A CLASSICAL THEOREM ON THE SINGULARITIES OF LEGENDRE SERIES IN C-3 AND ASSOCIATED SYSTEM OF HYPERBOLIC PARTIAL DIFFERENTIAL EQUATIONS

摘要

A classical theorem of Nehari relates the singularities of Legendre series expansions in C-z with those of associated Taylor's series in C-t. The generalization of Nehari's theorem is known for Legendre series in C-z1 x z2. In this paper, function theoretic methods develop the analogous relationships between the singularities of series expanded as triple products of Legendre polynomials in C-z1 x z2 x z3 and those of associated analytic functions in C-t. The singularities of these generalized Legendre series are determined by certain elliptic curves. Moreover, these series satisfy a system of hyperbolic partial differential equations (PDEs) in C-3 that are connected to Bochner's study of Poisson processes in R-2. [References: 18]