### 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.

#### Article information

Source
Adv. Oper. Theory, Volume 4, Number 1 (2019), 226-250.

Dates
Accepted: 15 July 2018
First available in Project Euclid: 27 July 2018

https://projecteuclid.org/euclid.aot/1532656922

Digital Object Identifier
doi:10.15352/aot.1804-1356

Mathematical Reviews number (MathSciNet)
MR3867343

Zentralblatt MATH identifier
06946452

#### Citation

Bond, Stuart; Taheri, Ali. Operators of Laplace transform type and a new class of hypergeometric coefficients. Adv. Oper. Theory 4 (2019), no. 1, 226--250. doi:10.15352/aot.1804-1356. https://projecteuclid.org/euclid.aot/1532656922

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