Associated Hermite Polynomials Related to Parabolic Cylinder Functions

In analogy to the role of Lommel polynomials ( ) Rn z ν in relation to Bessel functions ( ) J z ν the theory of Associated Hermite polynomials in the scaled form ( ) Hen z ν with parmeter ν to Parabolic Cylinder functions ( ) D z ν is developed. The group-theoretical background with the 3-parameter group of motions ( ) 2 M in the plane for Bessel functions and of the Heisenberg-Weyl group ( ) 2 W for Parabolic Cylinder functions is discussed and compared with formulae, in particular, for the lowering and raising operators and the eigenvalue equations. Recurrence relations for the Associated Hermite polynomials and for their derivative and the differential equation for them are derived in detail. Explicit expressions for the Associated Hermite 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.


Introduction
Let us motivate our intentions by an analogy of Bessel functions 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 polynomials 1 ( [ ] ) ( ) Whereas Lommel derives his polynomials by a somewhat cumbersome induction ( [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 function which in similar form was introduced by Hurwitz [4] (cited on page 302 in [2]).
By an analogous process one may relate the Parabolic Cylinder functions ( ) ) ( ) where ( ) ( ) denotes the Jacobi polynomials introduced in this form by Szegö [5] and here taken for argument 0 u = .The polynomials (1.4)

( )
He n z ν 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.

D z ν
The Weber equation with parameter ν ∈  ( ) with important application in physics (e.g., quantum mechanics) is satisfied, for example, by the following two independent solutions ( ) with the Wronski determinant 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, 0 z = .
From the two linearly independent solutions ( ) where we used in the second representation the Kummer transformation of the Confluent Hypergeometric function (e.g., [9], (chap.(6.3, Equation ( 7))) and with A second independent solution of the Weber equation can be constructed in various ways from superpositions of the kind with coefficients ( ) , , , α β γ δ but maximally only two of the four basic functions can be linearly independent solutions of the Weber equation.For example ( ) ( ) ( ) is such a relation between three of these functions [3] (chap.8.2.( 6)).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]) A few but very important special cases of these definitions which can be also and (e.g., [10], p. 309, 12.7 (iii)) ( ) is a standard notation for a category of Bessel functions (e.g., [3], chap.7.2.2.Equation ( 13), [11], p. 251) ( ) ( ) ( ) ( ) Erf u denotes the Error function defined by 2 ( ) 12) The function is the Complementary Error function.

Series Representations of the Parabolic Cylinder Functions
If we insert in (2.4) the well-known Taylor series of the Confluent Hypergeometric function ) where ( ) k ν − denotes the Pochhammer symbol according to the definition Using the first of the given representations of the Pochhammer symbol ( ) or separated in the even and odd part 3 The Pochhammer symbol ( ) k a 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). ( We gave here only Taylor series for ( )

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 ( ) Sections 2 and 3 one can derive formulae for the differentiation of this function.
The simplest form they take on if we differentiate ( ) or may check the known formulae (e.g., [3] [10]) These relations can be written We call 2 A , respectively 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-  forms a basis of a countably infinite irreducible unitary representation of this Lie algebra characterized by the index ν within the interval 0 1 ν ≤ < .The simplest eigenvalue equations for arbitrary ν are However, only the functions ( )  from these eigenfunctions prove themselves to be normalizable.Only for the irreducible representation with index 0 ν = with the basis functions ( ) ( )  onto functions ( ) From (4.2) we find first the derivative of ( ) and second, the 3-term recurrence relation Using the following general disentanglement of linear combinations of operators z and z ∂ ∂ [14] ( ) ( ) which can be proved by complete induction 4 one finds from (4.2) (see also (3.3) for representation with the Pochhammer symbol) For 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.
The second formula fails to act for nonnegative integers 0,1, Alternatively we find from (4.1) Inserting in the first of these formulae 0 ν = and in the second

Representation of Parabolic Cylinder Functions by Two Neighbored Basic Ones with the Recurrence Relation
Applying the recurrence relations (4.7) to ( ) with coefficients depending on variable z of the form (see also The general form of the polynomials ( ) 3) The inner sum in this expression can be expressed in two different ways (due to transformations relations) by the Jacobi polynomials ( ) ( ) For the initial members The given representations of ( ) He n z ν can be proved, for example, by complete 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 Applying this identity and substituting 1 ν ν → − in (5.2) this formula can be written in the form which, in particular, is advantageous in the special cases of integer ν , in partic- ular, 0 ν = 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 0 ν = the involved Jacobi polynomials in (5.4) are eva- luated to as one may see also from (5.3) and the sequence of polynomials ( ) He n z possesses the form This means that they possess a form of scaled Hermite polynomials ( ) The right-hand side may be considered as disentanglement in normal ordering of the operators on the left-hand side.Other forms of operator disentanglement can be found in [14].
In general, the Parabolic Cylinder function 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 ( )

D z
− as basis functions one may obtain more symmetric representations that we do not try to do here.

