. https://doi.org/10.4236/apm.2019.91002 functions Jν ( z) the theory of Associated Hermite polynomials in the scaled ν form Hen ( z) with parmeter ν to Parabolic Cylinder functions Dν ( z) is Received: November 26, 2018 Accepted: January 20, 2019 developed. The group-theoretical background with the 3-parameter group of Published: January 23, 2019 motions M (2) in the plane for Bessel functions and of the Heisen-
Copyright © 2019 by author(s) and berg-Weyl group W (2) for Parabolic Cylinder functions is discussed and Scientific Research Publishing Inc. compared with formulae, in particular, for the lowering and raising operators This work is licensed under the Creative and the eigenvalue equations. Recurrence relations for the Associated Her- Commons Attribution International License (CC BY 4.0). mite polynomials and for their derivative and the differential equation for http://creativecommons.org/licenses/by/4.0/ them are derived in detail. Explicit expressions for the Associated Hermite Open Access polynomials with involved Jacobi polynomials at argument zero are given and by means of them the Parabolic Cylinder functions are represented by two such basic functions.
Keywords Bessel Functions, Lommel Polynomials, Parabolic Cylinder Functions, Associated Hermite Polynomials, Jacobi polynomials, Recurrence Relations, Lowering and Raising Operators, Heisenberg-Weyl Group, Motion Group of Plane, Irreducible Representations
1. Introduction
Let us motivate our intentions by an analogy of Bessel functions Jν ( z) to Pa-
rabolic Cylinder functions Dν ( z) . Both sets of functions satisfy a certain second-order differential equation and a certain 3-term recurrence relation. The
3-term recurrence relation for the Bessel functions in the form Jν +n ( z) pro-
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vides the possibility to express it successively by the sum of two Bessel functions
with lower indices Jν +m ( z) and Jν +−m 1 ( z) , ( mn< ), and we may continue this
up to the case where we have presented Jν +n ( z) by a superposition of two Bes-
sel functions Jν ( z) and Jν −1 ( z) considered as our “basic” functions of the form
νν+1 Jν+nn( z) =−=∈ R( zz) J νν( ) R n−−11( z) J( zn) ,( 0,1,2, ,ν ) , (1.1)
ν −1 with coefficients R n ( z) which are polynomials of n-th degree of variable z called Lommel polynomials. Their explicit form is known [1] (cited according to Watson [2]). We find this in Watson [2] (from p. 294 on in very detailed form) and in Bateman and Erdélyi [3] (chap. 7.5.2, p. 43) with the explicit formula for the Lommel polynomials1 ( [µ] means integer part of µ ) n k 2 nk−2 ν (−1) (nk −) !( n +−−ν k 1!) 2 R.n ( z) ≡ ∑ (1.2) k =0 kn!( − 2 k) !(ν +− k 1!) z
Whereas Lommel derives his polynomials by a somewhat cumbersome induc- tion ([2], chap. 9.61.) Watson derives the Lommel polynomials from some (Laurent) series of products of Bessel functions that according to his statement is simpler. Furthermore, he discusses in chap. 9.7 (from p. 303 on) a related n zz2 function gznn,,νν≡ R ( ) which in similar form was introduced by 42 Hurwitz [4] (cited on page 302 in [2]). By an analogous process one may relate the Parabolic Cylinder functions
Dν +n ( z) to the sum of two Parabolic Cylinder functions Dν ( z) and Dν −1 ( z) considered as our “basic” functions of the form
νν+1 Dνν+nn( z) = He( zz) D( ) −ν Hen−−11( z) D ν( z) , (1.3)
ν with polynomials Hen ( z) of n-th degree in variable z which possess the expli- cit form
n k 2 (nk−)!( − 1) k (ν −+ 1!j) ν nk−2 Hen ( zz) = ∑∑− −−ν kj kj=0 (nk2) !( 1!) =0 jk!( − j) !2 (1.4) n 2 (nk− )! (−−ννkn, +− k) nk−2 = ∑ Pk ( 0,) z k =0 (nk− 2!)
(αβ, ) where Pk (u) denotes the Jacobi polynomials introduced in this form by Szegö [5] and here taken for argument u = 0 . The polynomials (1.4) are asso- ciated to the Hermite polynomials in a scaled form and are for ν = 0 identical
with the scaled Hermite polynomials usually denoted by Hen ( z) that means 0 Henn( zz) ≡ He ( ) . We call them Associated Hermite polynomials (scaled for
applications in connection with the Parabolic Cylinder functions Dν ( z) ). They
1We change slightly the standard notation of the Lommel polynomials [1] [2] [3] according to ν RRnn,ν ( zz) → ( ) for reason which is better to feel after the development of the analogous formal- ism for Parabolic Cylinder functions.
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were introduced together with some of their basic relations in [6]. One of the possible applications of Parabolic Cylinder functions is in quantum mechanics for the calculation of eigenfunctions of squeezing and rotation oper- ators in the two-dimensional phase plane that means for the general quadratic combinations of a pair of boson annihilation and creation operators and, more generally, for several pairs of boson operators. This is also connected with the reduction to two possible normal forms and one degenerate form of quadratic combinations of such operators. For squeezing operators the eigenfunctions are not normalizable and up to now are little known. The notions “Associated Hermite polynomials” and “Generalized Hermite polynomials” are not used in fully unique sense in literature, e.g., [7] [8]. How- ν ever, our notation Hen ( z) for them seems to be unique and specifies them. Our “Associated Hermite polynomials” are related to the Parabolic Cylinder functions in the analogous way as Lommel polynomials are related to Bessel functions. They are, in general, not orthogonal polynomials and satisfy a 4-th order differential equation. In the following Sections we develop more systematically some formalism for the Parabolic Cylinder functions in connection with their Associated Hermite polynomials. Later, after this we will come back again to the analogies between Bessel functions and Parabolic Cylinder functions with discussion of the group-theoretical background.
2. The Parabolic Cylinder Functions in the Form Dν (z)
The Weber equation with parameter ν ∈ ∂22z 1 − ++νν = 2 yz( ;) 0, (2.1) ∂z 42
with important application in physics (e.g., quantum mechanics) is satisfied, for example, by the following two independent solutions zz22 ν 1 yz1(ν ;) =−− exp 11 F ; ; , 4 22 2
zz2213−ν yz2(ν ;) = exp − z11 F ; ; , (2.2) 4 2 22
with the Wronski determinant ∂∂ Wyyzyzyzyzyz( ,)(νν ;) ≡−=( ;) ( νν ;) ( ;) ( ν ;) 1. (2.3) 12 1 ∂∂zz2 2 1 It is independent of variable z due to differential Equation (2.1) with no first-order derivative and with a second-order derivative without a coefficient in front depending on z as it is easily to derive. In addition, it is independent on parameter ν that is a special property of the two solutions (2.2). Due to inde- pendence of z we obtain the Wronski determinant (2.3) setting, for example, z = 0 .
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From the two linearly independent solutions yz1 (ν ; ) and yz2 (ν ; ) one
constructs by superposition as definition yz(ν ;D) ≡ ν ( z) the following solu- tion of the Weber Equation (2.1)
1 − ! ν 22 ν 2 zz 2 1 Dν ( z) ≡− 2 exp 11F;; − 4 1+ν 22 2 − ! 2 3 − ! 2 zz2 13−ν + 11F ;; 2 ν 2 22 −−1! 2 (2.4) 1 − ! ν 22 +ν 2 zz 2 11 = 2 exp 11F;; − 4 1+ν 22 2 − ! 2 3 − ! 2 z 2 ν 3 z + 11F1+− ; ; , 2 ν 22 2 −1!− 2
where we used in the second representation the Kummer transformation of the Confluent Hypergeometric function (e.g., [9], (chap. (6.3, Equation (7)))
z 11F(acz ;;) = e11 F( c −− ac ;; z) , (2.5) 13 and with −=! π, −=− !2π . A second independent solution of the 22 Weber equation can be constructed in various ways from superpositions of the kind
fz(να;) = Dνν( z) + β D( −+ z) γ D−− ν11( i z) + δ D−− ν( − i, z) (2.6)
with coefficients (αβγδ, ,,) but maximally only two of the four basic functions
(Dνν(±±zz) ,D−−1 ( i )) can be linearly independent solutions of the Weber equa- tion. For example ν ! π π Dν( z) =expiνν D−− νν11( i zz) +− expi D−−( − i) , (2.7) 2π 22
is such a relation between three of these functions [3] (chap. 8.2. (6)).
