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

Bond, Stuart and Taheri, Ali (2019) Operators of Laplace transform type and a new class of hypergeometric coefficients. Advances in Operator Theory, 4 (1). pp. 226-250. ISSN 2538-225X

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 representations as well as extensions of the differential identity to the generalised hypergeometric function ${}_pF_q({\bf a}; {\bf b}; z)$ are formulated and proved.