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

Document Type: Special issue: Trends in Operators on Banach Spaces

Authors

1 University of Sussex, UK

2 ‎University of Sussex, UK

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 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.

Keywords

Main Subjects

### Histroty

• Receive Date: 30 April 2018
• Revise Date: 15 July 2018
• Accept Date: 15 July 2018