Instead of Dν ( z) other authors use a fully equivalent form of basic Parabol-
ic Cylinder function with notation U,(µ z) related to Dν ( z) as follows (e.g., Temme [10] in chap. 12 of NIST Handbook [11] and also J. Miller [12] in the older Handbook by the editors Abramowitz and Stegun [13]) 1 Dν ( z) ≡ U −−νµ ,, z ⇔ U,( zz) ≡ D1 ( ) . (2.8) −−µ 2 2
A few but very important special cases of these definitions which can be also
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expressed by other Special functions are [10] (p. 309)
3 zzz2 22 =−= + D1( zz) U( 1, ) KK,31 2 22π 4444
zz2 D11( zz) = U( 0,) = K , − 242π 4
3 zz2 22 z D3( zz) = U( 1,) = K31− K , (2.9) − 2 2π 4444
and (e.g., [10], p. 309, 12.7 (iii))
1 z2 D0 ( zz) =−=− U , exp , 24
1 zz2 π D−1 ( zz) = U , = exp 1− Erf 2 422 (2.10) zz2 π ≡ exp Erfc . 42 2
Herein, Kν (u) is a standard notation for a category of Bessel functions (e.g., [3], chap. 7.2.2. Equation (13), [11], p. 251)
π II−νν(uu) − ( ) K,ν (u) ≡ (2.11) 2 sin (ν π)
and Erf (u) denotes the Error function defined by2
2 zt2 z ≡ −> Erf ∫ dta exp2 ,( 0) . (2.12) a πa2 0 a
The function Erfc(uu) ≡− 1 Erf ( ) is the Complementary Error function.
3. Series Representations of the Parabolic Cylinder Functions
If we insert in (2.4) the well-known Taylor series of the Confluent Hypergeome- tric function
(c−1!) ∞ ( ka +− 1!) zk 11F(acz ;;) ≡ ∑ , (3.1) (a−1!) k =0 ( kc +− 1!) k !
then we may derive two different series representations of the Parabolic Cylinder
functions Dν ( z) .
2Using Erdélyi, Vol. 2 [3] (chap. 8 together with chap. 9.9.) one has to pay attention that in relations to the Error function Erf ( x) the last is defined there under the same notation without the factor 2 in comparison to the modern definition (2.12) which is here given and which is used in Pro- π gram “Mathematica”.
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If we use the first representation in (2.4) we obtain a series representation which can be written in the following compact form
k k zz2 ν ∞ (−1) =−−2 ν Dν ( z) exp 2 π ∑ ( )k , (3.2) 4 k =0 k −−ν 1 2 k!!=Pochh. 2
−ν where ( )k denotes the Pochhammer symbol according to the definition
k (ka+−1!) (k−−νν1!) ( − 1) ! (a) ≡,. ⇒−( ν ) ≡ = (3.3) kk(ak−1!) (−−νν1!) ( −) !
−ν Using the first of the given representations of the Pochhammer symbol ( )k one finds from (3.2)
ν k k zz2 2 2 π ∞ (−1) (k −−ν 1!) Dν ( z) = exp − ∑ 4(−−ν 1!) k =0 k −−ν 1 2 k!! 2 (3.4) k k −ν 2 (−−1) 1! z 1 ∞ 2 k = − expν ∑ ( 2z) . 4!+1 k =0 k 22 (−−ν 1!)
These representations fail to act in the important special cases of non-negative integers ν =n = 0,1,2, due to infinities in numerators and denominator. Us- −ν ing the second of the given representations of the Pochhammer symbol ( )k one finds from (3.2) alternatively
2 ν ∞ ν k zz2 ! Dν ( z) = exp − 2 π ∑ . (3.5) 4 k =0 k −−ν 1 2 kk! !!(ν − ) 2
This representation fails to provide the result without limiting transitions for negative integers ν =−=−−−n 1,2,3, due to infinities in numerator and de- nominator. This is the main reason why we gave in (3.2) the representation by the Pochhammer symbol3. If we use the second representation in (2.4) we obtain from the series repre- sentation of the Confluent Hypergeometric function (3.1) a series representation 1 of Dν ( z) which can be written in the following form ( −=π ) 2
ν k z2 212 ∞ kk−νν +− ( 2z) Dν ( z) = exp ∑ cosπ !, (3.6) 41 k =0 22! k − ! 2
or separated in the even and odd part
3 The Pochhammer symbol (a)k is programmed in “Mathematica” as “Pochhammer [a,k]” in a way that it does not fail also in the mentioned special cases of failure of formulae (3.4) and (3.5).
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ν 2l 2 2 ∞ z 2 π l ν −1 ( 2z) Dν ( zl) = exp cosν ∑(−+ 1) ! 4 1 2l=0 2 ( 2!l) − ! 2 (3.7) 21l+ ∞ π l ν ( 2z) +sinν ∑( −+ 1) l ! . 2 l=0 2 ( 2l + 1!)
z2 We gave here only Taylor series for exp Dν ( z) since pure Taylor se- 4
ries of Dν ( z) are complicated and not favorable for applications.
The basic Parabolic Cylinder functions Dν ( z) were introduced in a way that they are vanishing for real zx= in the limit x → +∞ for arbitrary real indic- es ν ∈ .
4. Lowering and Raising Operators for the Indices of the Parabolic Cylinder Functions and Derivatives and Recurrence Relations
Practically, from every of the given explicit representations of Dν ( z) in last Sections 2 and 3 one can derive formulae for the differentiation of this function. z2 The simplest form they take on if we differentiate exp± Dν ( z) and find 4 or may check the known formulae (e.g., [3] [10]) ∂ zz22 exp Dνν( zz) =ν exp D−1 ( ) , ∂z 44 ∂ zz22 exp− Dνν( zz) =−− exp D+1 ( ) . (4.1) ∂z 44 These relations can be written z ∂ +Dν( zA) ≡= D νν( z) ν D,−1 ( z) 2 ∂z
z ∂ † −Dν( zA) ≡= D νν( z) D.+1 ( z) (4.2) 2 ∂z z ∂ z ∂ We call + the lowering and − the raising operator for the 2 ∂z 2 ∂z † Parabolic Cylinder functions Dν ( z) and denote them by A and A , respec- tively
zz∂∂† zz ∂∂ AA≡+ =+−ii , ≡− =−−ii , (4.3) 22∂∂zz 22 ∂∂ zz The operators A and A† are Hermitean adjoint ones to each other in spaces of functions of real variables z and they satisfy the following commutation rela- tions
† zz∂∂ † AA, ≡+, − = 1,[ AA ,1] = ,1 = 0. (4.4) 22∂∂zz
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These are the commutation relations for a Heisenberg-Weyl Lie algebra w(2) (corresponding to Lie group W (2) ) in representation by a pair of boson annihilation and creation operator in a space of functions fz( ) and each set of
the Parabolic Cylinder functions Dν ±n ( zn) ,( = 0,1,2,) forms a basis of a countably infinite irreducible unitary representation of this Lie algebra characte- rized by the index ν within the interval 01≤<ν . The simplest eigenvalue equations for arbitrary ν are † AADνν( z) =ν D.( z) (4.5)
However, only the functions Dn ( z) with n = 0,1,2, from these eigen- functions prove themselves to be normalizable. Only for the irreducible repre- sentation with index ν = 0 with the basis functions
D±n ( zn) ,( = 0,1,2,) onto functions Dn ( z) with n ≥ 0 acting onto them with the lowering operator A it arises the impression that this representation
breaks down with n = 0 for AzD00 ( ) = but coming from below and acting † with the raising operator A onto functions Dn ( z) with n < 0 we do not
have a breakdown at n = 0 and all basis functions D±n ( z) are related also in this representation.