Recurrence Relations for the Associated Hermite Polynomials
Recurrence relations for the Associated Hermite polynomials

( )
He n z ν were derived 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 polynomials and of their differentiation it is useful to know some algebraic relations for the Jacobi polynomials.First of all these are recurrence relations.Since beside the variable u we have in the Jacobi polynomials ( ) ( ) two kinds of indices, the lower index 0,1, 2, n =  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 and use now the representation (5.1) leading to the following form of the left-hand side of (6.1) If we apply for the Parabolic Cylinder functions on the right-hand side of (6.1) the same representation (5.1) we find Since the representation of the Parabolic Cylinder functions by two neighbored such functions is unique one finds by comparison of (6.2) with (6.3) the recurrence relation (compare also Drake [8] (Equation ( He He He . For 0 ν = it provides the recurrence relation for the polynomials ( ) He n z closely related to that for the usual Hermite polynomials ( ) He He where we applied the identity (D.3).If we now apply the recurrence relation (D.2) for the Jacobi polynomials ( ) ( ) with fixed lower indices n but varying upper indices with the substitutions , , Inserting this with 0 u = in (6.5) we find an identity of the Associated Her- mite polynomials in the form He He He He He He He .
where we used (7.3) for the transformation of the result.This means that the Associated Hermite polynomials satisfy the equation For 0 ν = we obtain the differential equation for modified Hermite polyno- mials The full elimination of the derivative ( ) He He He .
Calculating the derivative from (5.4) we find Using herein the identity (D.8) of Appendix D with the substitutions He !P 0 P 0 2 !

He
He .
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 possibilities with respect to the order of application which, clearly, have to lead to the same result.The first is and the second By transition to normal ordering (all powers of z stand in front of all powers where ( ) The Bessel functions ( ) 2, ; .
where the sequences of polynomials

( )
He n z ν of n-th degree are given by

Z Z
and the operator of the rotation by 3 Z then we have the commuta- tion relations for ( ) We now form the new Lie-group operators ( )

, ,
With respect to the Bessel functions ( ) J z ν we consider the following realization of the operators ( )

.
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 ( ) A. Wünsche DOI: 10.4236/apm.2019.9100216 Advances in Pure Mathematics vides the possibility to express it successively by the sum of two Bessel functions of n-th degree of variable 1 z − th degree in variable z which possess the explicit form

1 We
are asso- ciated to the Hermite polynomials in a scaled form and are for 0 ν = identical with the scaled Hermite polynomials usually denoted by change slightly the standard notation of the Lommel polynomials [1] [2] [3] according to

4 )
These representations fail to act in the important special cases of non-negative integers 0,1, 2, n ν = =  due to infinities in numerators and denominator.Using the second of the given representations of the Pochhammer symbol ( ) k

5 )
This representation fails to provide the result without limiting transitions for negative integers 1, 2, 3, n ν = − = − − −  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 representation of the Confluent Hypergeometric function (3.1) a series representation of a way that they are vanishing for real z x = in the limit x → +∞ for arbitrary real indices ν ∈  .
by A and † n z with 0 n ≥ acting onto them with the lowering operator A it arises the impression that this representation breaks down with 0 n = for ( ) 0 D 0 A z = but coming from below and acting with the raising operator † A onto functions ( ) D n z with 0 n < we do not have a breakdown at 0 n = and all basis functions ( ) D n z ± are related also in this representation.

1 ν = − and using ( 2 . 10 )
we find known formulae for the Parabolic Cylinder functions ( ) D z ν 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 generate the Parabolic Cylinder functions from two basic such functions by multiplication with a certain kind of polynomials which generalize the Hermite polynomials ( ) H n z in the scaled form usually denoted by ( ) He n z .
polynomials of degree n depending on ν as a parameter.For reason which we see below we call them associated Hermite polynomials.In the same way by the same recurrence relations (4.7) one may suc-

4 )
the sum consists only of the term to 0 k = and due to ( ) ( )

.
He n z .In the Appendices A and B we give some initial members of the sequences ( ) He n z ν for integer and for semi-integer ν in explicit form.In Appendix C we give explicit formulae for the Parabolic Cy-We mention that by means of the polynomials ( ) He n z the operator disentanglement formula (4.8) can be represented in the form ( 2 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 2) the 3-term relations reduce to 2-term relations, for example, in case of a zero of Optimization problems for relative extrema in some problems may lead to the search for zeros of functions in series of the form 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 functions is particularly interesting for the irreducible representation of 10) as such.This case concerns also the most important applications in quantum mechanics.In case of the irreducible representation of the Parabolic the recurrence relations for the Associated Hermite polynomials can be made using the recurrence relations for the Parabolic Cylinder functions (4.7) which by the substitution n ν ν → + we write for this purpose in the form c c u is a Hypergeometric function in its standard nota- tion and ( ) n ν the Pochhammer symbol.The polynomials ( )

8 ) 5 Therefore≡
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 Lommel polynomials involves the Hypergeometric function a a c c c u 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 paper.Beside the mentioned similarities in both systems there are also essential differences.The most striking one is that5 .A certain feeling why this is so one may attain if one compares or similar would be better instead of the notation for the Lommel polynomials based on literature.

1 I
of A + and A − becomes exchanged (beside sign changes) that preserves the commutation relations (9.3).The identity operator I in the realization = can be expressed by the Lie-algebra operators according to[16]

) 2 M
countably infinite irreducible representation of the (inhomogeneous) group of motions ( of the real two-dimensional Euclidean plane which can be represented as the semi-direct product to change the realizations of the lowering and raising operators and of the eigenvalue operator ( ) in comparison to (9.4) that we do not write down.members of the sequence of Parabolic Cylinder functions with negative integer indices represented in the form (5.2) or better (5.7) with 0 ν = and using (2.10) are (definition of Complementary Error function