Generalized Hermite polynomials associated with functions of parabolic cylinder

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Abstract

Associated Hermite polynomials Heνn(z) which generalize the usual (scaled) Hermite polynomials Hen(z) corresponding to ν=0 are introduced for the purpose to represent the raising and lowering of the indices of functions of the parabolic cylinder Dν(z) in finite integer steps. Properties of these polynomials such as recursion relations, explicit representations and the differential equation are derived. The generation of the associated Hermite polynomials from the usual Hermite polynomials by differential operators representable by means of the confluent hypergeometric function is given. An application for the explicit calculation of the functions of the parabolic cylinder for negative integer indices is discussed. Other applications are visible for the investigation of the zeros of the functions of the parabolic cylinder.

Introduction

The most interesting generalizations of well-established functions are usually born from necessities in applications when the known notions become too narrow. In case of present paper it was the problem to represent the functions of the parabolic cylinder Dν(z) (e.g. [24]) for negative integers ν=−1,−2,… in a closed way by superposition of the functions D0(z) and D−1(z) multiplied by some polynomials in z that is possible as one can inductively see examining D−2(z), D−3(z) and so on. Then it was found that the solution of the problem can be obtained in a wider range of more general problems and in this way we present here the solution. It is similar to the introduction of Lommel polynomials associated to Bessel functions [5], [12], [23].

Apart from the well-known embedding of Hermite polynomials into the confluent hypergeometric functions, e.g. [5], [12], [16], [21], [22], and the extension to multi-variable Hermite polynomials, e.g. [9], [10], [12] and to Hermite 2D polynomials [26], [27], [29], [30], some other specific generalizations were considered in literature. One such generalization are the Gould–Hopper generalized Hermite polynomials (see [21]; this work contains a very extensive bibliography of full-cited titles on 59 pages)Hmn(z,y)≡∑n/mk=0n!k!(n−mk)!ykzn−mk(m=1,2,…),which for m=1 contain the binomial expansion and for m=2 the usual Hermite polynomials in the transformed two-variable form of Appell and Kampé de Fériet [2]. They were a few times rediscovered and different aspects have been investigated, e.g. [4], [6], [8], [18]. Another generalization of the Hermite polynomials uses their relation to associated Laguerre–Sonin polynomials with semi-integer upper indices and with squared arguments [5] (Chapter V.2) but it seems to us that this generalization is not of very principal necessity. Properties of polynomials associated to the classical orthogonal polynomials and their fourth-order differential equations are considered in [3], [19], [20], [31] (see also [17]).

Section snippets

Some basic properties of functions of parabolic cylinder

We collect in this section some basic properties of the functions of the parabolic cylinder [12], [16], [24] which we need more or less in the following.

The functions of the parabolic cylinder Dν(z) are solutions of the following ordinary linear second-order differential equation for a complex variable z and with ν as a (complex) parameter (Weber equation)∂2∂z2−z24+ν+12fν(z)=0.It is important, for example, in scaled form in the discussion of the quantum-mechanical harmonic oscillator. There are

Recursion relations and lowering and raising operators

From definition (2.3) in connection with the differential equation (2.1), one finds the three-term recursion relationDν+1(z)−zDν(z)+νDν−1(z)=0and, furthermore, a three-term relation involving the first derivative of the functions of the parabolic cylinderDν+1(z)+2∂∂zDν(z)−νDν−1(z)=0,which can serve for the calculation of this derivative. From these relations, one can derive the following lowering and raising operations for the indices of the functions of the parabolic cylinderz2+∂∂zDν(z)=νDν−1

Polynomials associated to raising of indices of functions of the parabolic cylinder

We construct in this section a set of polynomials which is related to the raising of the indices of the functions of the parabolic cylinder. The starting point is the recursion relation (3.1) for functions of the parabolic cylinder which we write by substitution ν→ν+n in the formDν+n+1(z)=zDν+n(z)−(ν+n)Dν+n−1(z)n=0,1,…We can successively express all terms on the right-hand side by means of Dν(z) and Dν−1(z) that leads for the first two cases to the following explicit relationsDν+1(z)=zDν(z)−νD

Polynomials associated to lowering of indices of functions of the parabolic cylinder

Before we continue to derive properties of the associated Hermite polynomials Heνn(z), we show that the lowering of the indices of the functions of the parabolic cylinder can be solved by the same kind of polynomials as the raising of the indices.