From (4.2) we find first the derivative of Dν ( z) ∂ 1 Dν( z) = (ν D νν−+( zz) − D,( )) (4.6) ∂z 2 11 and second, the 3-term recurrence relation
Dν+−11( zz) −+ D νν( z) ν D( z) = 0. (4.7) Using the following general disentanglement of linear combinations of opera- ∂ tors z and [14] ∂z − nkk nk zz∂∂ n (−1!) n αβ α αβ−= β ∑ H,nk− (4.8) 2 ∂−z k =0 kn!( k) !2 2 αβ ∂z
which can be proved by complete induction4 one finds from (4.2) (see also (3.3) for representation with the Pochhammer symbol) n z ∂ DDνν+n ( zz) = − ( ) 2 ∂z k nk− n (−1!) n 1 z ∂k = ∑ Hnk− k D,ν ( z) k =0 kn!( − k) !2 2∂z
n (ν − n)!z ∂ Dνν−n ()zz= + D( ) ν !2∂z − n nk k (ν − n)! n nz!i ∂ (4.9) = − ( 1) ∑ Hink− k D.ν ( z) ν ! k =0 kn !( − k) !2 2 ∂z 1 ≡ −ν ( )n
4For αβ in this formula can be chosen an arbitrary sign but it has to be the same in all parts of the right-hand side.
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The second formula fails to act for nonnegative integers ν = 0,1, in cases of ν − 22n n zz∂ =−− Dνν+n ( zz) ( 1) expn exp D( ) , 44∂z (ν − n)! zz22 ∂n = − Dνν−n ( zz) exp n exp D( ) . (4.10) ν !4 ∂z 4 Inserting in the first of these formulae ν = 0 and in the second ν = −1 and using (2.10) we find known formulae for the Parabolic Cylinder functions Dν ( z) with positive and negative integer ν . The formulae up to this point are more or less known but are necessary to make the paper widely self-contained. In next Section we show that we can gen- erate the Parabolic Cylinder functions from two basic such functions by multip- lication with a certain kind of polynomials which generalize the Hermite poly- nomials Hn ( z) in the scaled form usually denoted by Hen ( z) . 5. Representation of Parabolic Cylinder Functions by Two Neighbored Basic Ones with the Recurrence Relation Applying the recurrence relations (4.7) to Dν +n ( z) one may successively re- duce it to a superposition of two basic functions Dν ( z) and Dν −1 ( z) with coefficients depending on variable z of the form νν+1 Dνν+nn( z) =−= He( z) D( z) νν Hen−−11( z) D ν( zn) ,( 0,1,2, ;∈) , (5.1) ν where Hen ( z) are sequences of polynomials of degree n depending on ν as a parameter. For reason which we see below we call them associated Hermite po- lynomials. In the same way by the same recurrence relations (4.7) one may suc- cessively reduce Dν −n ( z) to a superposition of the two basic functions Dν ( z) and Dν +1 ( z) with coefficients depending on variable z of the form (see also −ν (3.3) for Pochhammer symbol ( )n ) (ν − n)! nn−1 = − −−νν1 −− − Dν−nn( z) {( i) He( izz) Dνν( ) ( i) He n−+11( i z) D( z)} . (5.2) ν ! (−1)n = −ν ( )n ν The general form of the polynomials Hen ( z) of degree n is the following n k 2 (−1) k (nj −) !1!(ν −+ j) Heν zz= nk−2 n ( ) ∑∑kj− kj=00(nk− 2!) = jk!( −− j) !(ν 1) !2 n (5.3) kj 2 (−1) k ( − 1) (nj −−) !!( ν ) nk−2 = ∑∑ − z . − kj kj=00(nk2!) = jk!( − j) !( −−ν j) !2 The inner sum in this expression can be expressed in two different ways (due (αβ, ) to transformations relations) by the Jacobi polynomials Pk (u) [3] [5] taken for argument u = 0 or u = 3 as follows DOI: 10.4236/apm.2019.91002 23 Advances in Pure Mathematics A. Wünsche n 2 ν (nk− )! (−−ννkn, +− k) nk−2 Henk( zz) = ∑ P( 0) k =0 (nk− 2!) (5.4) n 2 (nk− )! 1 −−ν −− = ( kn,1) nk−2 ∑ k Pk ( 3.) z k =0 (nk− 2!) 2 ν For the initial members He0 ( z) of the sequences of polynomials (5.3) or (αβ, ) (5.4) the sum consists only of the term to k = 0 and due to P10 ( z) = for arbitrary values of variable z and parameters (αβ, ) we find ν He0 ( z) = 1. (5.5) ν The given representations of Hen ( z) can be proved, for example, by com- plete induction after derivation of the recurrence relations (6.4). It is remarkable that the upper index ν of the Associated Hermite polynomials denoting a pa- rameter is only contained in the Jacobi polynomials in the representation (5.4). The Associated Hermite polynomials satisfy the following symmetry property νnn−−n νν nn−− ν Hen( z) =−( i) He nn( i zz) = i He( −=− i) ( 1) He n( − z) . (5.6) Applying this identity and substituting νν→−1 in (5.2) this formula can be written in the form (ν −−1!n) nn−νν−1 −+1 Dν−− ( z) =( −i) He( iz) Dνν− ( z) −−( i) He− ( izz) D ( ) 1n(ν −1!) { nn11} (5.7) n (−ν )! νν−−nn =(−1) { Henn( z) Dνν−−11( z) −≥ He( zz) D( )} ,( n 0) , (n −ν )! which, in particular, is advantageous in the special cases of integer ν , in partic- ular, ν = 0 in which factors in front of the right-hand side of (5.2) become un- determined without limiting considerations or without using the Pochhammer symbol. In the special case ν = 0 the involved Jacobi polynomials in (5.4) are eva- luated to k (−kn, − k) 1 (−k,1 −− n ) (−1!) n P( 0P) = ( 3) = , (5.8) kk2kk2!kn( − k) ! 0 as one may see also from (5.3) and the sequence of polynomials Hen ( z) pos- sesses the form n k 2 nk−2 0 (−1!) n zz1 Hen( zz) = ∑ = Hnn≡ He( ) . (5.9) − kn k =0 kn!( 2! k) 2 ( 2 ) 2 This means that they possess a form of scaled Hermite polynomials Hn ( z) which is usually denoted by Hen ( z) . In the Appendices A and B we give some ν initial members of the sequences Hen ( z) for integer and for semi-integer ν in explicit form. In Appendix C we give explicit formulae for the Parabolic Cy- linder functions Dν ( z) with integer ν = ±n related to the basic functions DOI: 10.4236/apm.2019.91002 24 Advances in Pure Mathematics A. Wünsche D0 ( z) and D−1 ( z) . We mention that by means of the polynomials Hen ( z) the operator disentanglement formula (4.8) can be represented in the form (αγ→ 2 ) nkk ∂∂ n (−1!) n nk− γ z γ−= β γβ β z ∑ ( ) Henk− . (5.10) ∂−z k =0 kn!!( k) γβ ∂z The right-hand side may be considered as disentanglement in normal order- ing of the operators on the left-hand side. Other forms of operator disentangle- ment can be found in [14]. In general, the Parabolic Cylinder function Dν ( z) can be composed according to (5.1) and (5.2) only from two basic Parabolic Cylinder functions where it is also possible to develop formulae with steps of ν greater than 1. In special cases of zeros of one of the functions Dν ( z) or Dν −1 ( z) in (5.1) and analogously in (5.2) the 3-term relations reduce to 2-term relations, for example, in case of a zero of Dν −1 ( z) for a certain zz= 0 according to ν Dννν−+10( z) =⇒= 0 : Dnn( z0) He( zz0) D( 0) . (5.11) Optimization problems for relative extrema in some problems may lead to the ∞ search for zeros of functions in series of the form ∑n=0cznnDν + ( ) with coeffi- cients cn to determine and for such and also other cases it would be useful to know also generating functions for the Associated Hermite