JOURNAL OF CONCRETE AND APPLICABLE MATHEMATICS VOLUME 3- 2005

504 Pages·7.786 MB·English

VOLUME 3,NUMBER 1 JANUARY 2005

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JOURNAL OF CONCRETE AND APPLICABLE MATHEMATICS,VOL.3,NO.1,9-29,2005,COPYRIGHT 2005 EUDOXUS PRESS,LLC9
Hermite-Kamp´e de F´eriet polynomials
and solutions of Boundary Value Problems
in the half-space
Giulia Maroscia
Universit`a di Roma “La Sapienza”, Dipartimento di Metodi e Modelli Matematici
per le Scienze Applicate, Via A.Scarpa, 16, 00161 Roma, Italia
e-mail: [email protected]
Paolo E. Ricci
Universit`a di Roma “La Sapienza”, Dipartimento di Matematica, P.le A. Moro, 2 00185
Roma, Italia - e-mail: [email protected]
Abstract
ExplicitsolutionsofmultidimensionalcanonicalBVPofparabolic,hyperbolic
and elliptic type in the half-space are derived by using operational methods and
series expansions in terms of the Hermite-Kamp´e de F´eriet polynomials.
2000 Mathematics Subject Classification. 33C45, 44A45, 35G15.
Key words and phrases. Hermite-Kamp´e de F´eriet (or Gould-Hopper) polynomials,
Operational calculus, Boundary value problems.
1 Introduction
In a preceding article [5], the two-dimensional polynomials considered by Hermite, and
subsequently studied by P. Appell and J. Kamp´e de F´eriet [1], H.W. Gould and A.T.
Hopper [12], (see also: [16], p. 76, Eq. 1.9 (6)), G. Dattoli et al. [7], were stressed in
order to obtain explicit solutions of all the canonical (parabolic, hyperbolic or elliptic)
BVP in the half-plane.
For shortness, in the following, we will use the abbreviation H-KdF to denote the
Hermite-Kamp´e de F´eriet polynomials.
In the present article, by using the same operational approach recalled in [6], we
presentheresomegeneralizationsoftheresultsobtainedin[5]tothecaseofthesolution
of all the canonical BVP in the half-space.
According to our results, the two-dimensional H-KdF polynomials appear as the
natural tool for representing the solutions of all these problems.
It is worth noting that the use of operational methods in this framework naturally
leadstotheconsiderationoffractionaloperators. Howeverinalltheconsideredcases, it
10 G.MAROSCIA,P.RICCI
is possible to exploit the peculiarity of problems in order to avoid fractional derivatives,
sothattherelevantsolutionsarealwaysexpressedintermsofsuitableseriesexpansions
for which a general convergence criterion is proved.
Theconvergenceconditionsonthedataexpressedbythiscriterionarenotnecessary.
It should be interesting to find more general conditions to be satisfied by the data in
order to ensure convergence for the formal expansions, but the main motivation of this
article is to show the relevance of the H-KdF polynomials in the explicit solutions of
the considered problems.
We recall that, in the two-dimensional case, the H-KdF polynomials are closely re-
lated to the classical Hermite ones, and consequently to the Gauss’ normal distribution
which is the weight of the Hermite polynomials.
We conjectured in [5] that this was the hidden motivation of the possibility to
express the solutions of all the canonical BVP.
In any case, the importance of the H-KdF polynomials in the framework of the
multidimensional or multi-index special functions was recently recognized (see e.g. [7],
[4], [10]), and their importance even in the solution of BVP seems to be another
property which is worthy to be noted (see e.g. [5], [2], [15]).
2 Hermite-Kamp´e de F´eriet polynomials
We recall the definitions of the H-KdF polynomials, in the two-dimensional case.
Definition 2.1 The H-KdF polynomials in two variables H(2)(x,y) are defined by
m
[m]
(cid:88)2 ynxm−2n
H(2)(x,y) := m! (2.1)
m n!(m−2n)!
n=0
Definition 2.2 The H-KdF polynomials in two variables H(j)(x,y) are defined by
m
[m]
(cid:88)j ynxm−jn
H(j)(x,y) := m! (2.2)
m n!(m−jn)!
n=0
In a number of articles by G. Dattoli et al., (see e.g. [7], [8],[9]), by using the
so called monomiality principle, the following properties for the two-variable H-KdF
polynomials H(j)(x,y), j ≥ 2 have been recovered.
m
• Operational definition
(cid:195) (cid:33)
Hm(j)(x,y) = ey∂∂xjjxm = x+jy∂∂xjj−−11 m(1). (2.3)