2023/06/05 by Howard S. Cohl, Roberto S. Costas-Santos, Cohl, Howard S. +7 · 1 citation
Mathematics · Physics and Astronomy · #Advanced Mathematical Identities #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Mathematical functions and polynomials #Quantum Mechanics and Non-Hermitian Physics
paper · pdf · doi:10.48550/arxiv.2306.03035
openalex publication_date 2023/06/05 · openalex created_date 2023/06/07 · openalex updated_date 2026/07/28
In this paper we review and derive hyperbolic and trigonometric double summation addition theorems for Jacobi functions of the first and second kind. In connection with these addition theorems, we perform a full analysis of the relation between symmetric, antisymmetric and odd-half-integer parameter values for the Jacobi functions with certain Gauss hypergeometric functions which satisfy a quadratic transformation, including associated Legendre, Gegenbauer and Ferrers functions of the first and second kind. We also introduce Olver normalizations of the Jacobi functions which are particularly useful in the derivation of expansion formulas when the parameters are integers. We introduce an application of the addition theorems for the Jacobi functions of the second kind to separated eigenfunction expansions of a fundamental solution of the Laplace-Beltrami operator on the compact and noncompact rank one symmetric spaces.