polynomials and in case of Bessel functions for the Lommel polynomials. The use of two neighbored basic Parabolic Cylinder function as basis func- tions is particularly interesting for the irreducible representation of Dν ( z) with integer indices ν = ±n using in this case D0 ( z) and D−1 ( z) according to (2.10) as such. This case concerns also the most important applications in quantum mechanics. In case of the irreducible representation of the Parabolic Cylinder functions with semi-integer indices one cannot find according to (2.9) two equally well appropriate neighbored basic functions but it is well possible that by choosing D1 ( z) and D 3 ( z) as basis functions one may obtain more − symmetric representations2 that we 2do not try to do here. 6. Recurrence Relations for the Associated Hermite Polynomials ν Recurrence relations for the Associated Hermite polynomials Hen ( z) were de- rived in [6]. For some completeness of the description of these polynomials we partially repeat this here with some modifications. As preparation for the calculation of relations for the Associated Hermite po- lynomials and of their differentiation it is useful to know some algebraic rela- tions for the Jacobi polynomials. First of all these are recurrence relations. Since (αβ, ) beside the variable u we have in the Jacobi polynomials Pn (u) two kinds of indices, the lower index n = 0,1,2, of the degree of the polynomials and two upper indices which may take on arbitrary real (or even complex) numbers there exists a great variety of such relations. We collect some of the most basic ones in DOI: 10.4236/apm.2019.91002 25 Advances in Pure Mathematics A. Wünsche Appendix D. The derivation of the recurrence relations for the Associated Hermite poly- nomials can be made using the recurrence relations for the Parabolic Cylinder functions (4.7) which by the substitution νν→+n we write for this purpose in the form Dνν++nn11( zz) = D+ ( z) −+( nν ) D, ν+− n( z) (6.1) and use now the representation (5.1) leading to the following form of the left-hand side of (6.1) νν+1 Dν++nn11( z) = He+( zz) D νν( ) −ν Hen( z) D−1( z) . (6.2) If we apply for the Parabolic Cylinder functions on the right-hand side of (6.1) the same representation (5.1) we find νν Dνν++nn11( z) =( zHe ( zn) −+( ν )He n−( z)) D ( z) (6.3) νν++11 −νν( zHenn−1( zn) −+( )He−−21( z)) Dν ( z) . Since the representation of the Parabolic Cylinder functions by two neigh- bored such functions is unique one finds by comparison of (6.2) with (6.3) the recurrence relation (compare also Drake [8] (Equation (1.1)) νν ν Henn+−11( zz) = He ( z) −+( nν )He n( z) . (6.4) For ν = 0 it provides the recurrence relation for the polynomials Hen ( z) closely related to that for the usual Hermite polynomials Hn ( z) . We now calculate νν−1 Henn++11( zz) − He ( ) n+1 2 (nl+−1!) = (−ν −+lnl1, + ν −) − (− νν −lnl, + −+ 1) nl+−12 ∑ (Pll( 0P) ( 0)) z l=0 (nl+−12!) n+1 2 (6.5) (nl+−1!) (−νν −+ln1, + −+ l 1) nl+−12 = ∑ P0l−1 ( ) z l=1 (nl+−12!) n−1 2 (nk− )! (−−ννkn, +− k) nk−−12 = ∑ Pk ( 0,) z k =0 (nk−−12) ! where we applied the identity (D.3). If we now apply the recurrence relation (D.2) (αβ, ) for the Jacobi polynomials Pn (u) with fixed lower indices n but varying upper indices with the substitutions nk→,,ανβ →− − k → n + ν − k then from this follows (−−ννkn, +− k) (−−ννkn,1 −+− k) (−−− νν1,kn +− k) (nk−=)Pk( u) ( n+νν) P kk(u) − P.(u) (6.6) Inserting this with u = 0 in (6.5) we find an identity of the Associated Her- mite polynomials in the form νν−+11 ν ν Henn++11( z) −=+− He ( zn) ( νν)He n − 1( z) He n − 1( z) νν ν+1 (6.7) =−−zzHenn( ) He+−11( z) ν He n( z) . DOI: 10.4236/apm.2019.91002 26 Advances in Pure Mathematics A. Wünsche from which using (6.4) follows ννν−+11 Hennn+−11( z) +=ν He( zz) He( z) . (6.8) In next Section we derive formulae for the differentiation of the Associated Hermite polynomials from which we finally develop the differential equation for these polynomials. 7. Derivative of the Associated Hermite Polynomials and Differential Equations ν By differentiation of the Associated Hermite polynomials Hen ( z) in the re- presentation (5.4) we find n−1 2 ∂ ν (nk− )! (−−ννkn, +− k) nk−−12 Henk( zz) = ∑ P( 0) . (7.1) ∂zk =0 ( nk−−12) ! The right-hand side is the same as the last line on the right-hand side of (6.5) and therefore we may write the result of differentiation in the forms ∂ νν−1 ν He( z) = He++( zz) − He ( ) ∂z nn11 n νν+1 =+−(nzzνν)Henn−−11( ) He ( ) (7.2) νν ν+1 =−−zzHenn( ) He+−11( z) ν He n( z) . From this follows by index substitutions ∂ ν νν+1 Hen+1 ( zn) =( ++1νν) Henn( z) − He( z) , ∂z ∂ ν++11 νν He− ( zz) = He( ) − He( z) . (7.3) ∂z n1 nn By repeated differentiation using the representation in last line of (7.2) we ob- tain 2 ∂∂ν νν ∂∂ ν ν+1 He( zz) = He( z) +− He( z) He +−( z) −ν He ( z) ∂z2 n∂z nn ∂∂ zz n11 n (7.4) ∂∂νν+1 =−−znHenn( z) 2ν He−1 ( z) , ∂∂zz where we used (7.3) for the transformation of the result. This means that the Associated Hermite polynomials satisfy the equation ∂∂2 ∂ −+νν =−ν +1 2 z nH eznn( ) 2 He−1 ( z) . (7.5) ∂z ∂∂zz For ν = 0 we obtain the differential equation for modified Hermite polyno- 0 mials HHeznn( ) ≡ ez( ) . ∂ ν +1 The full elimination of the derivative He − ( z) by means of the last of ∂z n 1 Equations (7.3) provides ∂∂2 − ++νννν = +1 2 zn2 Henn( z) 2 He( z) . (7.6) ∂z ∂z DOI: 10.4236/apm.2019.91002 27 Advances in Pure Mathematics A. Wünsche On the other side one obtains ∂∂2 + ++νννν = + −1 2 zn2 Henn( z) 2( n) He( z) . (7.7) ∂z ∂z Under the assumption that both Equations (7.6) and (7.7) are correct (for (7.6) it is proved) one finds from these equations by forming the sum ∂2 ++νν = + νν νν−+11 + 2 n2 Hen( zn) ( )He nn( z) He( z) , (7.8) ∂z and for the difference ∂ ν νν−+ zHe ( zn) =+−( νν)He11( z) He( z) . (7.9) ∂z n nn Calculating the derivative from (5.4) we find n 2 ∂ ν (nk− )! (−−ννkn, +− k) nk−2 zHenk( z) = ∑ (nk− 2) P( 0) z . (7.10) ∂−zk =0 ( nk2!) Using herein the identity (D.8) of Appendix D with the substitutions nk→,,ανβ →− − k → n + ν − k one obtains from (7.9) ∂ zzHeν ( ) ∂z n n 2 (nk− )! =+−νν(−νν +−1,kn + −− 1 k) (−νν −−1,kn + +− 1 k) nk−2 ∑ ((nz)P0P0kk( ) ( )) (7.11) k =0 (nk− 2!) νν−+11 =+−(nzzνν)Henn( ) He( ) . Thus relations (7.8) and (7.9) and the equivalent relations (7.6) and (7.7) are proved. Now we are able to derive the differential equations for which the Associated Hermite polynomials are solutions. Using (7.6) and (7.7) we have two possibili- ties with respect to the order of application which, clearly, have to lead to the same result. The first is ∂∂22∂∂ = + ++ν + − ++νννν − +1 0 22zn2( 1)zn2 Henn( z) 2 He ( z) ∂∂zz∂∂zz ∂∂22 ∂∂ = + ++ν + − ++ννν − ++ ν 22zn2( 1) zn2 4( n 1) Hen ( z) , ∂∂zz∂∂zz (7.12) and the second ∂∂22∂∂ = − ++ν − + ++ νννν − + −1 0 22zn2( 1)zn2 Henn( z) 2( n) He ( z) ∂∂zz∂∂zz ∂∂22 ∂∂ = − ++ν − + ++νν − − + νν 22zn21( ) zn241 ( )( n) He.n ( z) ∂∂zz∂∂zz (7.13) By transition to normal ordering (all powers of z stand in front of all powers DOI: 10.4236/apm.2019.91002 28 Advances in Pure Mathematics A. Wünsche ∂ of the differential operator ) we find the following differential equation of ∂z ν 4-th order for which Hen ( z) are solutions ∂42∂∂ + +−ν 2 − ++ν = 42(2(n 2 ) z) 3 z nn( 2) Hen ( z) 0. (7.14) ∂∂zz∂z 0 In the special case ν = 0 and thus Henn( zz) ≡ He ( ) the operator of this differential equation factorizes as follows ∂∂22 ∂∂ +zn ++2 − zn +He( z) = 0, (7.15) ∂∂zz22∂∂zzn =0 where Hen ( z) according to (7.5) satisfies already the shown differential equa- tion of 2-nd order. 