Tusi Mathematical Research Group (TMRG)Advances in Operator Theory2538-225X4120190101Operators of Laplace transform type and a new class of hypergeometric coefficients2262506577210.15352/aot.1804-1356ENStuart BondUniversity of Sussex, UKAli TaheriUniversity of Sussex, UKJournal Article20180430A differential identity on the hypergeometric function ${}_2F_1(a,b;c;z)$ unifying and extending certain spectral results on the scale of<br /> Gegenbauer and Jacobi polynomials and leading to a new class of hypergeometric related scalars $mathsf{c}_j^m(a,b,c)$ and<br /> polynomials $mathscr{R}_m=mathscr{R}_m(X)$ is established. The Laplace-Beltrami operator on a compact rank one symmetric<br /> space is considered next and for operators of the Laplace transform type by invoking an operator trace relation, the Maclaurin spectral<br /> coefficients of their Schwartz kernel are fully described. Other representations as well as extensions of the differential<br /> identity to the generalised hypergeometric function ${}_pF_q({bf a}; {bf b}; z)$ are formulated and proved.http://www.aot-math.org/article_65772_1103f2a4dda7ee03f7174c00c3490a99.pdf