Hypergeometric functions and the Tricomi operator.pdf

a r X i v : m a t h / 0 3 1 0 4 8 0 v 1 [ m a t h .A P ] 3 0 O c t 2 0 0 3 Hypergeometric functions and the Tricomi operator J. Barros-Neto? Rutgers University, Hill Center 110 Frelinghuysen Rd, Piscataway, NJ 08854-8019 e-mail: jbn@math.rutgers.edu Fernando Cardoso? Departmento de Matema?tica, Universidade Federal de Pernambuco 50540-740 Recife, Pe, Brazil e-mail: fernando@dmat.ufpe.br Abstract In this paper we show how certain hypergeometric functions play an important role in finding fundamental solutions for a generalized Tricomi operator. 1 Introduction In this article we consider the operator T = y?x + ? 2 ?y2 , (1.1) in Rn+1, where ?x = ∑n j=1 ?2 ?x2j , n ≥ 1. This is a natural generalization of the classical Tricomi operator in R2 already considered by us in the article [2] where it was called generalized Tricomi operator. ?Partially supported by NSF, Grant # INT 0124940 ?Partially supported by CNPq (Brazil) 1 In that article we obtained, by the method of partial Fourier transforma- tion, explict expressions for fundamental solutions to T , relative to points on the hyperplane y = 0. That lead us to calculate inverse Fourier trans- forms of Bessel functions which, in turn, revealed the importance of certain hypergeometric functions (depending on the “space dimension” n) that are intimately related to the operator T . In the present article we look for fundamental solutions of T relative to an arbitrary point (x0, y0), located in the hyperbolic region (y 0) of the op- erator, and which are supported by the “forward” characteristic conoid of T with vertex at (x0, y0).We follow the method of S. Delache and J. Leray in [5] where they introduced hypergeometric distributions, a notion also considered by I. M. Gelfand and G. E. Shilov in [7]. The plan of this article is the following. In Section 2 we deal with pre- liminary material that is needed throughout the paper. Hypergeometric dis- tributions are introduced in Section 3 where we obtain the basic for