8. Analogies and Differences between the Lommel Polynomials for the Bessel Functions to the Associated Hermite Polynomials for the Parabolic Cylinder Functions In the Introduction we already mentioned shortly the analogies between the ν Lommel polynomials R n ( z) in relation to the Bessel functions to the Asso- ν ciated Hermite polynomials Hen ( z) in relation to the Parabolic Cylinder func- tions. We now come back to these analogies with more details. The Bessel functions Jν ±n ( z) can be generated from two arbitrary basic functions Jν ( z) and Jν 1 ( z) according to the relations ([2] [3]; see also foot- note 1) νν+1 Jν+nn( z) = RJ( zz) νν( ) − R n−−11( z) J( z) , n −−νν1 Jν−nnn( z) =(−+ 1R) { ( zz) Jνν( ) R−+11( z) J( z)} , (8.1) ν −1 where the sequences of polynomials R n of n-th degree in variable z , called Lommel polynomials, are given by n k 2 nk−2 ν (−1) (nk −) !( n +−−ν k 1!) 2 Rn ( zn) ≡≥∑ ,( 0,) k =0 kn!( − 2 k) !(ν +− k 1!) z n (8.2) 21 − nn 2 =(ν) 23F ,− ; −nnz ,νν ,1 −− ; − ,( n ≥ 0) , n z 22 (n+−ν 1!) = (ν −1!) where 23F(aacccu 1 , 2 ;,,; 1 2 3 ) is a Hypergeometric function in its standard nota- ν ν tion and ( )n the Pochhammer symbol. The polynomials R n ( z) possess the symmetry property νnn11−−nn νν−− ν Rn( z) =−( 1R) nn( zz) = R( −=−) ( 1R) n( − z) . (8.3) The Bessel functions Jν ( z) satisfy the differential equation DOI: 10.4236/apm.2019.91002 29 Advances in Pure Mathematics A. Wünsche 2 ∂ 22 0J=zz +−ν ν ( z) ∂z (8.4) ∂∂22 22 =zz ++νν zz +−J,ν ( z) ∂∂zz with two commuting operators as factors in front. In contrast to the differential equation for the Parabolic Cylinder function the parameter ν of the Bessel function is here involved quadratically that invites to the given factorization. The Parabolic Cylinder functions Dν ( z) can be generated from two arbi- trary basic functions Dν ( z) and Dν 1 ( z) according to the relations νν+1 Dνν+nn( z) =−= He( z) D( z) νν Hen−−11( z) D ν( zn) ,( 0,1,2, ;∈) , (ν − n)! nn−−νν1 −1 − Dν−( z) =( −i) He( izz) Dνν( ) −−( i) He−+( i z) D( z) , nnν ! { n11} (8.5) (n =0,1,2, ;ν ∈ ) . ν where the sequences of polynomials Hen ( z) of n-th degree are given by n k 2 (−1) k (nj −) !1!(ν −+ j) Heν zz= nk−2 n ( ) ∑∑kj− kj=00(nk− 2!) = jk!( −− j) !(ν 1) !2 (8.6) n 2 (nk− )! (−−ννkn, +− k) nk−2 = ∑ Pk ( 0.) z k =0 (nk− 2!) They possess the symmetry property νnn−−n νν nn−− ν Hen( z) =−( i) He nn( i zz) = i He( −=− i) ( 1) He n( − z) . (8.7) The Parabolic Cylinder functions Dν ( z) satisfy the differential equation ∂22z 1 = − +−ν 02 D.ν ( z) (8.8) ∂z 42 For the Lommel polynomials and for the Associated Hermite polynomials we know the explicit formulae ((8.2) and (5.3)). Since the formula for the Lom- mel polynomials involves the Hypergeometric function 23F(aacccu 1 , 2 ;,,; 1 2 3 ) which satisfies a differential equation of 4-th order the same should be true for the Lommel polynomials. However, we did not find it in literature and did not try to calculate it up to now. The derivation of the differential equation for the Associated Hermite polynomials (7.14) takes on an essential part of present pa- per. Beside the mentioned similarities in both systems there are also essential dif- ν ferences. The most striking one is that Hen ( z) are polynomials of n-th 1 degree in z whereas Rν ( z) are polynomials of n-th degree in , explicitly n z given in (8.2)5. A certain feeling why this is so one may attain if one compares 5 νν1 Therefore, a notation RRnn( z) ≡ or similar would be better instead of the notation z ν RRnn( zz) ≡ ,ν ( ) for the Lommel polynomials based on literature. DOI: 10.4236/apm.2019.91002 30 Advances in Pure Mathematics A. Wünsche the two most important relations for the two systems of functions Dν ( z) and Jν ( z) beside the already written down differential equations that are the diffe- rentiations and the recurrence relations ∂∂11 Dν( z) =−=−(ν D ν−( z) D ν +( z)) ,J ν( z) ( J νν−+( zz) J( )) , ∂∂zz221 1 11 (8.9) 1 z Dν( z) =+=+(ν D ν−( z) D ν +( z)) ,J ν( z) ( J νν−+( zz) J( )) . z 1 1 2ν 11 As essential difference we see that on the right-hand sides of the recurrence relations the variable z stands for the Parabolic Cylinder functions Dν ( z) in the denominator and for the Bessel functions Jν ( z) in the numerator that ex- plains this. Other essential differences of the two considered systems of functions are the underlying symmetry groups. Let us make a few remarks about this in the next Section. 9. Group-Theoretical Background of the Bessel Functions Whereas the group-theoretical background of the Parabolic Cylinder functions as shown is the Heisenberg-Weyl group W (2) (see (4.3) and (4.4)) the group-theoretical background of the Bessel functions is the group of motions M (2) of the two-dimensional Euclidean plane (Vilenkin [15], chap. IV). The generators of this group consist of two commuting translations T2 in two in- dependent directions of the plane and the one-parameter group of rotations SO(2) in the plane which last does not commute with the translations (e.g., [15] [16] and [17]). Since this background is not so very well known let us give the most important relations in connection to the Bessel functions which show this. If we denote the two commuting operators of the Lie algebra of translations by (ZZ12, ) and the operator of the rotation by Z3 then we have the commuta- 6 tion relations for ( XXX123,,) [ZZ12,] = 0,[ ZZ23 ,] = Z 1 ,[ ZZ 31 ,] = Z 2 . (9.1) We now form the new Lie-group operators ( AAA+−,,3 ) AZZAZZAZ+−=+=−=12i, 123 i, i, 3 AA+− AA Z==+−, Z−= i+− , ZA− i, (9.2) 1222 33 which satisfy the commutation relations [ AA+−,] == 0,,,,[ AA33+] += A +[ AA −] − A − . (9.3) With respect to the Bessel functions Jν ( z) we consider the following reali- zation of the operators ( AAA+−,,3 ) ∂∂2 1 2 A+ = zz +− , zz∂∂ z 6 We use the notations of Barut and Raçzka [16] (chap. 14, §3). Vilenkin [15] uses (aaa123,,) and Miller [17] (PPM12,, ) . DOI: 10.4236/apm.2019.91002 31 Advances in Pure Mathematics A. Wünsche ∂∂2 1 2 A− = zz ++ , zz∂∂ z 2 ∂ 2 Az3 = + z, (9.4) ∂z or equivalently for (ZZZ123,,) 22 1 ∂22 ∂∂ Z1= zzZZ +=,23 i, =−+ i zz . (9.5) zz∂ ∂∂ z z According to (8.4) the Bessel functions Jν ( z) are eigenfunctions of the op- 2 ∂ 2 erator zz+ to the eigenvalue ±ν and one verifies ∂z ν ∂ Az++Jν( ) =−= Jνν( z) J,1 ( z) zz∂ ν ∂ Az−−Jν( ) =+= Jνν( z) J,1 ( z) zz∂ Az3 Jνν( ) =ν J.