Advances in Operator Theory

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

Stuart Bond and Ali Taheri

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‎‎‎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

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

Received: 30 April 2018
Accepted: 15 July 2018
First available in Project Euclid: 27 July 2018

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 47F05: Partial differential operators [See also 35Pxx, 58Jxx] (should also be assigned at least one other classification number in section 47)
Secondary: 33C05‎ ‎33C20‎ ‎ 33C45‎‎ ‎ 47B25‎ 47D06‎ 47E05‎ ‎58J35 ‎

Schwartz kernel‎ operator of Laplace transform type‎ ‎Laplace-Beltrami operator ‎‎hypergeometric function ‎ Maclaurin spectral function ‎symmetric space‎


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.

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