Operators of Laplace transform type and a new class of hypergeometric coefficients

Abstract

‎‎‎A differential identity on the hypergeometric function ${}_2F_1(a,b;c;z)$ unifying and extending certain spectral results on the scale of‎ ‎Gegenbauer and Jacobi polynomials and leading to a new class of hypergeometric related scalars $\mathsf{c}_j^m(a,b,c)$ and‎ ‎polynomials $\mathscr{R}_m=\mathscr{R}_m(X)$ is established‎. ‎The Laplace-Beltrami operator on a compact rank one symmetric‎ ‎space is considered next‎, ‎and for operators of the Laplace transform type by invoking an operator trace relation‎, ‎the Maclaurin spectral‎ ‎coefficients of their Schwartz kernel are fully described‎. ‎Other related representations as well as extensions of the differential‎ ‎identity to the generalized hypergeometric function ${}_pF_q(\textbf{a}; \textbf{b}; z)$ are formulated and proved‎.