( z) (9.6) If we change the sign of ν then the transformation νν→− leads to the transformations A± →− AA , 33 →− A and the role of A+ and A− becomes exchanged (beside sign changes) that preserves the commutation relations (9.3). The identity operator I in the realization I = 1 can be expressed by the Lie-algebra operators according to [16] 221 IZ=+=( ) ( Z) ( AAAA+− + −+) == AAAA +− −+ 122 2 2 2 1 ∂∂2 =zz +− (9.7) zz∂ ∂z2 2 11∂ ∂∂2 =zz +−2 − =1. 22 zz∂∂z zz ∂ Similar to the system of Parabolic Cylinder functions Jν ±n ( z) connected with the Heisenberg-Weyl group W (2) each (complete in some sense) system of Bessel functions Jν ±n ( z) with 01≤<ν and n = 0,1,2, realizes a coun- tably infinite irreducible representation of the (inhomogeneous) group of mo- tions M (2) of the real two-dimensional Euclidean plane which can be represented as the semi-direct product MT(22) = ( )⊃× SO(2) of the two-dimensional (com- mutative) translation group T (2) with the one-parameter rotation group SO(2) in the plane. In application to systems of modified Bessel functions Iν ( z) one has slightly to change the realizations of the lowering and raising operators and of the ei- genvalue operator ( AAA+−,,3 ) in comparison to (9.4) that we do not write down. DOI: 10.4236/apm.2019.91002 32 Advances in Pure Mathematics A. Wünsche 10. Conclusions In analogy to the known (but not well-known) relation of Bessel functions to Lommel polynomials we developed a similar concept for the relation of Parabol- ic Cylinder functions in the form Dν ( z) to polynomials which we call Asso- ν ciated Hermite polynomials and denote them by Hen ( z) . These polynomials obey a fourth-order differential equation and their explicit form can be concisely (αβ, ) expressed using the Jacobi polynomials Pk (u) taken for argument u equal to zero. Recurrence relations for these polynomials are derived. For the proof of the differential equation we give in Appendix D contiguous relations in the most symmetrical form concerning the parameters (αβ, ) and derive from them an identity (D.8) for Jacobi polynomials which is only true for argument u equal to zero but likely can be generalized to general argument u with coeffi- cients depending on u. In Appendices A-B we write down their explicit form for some initial members n and for some integer and semi-integer parameter ν . In Appendix C we represent the Parabolic Cylinder functions Dν ( z) for (posi- tive and negative) integers ν by means of two basic such functions D0 ( z) and D−1 ( z) and the introduced Associated Hermite polynomials. Our motivation for the introduction of the Associated Hermite polynomials in 2000 was analogies to the Lommel polynomials in relation to the Bessel func- tions. The Parabolic Cylinder functions together with the Associated Hermite polynomials possess as group-theoretical background the Heisenberg-Weyl group W (2) . For the Bessel functions together with the Lommel polynomials the group-theoretical background is the group of motions M (2) with the two (commuting) translations and the one-parameter group of rotations in the plane. We made a comparison of formulae of both groups in relation to the Bessel and to the Parabolic Cylinder functions and gave explicitly the basic operators of their Lie algebras (in particular, lowering and raising operators) in representa- tion related to these functions. Besides great analogies between the kind of for- mulae for these two sets of functions there are also essential differences. One 1 such difference is that the Lommel polynomials are polynomials of variable z whereas the Associated Hermite polynomials are polynomials in z. This is dis- cussed and presented by the corresponding formulae. We mention that in recent time the Parabolic Cylinder function are often used in a form denoted by U,(µ z) closely and equivalently related to the here used older form Dν ( z) by relation (2.8). Though some symmetries of U,(µ z) are 1 related to the parameter value µ = 0 and of Dν ( z) to the value ν = − the 2 most important applications of the Parabolic Cylinder function are connected with integer values of ν in case of Dν ( z) in opposition to semi-integer val- ues of µ in case of U,(µ z) . Finally, the normalizable Parabolic Cylinder functions are representable as superpositions of Dn ( zn) ,( = 0,1,2,) that is very important for a rational representation of many formulae in quantum me- chanics and we prefer this. DOI: 10.4236/apm.2019.91002 33 Advances in Pure Mathematics A. Wünsche Conflicts of Interest The author declares no conflicts of interest regarding the publication of this pa- per. References [1] Von Lommel, E.C.J. (1871) Zur Theorie der Bessel’schen Funktionen. Mathema- tische Annalen, IV, 103-116. https://doi.org/10.1007/BF01443302 [2] Watson, G.N. (1944) (A Treatise on the) Theory of Bessel Functions. 2nd Edition, Cambridge University Press, Cambridge. [3] Erdélyi, A. (1974) Higher Transcendental Functions, Vol. 2. Nauka, Moskva. [4] Hurwitz, A. (1889) Über die Nullstellen der Bessel’schen Funktionen. Mathema- tische Annalen, XXXIII, 246-266. [5] Szegö, G. (1959) Orthogonal Polynomials. 2nd Edition, American Mathematical Society, New York. [6] Wünsche, A. (2003) Generalized Hermite Polynomials Associated with Functions of Parabolic Cylinder. Applied Mathematics and Computation, 141, 197-213. https://doi.org/10.1016/S0096-3003(02)00333-8 [7] Dattoli, G., Chiccoli, C., Lorenzutta, S., Maino, G. and Torre, A. (1994) Theory of Generalized Hermite Polynomials. Computers and Mathematics with Applications, 28, 71-83. https://doi.org/10.1016/0898-1221(94)00128-6 [8] Drake, D. (2009) The Combinatorics of Associated Hermite Polynomials. European Journal of Combinatorics, 30, 1005-1021. https://doi.org/10.1016/j.ejc.2008.05.009 [9] Erdélyi, A. (1974) Higher Transcendental Functions, Vol. 1. Nauka, Moskva. [10] Temme, N.M. (2010) Parabolic Cylinder Functions, Chapter. 