### A formulation of the Jacobi coefficients $c^l_j(\alpha, \beta)$ via Bell polynomials

#### Abstract

The Jacobi polynomials $(\mathscr{P}^{(\alpha, \beta)}_k: k \ge 0, \alpha, \beta >-1)$ are deeply intertwined with the Laplacian on compact rank one symmetric spaces. They represent the spherical or zonal functions and as such constitute the main ingredients in describing the spectral measures and spectral projections associated with the Laplacian on these spaces. In this note we strengthen this connection by showing that a set of spectral and geometric quantities associated with Jacobi operator fully describe the Maclaurin coefficients associated with the heat and other related Schwartzian kernels and present an explicit formulation of these quantities using the Bell polynomials.

#### Article information

Source
Adv. Oper. Theory, Volume 2, Number 4 (2017), 506-515.

Dates
Accepted: 28 July 2017
First available in Project Euclid: 4 December 2017

https://projecteuclid.org/euclid.aot/1512431725

Digital Object Identifier
doi:10.22034/aot.1705-1163

Mathematical Reviews number (MathSciNet)
MR3730044

Zentralblatt MATH identifier
06804225

#### Citation

Day, Stuart; Taheri, Ali. A formulation of the Jacobi coefficients $c^l_j(\alpha, \beta)$ via Bell polynomials. Adv. Oper. Theory 2 (2017), no. 4, 506--515. doi:10.22034/aot.1705-1163. https://projecteuclid.org/euclid.aot/1512431725

#### References

• R. O. Awonusika and A. Taheri, Harmonic analysis on symmetric and locally symmetric spaces, preprint.
• R. O. Awonusika and A. Taheri, Spectral invariants on compact symmetric spaces: From heat trace to functional determinants, preprint.
• E. T. Bell, Exponential polynomials, Ann. of Math. 35 (1934), 258–277.
• M. Craioveanu, M. Puta, and Th. M. Rassias, Old and new aspects in spectral geometry, Mathematics and Its Applications, 534, Kluwer Academic Publishers, 2001.
• I. S. Gradshtejn and I. M. Ryzhik, Table of integrals series and products, Academic Press, 2007.
• S. Helgason, Topics in harmonic analysis on homogeneous Spaces, Birkhäuser, 1981.
• T. H. Koornwinder, The addition formula for Jacobi polynomials: 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.
• B. Osgood, R. Phillips, and P. Sarnak, Extremals and determinants of Laplacians, J. Funct. Anal. 80 (1988), no. 1, 148–211.
• P. Sarnak, Determinants of Laplacians, Comm. Math. Phys. 110 (1987), no. 1, 113–120.
• A. Taheri, Function spaces and partial differential equations. Vol.1 & Vol. 2, Oxford Lecture Series in Mathematics and its Applications, 40 & 41, Oxford University Press, Oxford, 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, 2009.