Computer-Assisted Proofs of Some Identities for Bessel Functions of Fractional Order

TL;DR

This paper uses computer algebra algorithms to prove identities involving Bessel functions of fractional order, restoring proofs that were previously lost in major mathematical references.

Contribution

It introduces a novel application of symbolic summation and generating functions to re-establish known but unproven identities for Bessel functions of fractional order.

Findings

Successfully proved identities from the Handbook of Mathematical Functions and DLMF.

Demonstrated the effectiveness of computer algebra in restoring lost proofs.

Provided new proofs for classical identities involving special functions.

Abstract

We employ computer algebra algorithms to prove a collection of identities involving Bessel functions with half-integer orders and other special functions. These identities appear in the famous Handbook of Mathematical Functions, as well as in its successor, the DLMF, but their proofs were lost. We use generating functions and symbolic summation techniques to produce new proofs for them.