12. In: NIST Hand- book of Mathematical Functions, Cambridge University Press, Cambridge, New York, 303-319. [11] Olver, F.W.J., Lozier, D.W., Boisvert, R.F. and Clark, Ch.W. (Eds.) (2010) NIST Handbook of Mathematical Functions. Cambridge University Press, Cambridge, New York. [12] Miller, J.C.P. (1964) Parabolic Cylinder Functions, Chapter. 19. In: Handbook of Mathematical Functions, National Bureau of Standards, Washington, 495-531. [13] Abramowitz, M. and Stegum, I.A. (Eds.) (1964) Handbook of Mathematical Func- tions. National Bureau of Standards, Washington. [14] Wünsche, A. (1999) Ordered Operator Expansions and Reconstruction from Or- dered Moments. Journal of Optics B: Quantum and Semiclassical Optics, 1, 264-288. https://doi.org/10.1088/1464-4266/1/2/010 [15] Vilenkin, N.Ya. (1965) Spetsialnyje funkzii i teorija predstavlyenija grupp. [Special Functions and the Theory of Representations of Groups.] Nauka, Moskva. [16] Barut, A.O. and Raçzka, R. (1977) Theory of Group Representations and Applica- tions. PWN-Polish Scientific Publishers, Warszawa. [17] Miller Jr., W. (1977) Symmetry and Separation of Variables. Addison-Wesley, Reading (Mass.). [18] Wünsche, A. (2015) Operator Methods and SU(1,1) Symmetry in the Theory of Ja- cobi and of Ultraspherical Polynomials. Applied Mathematics, 7, 213-261. DOI: 10.4236/apm.2019.91002 34 Advances in Pure Mathematics A. Wünsche Appendix A. Sequences of Associated Hermite Polynomials ν Hen (z) with Integer Parameter ν For convenience we compile here some first members of the sequences of poly- ν nomials form Hen ( z) in explicit form. (1) The first 12 initial members of the sequence of polynomials Hen ( z) are 1 He0 ( z) = 1, 1 He1 ( zz) = , 12 He2 ( zz) = − 2, 13 He3 ( zz) = − 5 z , 1 42 2 He4 ( zz) = − 9 z += 8( z − 1)( z + 1)( z − 8) , 1 53 2 2 He5 ( z) =−+= z 14 z 33 z zz( − 3)( z − 11) , He1( zz) =−+− 642 20 z 87 z 48, 6 He1( zz) =−+− 75 27 z 185 z 3 279 z , 7 1 86 4 2 He8 ( zz) =−+ 35 z 345 z − 975 z + 384, He1( zz) =−+− 97 44 z 588 z 5 2640 z 3 + 2895 z , 9 He1 ( zz) =−+−1054 z 8 938 z 6 6090 z4 + 12645 z2 − 3840, 10 1 11 9 7 5 3 He11 ( zz) =−+65 z 1422 z − 12558 z + 41685 z − 35685 z . (A.1) The sequence of polynomials −−11nn n 1 Henn( zz) = i He( −=− i) ( i) He n( i z) , (A.2) 1 is obtained from the sequence of polynomials Hen ( z) in (A.1) by substituting all minus signs on the right-hand side by plus signs. 0 The first 12 members of the sequence of polynomials Henn( zz) ≡ He ( ) are 0 He0 ( z) = 1, 0 He1 ( zz) = , 02 He2 ( zz) = − 1, He03( zz) = − 3 z , 3 He0( zz) =−+ 42 6 z 3, 4 He0( zz) =−+ 53 10 z 15 z , 5 He0( zz) =−+− 64 15 z 45 z 2 15, 6 He0( zz) =−+ 75 21 z 105 z 3 − 105 z , 7 He0( zz) =−+− 86 28 z 210 z 4 420 z 2 + 105, 8 He0( zz) =−+ 97 36 z 378 z 5 − 1260 z 3 + 945 z , 9 He0 ( zz) =−+1045 z 8 630 z6 − 3150 z4 + 4725 z2 − 945, 10 0 11 9 7 5 3 He11 ( zz) =−+−55 z 990 z 6930 z + 17325 z − 10395 z . (A.3) −n The sequence of polynomials Hen ( z) related to these polynomials are sep- arately written down in explicit form. DOI: 10.4236/apm.2019.91002 35 Advances in Pure Mathematics A. Wünsche −1 The first 12 members of the sequence of polynomials Hen ( z) are −1 He0 ( z) = 1, −1 He1 ( zz) = , −12 He2 ( zz) = , −13 He3 ( zzz) = − , −1 42 He4 ( zz) = − 3 z , −1 53 He5 ( zz) =−+ 6 z 3 z , −1 642 He6 ( zz) =−+ 10 z 15 z , −1 75 3 He7 ( zz) =−+− 15 z 45 z 15 z , −1 86 4 2 He8 ( zz) =−+− 21 z 105 z 105 z , −1 97 5 3 He9 ( zz) =−+− 28 z 210 z 420 z + 105 z , He−1( zz) =−+− 1036 z 8 378 z 6 1280 z4 − 945 z 2 , 10 −1 11 9 7 5 3 He11 ( zz) =−+−45 z 630 z 3150 z + 4725 z − 945 z . (A.4) −1 0 Generally Hen ( z) is related to Hen ( z) by −10 Henn( zz) = He−1 ( z) . (A.5) The sequence of polynomials 11−−nn n − 1 Henn( zz) = i He( −=− i) ( i) He n( i z) , (A.6) −1 is obtained from the sequence of polynomials Hen ( z) in (4) by substituting all minus signs on the right-hand side by plus signs. The first 12 members of the series of polynomials −nn00n Henn( zz) = i He( −=− i) ( i) He n( i z) , (A.7) are explicitly 0 He0 ( z) = 1, −1 He1 ( zz) = , −22 He2 ( zz) = + 1, −33 He3 ( zz) = + 3 z , −4 42 He4 ( zz) =++6 z 3, −5 53 He5 ( zz) =++ 10 z 15 z , −6 64 2 He6 ( zz) =+++ 15 z 45 z 15, −7 75 3 He7 ( zz) =+++21 z 105 z 105 z , −88642 He8 ( zz) =++++ 28 z 210 z 420 z 105, −9 97 5 3 He9 ( zz) =+++ 36 z 378 z 1260 z + 945 z , −10 10 8 6 4 2 He10 ( zz) =+++45 z 630 z 3150 z + 4725 z + 945, −11 11 9 7 5 3 He11 ( zz) =+++55 z 990 z 6930 z + 17325 z + 10395 z . (A.8) −m The polynomials Hen ( z) with non-positive upper integer index −m =0, −− 1, 2, DOI: 10.4236/apm.2019.91002 36 Advances in Pure Mathematics A. Wünsche are factorable and satisfy the following relation −−mm0 Hen ( z) = Hem( z) He nm− ( z) ,( nm−≥ 0) , (A.9) for example −10 Henn( zz) = He−1 ( z) , −2 20 Henn( zz) =( +1) He−2 ( z) , −33 0 Henn( zzz) =( + 3) He−3 ( z) , −4 42 0 Henn( zzz) =++( 6 3) He−4 ( z) . (A.10) 0 The scaled Hermite Hermite polynomials Henn( zz) ≡ He ( ) are connected with the usual Hermite polynomials Hn ( z) by 1 z n ≡ ⇔= Henn( z) n H , H n( zz) ( 2) He n( 2) , (A.11) ( 2 ) 2 and play often a role in applications. Appendix B. Sequences of Associated Hermite Polynomials ν Hen (z) with Semi-Integer Parameter ν 1 2 The first 12 initial members of the sequence of polynomials Hen ( z) are 1 2 He0 ( z) = 1, 1 2 He1 ( zz) = , 1 3 He 2 ( zz) =2 − , 2 2 1 2 3 He3 ( zz) = − 4 z , 1 15 21 He 2 ( zz) =−+42 z , 4 24 1 93 He2 ( zz) =−+53 12 z z , 5 4 1 35 129 231 He 2 ( zz) =−+64 z z 2 −, 6 22 8 1 285 He2 ( zz) =−+75 24 z z 3 −180 z , 7 2 1 63 1095 2655 3465 He 2 ( zz) =−+−86 z z 4 z 2 +, 8 2 4 4 16 1 1911 27945 He2 ( zz) =−+97 40 z z 5 −1875 z 3 + z, 9 4 16 1 99 35805 128835 65835 He 2 ( zz) =−+10 z 8 777 z6 − z4 + z2 − , 10 2 8 16 32 1 443835 81585 He2 ( zz) =−+1160 z 9 1197 z7 − 9492 z5 + z2 − z. (B.1) 11 16 4 DOI: 10.4236/apm.2019.91002 37 Advances in Pure Mathematics A. Wünsche 1 − 2 The first 12 initial members of the sequence of polynomials Hen ( z) are 1 − 2 He0 ( z) = 1, 1 − 2 He1 ( zz) = , 1 − 1 He 2 ( zz) =2 − , 2 2 1 − 2 3 He3 ( zz) = − 2 z , 1 − 95 He 2 ( zz) =−+42 z , 4 24 1 − 33 He 2 ( zz) =−+538 z z , 5 4 1 − 25 57 45 He 2 ( zz) =−+−642 z z , 6 228 1 − 145 He 2 ( zz) =−+7518 z z 3 −51 z , 7 2 1 − 49 615 945 585 He 2 ( zz) =−+−86 z z 4 z 2 +, 8 2 4 4 16 1 − 1155 6705 He 2 ( zz) =−+9732 z z 5 −780 z 3 + z , 9 4 16 1 − 81 16695 38835 9945 He 2 ( zz) =−+−10 z 8 497 z6 z4 + z2 −, 10 2 8 16 32 1 − 157395 34335 He 2 ( zz) =−+−1150 z 9 801 z 7 4830 z5 + z3 − z. (B.2) 11 16 8 3 − 