Starting point of the derivations is here the same recursion relation (3.1) which we write for our purpose in the formνDν−1(z)=zDν(z)−Dν+1(z),n=0,1,…for arbitrary ν or by substitution ν→ν−n(ν−n)Dν−n−1(z)=zDν−n(z)−Dν−n+1(z).Similar to the case of

Further relations and differential equation for the associated Hermite polynomials

The recursion relation (4.10) connects associated Hermite polynomials with equal upper index ν. In addition, there exists a relation which connects associated Hermite polynomials with different upper indices. We consider for this purpose the difference Heνn+1(z)−Heν−1n+1(z) for which we find from the explicit representation (4.12) after substitution j→j+1 and k→k+1 of the summation indicesHeνn+1(z)−Heν−1n+1(z)=−∑k=0(n−1)/2(−1)k(n−1−2k)!∑kj=0(n−j)!(ν−1+j)!j!(k−j)!(ν−1)!2k−jzn−1−2kthat leads to

Generation of associated Hermite polynomials from usual Hermite polynomials

The first of the Eqs. (6.8) gives an operator for raising the upper indices ν of the associated Hermite polynomials Heνn(z). We can begin the procedure of raising the upper indices with ν=0 and come to the following relationHeνn(z)=∑n/2j=0(−1)j(n−j)!(ν−1+j)!j!n!(ν−1)!∂2j∂z2jHen(z).The proof uses the formula of the differentiation of the usual scaled Hermite polynomials∂m∂zmHen(z)=∑(n−m)/2l=0(−1)ln!l!(n−m−2l)!2lzn−m−2l=n!(n−m)!Hen−m(z).By inserting this relation in (7.1) in the special cases m=2j

Special cases of the associated Hermite polynomials

In Section 4 (Eq. (4.13)), we have given the explicit form of the first eight initial associated Hermite polynomials Heνn(z). We consider now some special cases. For ν=0, we obtain the usual scaled Hermite polynomials He0n(z)=Hen(z) with the following first 8 casesHe00(z)=1,He01(z)=z,He02(z)=z2−1,He03(z)=z3−3z,He04(z)=z4−6z2+3,He05(z)=z5−10z3+15z,He06(z)=z6−15z4+45z2−15,He07(z)=z7−21z5+105z3−105z.For ν=1, we find from , He1n(z)=∑n/2k=0=(−1)k(n−2k)!∑kj=0(n−j)!(k−j)!2k−jzn−2kwith the following

Some applications

We now discuss some applications of the derived relations between functions of the parabolic cylinder with different indices.

In the special case ν=0, from (4.3) followsDn(z)=He0n(z)D0(z)=Hen(z)exp−z24.This is well known. We consider now an arbitrary zero z=z0 of the function Dν−1(z) that meansDν−1(z0)=0.Then from (4.3) followsDν+n(z0)=Heνn(z0)Dν(z0).

We now consider the special case ν=−1 of relation (5.4) for which followsD−1−n(z)=(−1)nn!{(−i)nHe0n(iz)D−1(z)−(−i)n−1He1n−1(iz)D0(z)}=(−1)nn!(−i)nHe

Conclusion

We have introduced associated Hermite polynomials and discussed some of their most important properties such as recursion relations, the differential equation which they obey and explicit representations. The derivation of generating functions was not successful up to now and concerning orthogonality or biorthogonality relations is maybe not spoken the last word. The associated Hermite polynomials Heνn(z) are related to the functions of the parabolic cylinder Dν(z) in a similar way as the

Acknowledgements

The author is very grateful to Professors G. Dattoli from Frascati. A. Ronveaux from Bruxelles, and H.M. Srivastava from Victoria (British Columbia) for valuable discussions and for urging him after his Workshop lecture to calculate the differential equation for the associated Hermite polynomials Heνn(z). In particular, he thanks Professor A. Ronveaux who transmitted him very quickly after the Workshop exactly the true equation (6.11) with support on the paper [31].

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