Summed series involving \,_{1}F_{2} hypergeometric functions
Jack C. Straton

TL;DR
This paper extends previous work on series involving _{1}F_{2} hypergeometric functions by deriving new doubly and trebly infinite summed series from Chebyshev and Gegenbauer polynomial expansions of Bessel functions.
Contribution
It introduces methods to generate additional infinite series involving _{1}F_{2} hypergeometric functions from polynomial expansions of Bessel functions, expanding the scope of summable series.
Findings
Derived doubly infinite summed series from Chebyshev expansions
Established trebly infinite summed series from Gegenbauer expansions
Connected series values to inverse powers of primes
Abstract
In a prior paper we found that the Fourier-Legendre series of a Bessel function of the first kind J_{N}\left(kx\right) and of a modified Bessel functions of the first kind I_{N}\left(kx\right) lead to an infinite set of series involving \,_{1}F_{2} hypergeometric functions (extracted therefrom) that could be summed, having values that are inverse powers of the eight primes 1/\left(2^{i}3^{j}5^{k}7^{l}11^{m}13^{n}17^{o}19^{p}\right) multiplying powers of the coefficient k, for the first 22 terms in each series. The present paper shows how to generate additional, doubly infinite summed series involving \,_{1}F_{2} hypergeometric functions from Chebyshev polynomial expansions of Bessel functions, and trebly infinite sets of summed series involving \,_{1}F_{2} hypergeometric functions from Gegenbauer polynomial expansions of Bessel functions.
Peer Reviews
No public reviews on file for this paper yet. If you reviewed it on a platform where reviews are public (OpenReview, ICLR, NeurIPS, ICML), you can paste yours below so the community can read it here.
Videos
No videos yet. Explain this paper in a talk, walkthrough, or lecture? Add one.
Taxonomy
TopicsMathematical functions and polynomials · Advanced Mathematical Identities · Functional Equations Stability Results