2 The first 12 initial members of the sequence of polynomials Hen ( z) are 3 − 2 He0 ( z) = 1, 3 − 2 He1 ( zz) = , 3 − 1 He 2 ( zz) =2 + , 2 2 3 − 2 3 He3 ( zz) = , 3 − 33 He 2 ( zz) =−−42 z , 4 24 3 − 3 He 2 ( zz) =−−534 z z , 5 4 3 − 15 9 21 He 2 ( zz) =−++6 z 42 z , 6 228 3 − 45 He 2 ( zz) =−++7512 z z 3 6 z , 7 2 3 − 35 255 75 231 He 2 ( zz) =−+−−86 z z 42 z , 8 2 4 4 16 DOI: 10.4236/apm.2019.91002 38 Advances in Pure Mathematics A. Wünsche 3 − 567 855 He 2 ( zz) =−+−9724 z z 5165 z 3 − z , 9 4 16 3 − 63 5145 1395 3465 He 2 ( zz) =−+−10 z 8 273 z6 z4 + z2 +, 10 2 8 16 32 3 − 23835 1125 He 2 ( zz) =−+−1140 z 9 477 z7 1848 z5 + z3 + z. (B.3) 11 16 2 Appendix C. Sequence of Parabolic Cylinder Functions with Integer Indices The first 12 members of the sequence of Parabolic Cylinder functions with non-negative integer indices are z2 D0 ( z) = exp − , 4 z2 D1 ( zz) = exp − , 4 2 z 2 D2 ( zz) =−− exp( 1) , 4 2 z 3 D3 ( z) =−− exp( zz3) , 4 2 z 42 D4 ( z) = exp −( zz −+6 3) , 4 2 z 53 D5 ( z) = exp −( zzz −+10 15) , 4 2 z 64 2 D6 ( z) = exp −( zz −+−15 45 z 15) , 4 2 z 75 3 D7 ( z) =− exp( zz −+21 105 z − 105 z) , 4 2 z 86 4 2 D8 ( z) =− exp( zz −+28 210 z − 420 z + 105) , 4 2 z 97 5 3 D9 ( z) =− exp( zz −+36 378 z − 1260 z + 945 z) , 4 2 z 10 8 6 4 2 D10 ( z) =− exp( zz −+45 630 z − 3150 z + 4725 z − 945) , 4 2 z 11 9 7 5 3 D11 ( z) =− exp( zz −+55 990 z − 6930 z + 17325 z − 10395 z) . (C.1) 4 The first 12 members of the sequence of Parabolic Cylinder functions with negative integer indices represented in the form (5.2) or better (5.7) with ν = 0 and using (2.10) are (definition of Complementary Error function Erfc(uu) ≡− 1 Erf ( ) ) DOI: 10.4236/apm.2019.91002 39 Advances in Pure Mathematics A. Wünsche 1 zz2 π D−1 ( z) = exp Erfc , 0! 4 2 2 1 z22π zz D−2 ( zz) =−exp Erfc+− exp , 1! 4 2 2 4 22 1 zzπ 2 z D−3 ( z) = exp Erfc( zz+− 1exp) − , 2! 4 2 2 4 22 1 zzπ 32 z D−4 ( z) =−exp Erfc( zz++ 3) exp − ( z +2) , 3! 4 2 2 4 22 1 zzπ 42 z 3 D−5 ( z) = exp Erfc( zz+ 6 +− 3exp) − ( zz +5) , 4! 4 2 2 4 2 1 zzπ 53 D−6 ( z) =− exp Erfc( zzz++ 10 15 ) 5! 4 2 2 z2 +exp −( zz42 ++9 8) , 4 2 1 zzπ 64 2 D−7 ( z) = exp Erfc( zzz+++ 15 45 15) 6! 4 2 2 z2 −−exp( zzz53 ++14 33) , 4 2 1 zzπ 75 3 D−8 ( z) =− exp Erfc( zz++ 21 105 z + 105 z) 7! 4 2 2 z2 +−exp( zzz642 +++20 87 48) , 4 2 1 zzπ 8642 D−9 ( z) = exp Erfc( zz++++ 28 210 z 420 z 105) 8! 4 2 2 z2 −−exp( zz75 ++27 185 z 3 + 279 z) , 4 2 1 zzπ 97 5 3 D−10 ( z) =− exp Erfc( zz++ 36 378 z + 1260 z + 945 z) 9! 4 2 2 z2 +−++++exp( zz864235 345 z 975 z 384) , 4 2 1 zzπ 10 8 D−11 ( z) = exp Erfc( zz+ 45 10! 4 2 2 ++630zzz642 3150 + 4725 + 945) z2 −−exp( zz97 ++44 588 z 5 + 2640 z 3 + 2895 z) , 4 2 1 zzπ 11 9 D−12 ( z) =−+exp Erfc( zz55 11! 4 2 2 ++990zz75 6930 + 17325 z 3 + 10395 z) (C.2) z2 +−exp( zz10 ++54 8 938 z 6 + 6090 z4 + 12645 z2 + 3840) . 4 DOI: 10.4236/apm.2019.91002 40 Advances in Pure Mathematics A. Wünsche In applications (e.g., quantum mechanics) one has often to do with the scaled n ′ Parabolic Cylinder functions Dnn( zz) ≡ ( 2D) ( 2) which we do not expli- citly write down. Appendix D. Some Algebraic Relations for the Jacobi Polynomials We collect here a few algebraic relations for the Jacobi polynomials which are interesting for the derivation of relations between the Associated Hermite poly- nomials. The only general basic recurrence relation for the Jacobi polynomials with stable upper indices (αβ, ) is the three-term relation (αβ, ) 0= 2(nn + 1)( +++αβ 12)( n ++ αβ) Pn+1 ( u) 22(αβ, ) −(2n +++αβ 12)(( nn ++ αβ)( 2 +++ αβ 2) u + α − β) Pn ( u) (D.1) (αβ, ) +2(nn +α)( + β)(2 n +++ αβ 2P) n−1 (u) , connecting neighbored terms with lower indices (n+−1, nn , 1) is well known (e.g., [3], chap. 10.8. (11)). In addition there are possible a number of so-called contiguous relations where the upper indices (αβ, ) are involved with differ- ences by integers. We give in the following a few basic ones of them. There exists the following recurrence relations with only one lower index n but with unstable upper indices (αβ, ) and constant coefficients (see also [3] (10.8) Equations (34)-(37)) (αβ, ) (αβ,−− 1) (α1, β) (n++αβ)Pnn( u) =( n +β) P( un) +( +α) P, n( u) (D.2) and the following recurrence relation with two lower indices n and n −1 and unstable upper indices (αβ, ) (αβ,−− 1) (α1, β) Pnn−1 (uuu) = P( ) − P. n( ) (D. 3) Both can be directly proved using basic definitions of the Jacobi polynomials or their explicit representations (e.g., [5] [18]). A recurrence relation with only one lower index but contiguous upper indices and with coefficients which de- pend on the variable u is (αβ−−1, 1) 11−+uu(αβ,− 1) (αβ−1, ) P(uuu) = P( ) + P.( ) (D.4) n22 nn It also can be proved from the definition or from the basic explicit representa- tions of the Jacobi polynomials. By linear combination of the above three conti- guous relations which are complete as the fundamental ones we may obtain some other but no more fundamental forms. All these identities are true for general argument u. We now derive an identity which is only true for the special argument u = 0 of the Jacobi polynomials. Setting u = 0 in the identity (D.4) we find (αβ−−1, 1) 1 (αβ,− 1) (αβ−1, ) P( 0) = P( 0) + P( 0.) (D.5) n2 ( nn) DOI: 10.4236/apm.2019.91002 41 Advances in Pure Mathematics A. Wünsche and if we substitute first the parameter αα→+1 and second the parameter ββ→+1 we obtain from (5) the two identities (αβ,1−) (α+−1,1 β ) (αβ,) 2Pnn( 0) = P( 0) + P n( 0) , (αβ−1, ) ( αβ, ) ( αβ−+1, 1) 2Pn( 0) = P nn( 0) + P( 0) . (D.6) The contiguous relation (D.2) specialized for argument u =0 can be writ- ten (αβ, ) (αβ,−− 1) (α1, β) 2(n++αβ) Pnn( 0) =( nn +β) 2P( 0) +( +α) 2P n( 0) . (D.7) Substituting now the Jacobi polynomials on the right-hand side according to the identities (D.6) one obtains a relation which by some shortening of expres- sions on both sides can be represented in the form (αβ, ) (α+−1, β 1) (α−+1, β 1) (αβ+)P0P0nn( ) =+(nnβ) ( ) ++( α) P0. n( ) (D.8) As already announced by the derivation this identity is specific for the argu- ment u = 0 of the Jacobi polynomials and cannot be generalized to arbitrary u ≠ 0 by simply substituting only the argument zero of the Jacobi polynomials by an arbitrary one. We mention in addition the following general relation between Jacobi poly- nomials with different arguments [5] (chap. IV, (4.22.1)) n (αβ, ) 13+−uu(α,2−n −−− α β 1) n (βα, ) Pnn(uu) =P =−−( 1P) n( ) . (D.9) 21 + u It is applied in (5.4) to get two different representations of the formula for ν Hen ( z) . DOI: 10.4236/apm.2019.91002 42 Advances in Pure Mathematics