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

Permanent link to this document
https://projecteuclid.org/euclid.aot/1532656922

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

Mathematical Reviews number (MathSciNet)
MR3867343

Zentralblatt MATH identifier
06946452

Subjects
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 ‎

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

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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References

  • M. Abramowitz and I. A. Stegun, Handbook of mathematical functions with formulas, graphs, and mathematical tables, Applied Mathematics Series 55, 1983.
  • G. E. Andrews, The theory of partitions, Encyclopedia of Mathematics and its Applications 2, Cambridge University Press, 1976.
  • G. E. Andrews, R. Askey, and R. Roy, Special functions, Encyclopedia of Mathematics and its Applications 71, Cambridge University Press, Cambridge, 1999.
  • R. O. Awonusika and A. Taheri, On Jacobi polynomials $(P_k^{(\alpha, \beta)} : \alpha, \beta>-1)$ and Maclaurin spectral functions on rank one symmetric spaces, J. Anal. 25 (2017), 139–166.
  • R.O. Awonusika and A. Taheri, On Gegenbauer polynomials and coefficients $c_j^l(\nu)$, Results Math. 72 (2017), 1359–1367.
  • R. O. Awonusika and A. Taheri, A spectral identity on Jacobi polynomials and its analytic implications, Canad. Math. Bull., to appear.
  • D. Bakry, I. Gentil, and M. Ledoux, Analysis and geometry of Markov diffusion operators, Grundlehren der Mathematischen Wissenschaften 348, Springer, 2008.
  • W. Beckner, Sobolev inequalities, the Poisson semigroup and analysis on the sphere ${\mathbb S}^n$, Proc. Nat. Acad. Sci. USA 89 (1992), 4816–4819.
  • E. T. Bell, Exponential polynomials, Ann. of Math. 35 (1934), 258–277.
  • M. Berger, P. Gauduchon, and E. Mazet, Le Spectre Dúne Variétè Riemannienne, Springer, 1971.
  • R. S. Cahn and J. A. Wolf, Zeta functions and their asymptotic expansions for compact symmetric spaces of rank one, Comm. Math. Helv. 51 (1976), 1–21.
  • S. Day and A. Taheri, A formulation of the Jacobi coefficients via Bell polynomials, Adv. Oper. Theory 2 (2017), 506–515.
  • S. Day and A. Taheri, Semigroup asymptotics, Funk-Hecke identity and the Gegenbauer coefficients associated with the spherical Laplacian, Rocky Mount. J. Math., To appear 2018.
  • J. Dolbeault, M. J. Esteban, M. Kowalczyk, and M. Loss, Sharp interpolation inequalities on the sphere: New methods and consequences, Chinese Ann. Math. 34 (2013), 99–112.
  • N. Dunford and J. T. Schwartz, Linear operators I-III, Wiley Classics in Mathematics, Wiley-Blackwell, New Ed 1988.
  • A. Erdélyi, Higher transcendental functions, McGraw-Hill, 1953.
  • G. Gasper and M. Rahman, Basic hypergeometric series, Encyclopedia of Mathematics and its Applications 96, Cambridge University Press, 2004.
  • S. Helgason, Eigenspaces of the Laplacian; integral representations and irreducibility, J. Funct. Anal. 17 (1974), 328–353.
  • S. Helgason, Differential geometry, Lie groups, and symmetric spaces, Academic Press, 1978.
  • T. H. Koornwinder, The addition formula for Jacobi polynomials: $\textbf{I}$ Summary of results, Indag. Math. 34 (1974), 188–191.
  • T. H. Koornwinder, A new proof of a Paley-Wiener type theorem for the Jacobi transform, Ark. Matematik 13 (1975), 145–159.
  • J. Letessier, G. Valent, and J. Wimp, Some differential equations satisfied by hypergeometric functions, International Series of Numerical Mathematics 119, Birkhauser, 1994.
  • H. McKean and I. M. Singer, Curvature and the eigenvalues of the Laplacian, J. Diff. Geom. 1 (1967), 43–69.
  • H. Mulholland, An asymptotic expansion for $\sum(2n+1)e^{-\sigma(n+1/2)^2}$, Proc. Camb. Phil. Soc. 24 (1928), 280–289.
  • B. Osgood, R. Phillips, and P. Sarnak, Extremals and determinants of Laplacians, J. Funct. Anal. 80 (1988), 148–211.
  • I. Polterovich, Heat invariants of Riemannian manifolds, Israel J. Math. 119 (2000), 239–252.
  • P. Sarnak, Determinants of Laplacians, Comm. Math. Phys. 110 (1987), no. 1, 113–120.
  • R. Seeley, Complex powers of an elliptic operator, In: Singular Integrals, Proc. Sympos. Pure Math., Chicago, III, pp. 288-307, Amer. Math. Soc. Providence, R.I., 1966.
  • A. Taheri, Function spaces and partial differential equations I & II, Oxford Lecture Series in Mathematics and its Applications 40 & 41, Oxford University Press, 2015.
  • N. J. Vilenkin, Special functions and the theory of group representations, Translations of Mathematical Monographs 22, Amer. Math. Soc., 1968.
  • V. V. Volchkov and V. V. Volchkov, Harmonic analysis of mean periodic functions on symmetric spaces and the Heisenberg group, Springer Monographs in Mathematics, Springer-Verlag London, Ltd., London, 2009.
  • G. Warner, Harmonic analysis on semisimple Lie groups I & II, Springer, 1972.