Non-integer extension of integral identity (47) and integrability of Heine’s second term to remove integer restrictions

Construct a non-integer extension of the integral identity given in equation (47) for the integral ∫_0^π cos^m(θ) cos(nθ) dθ—currently used only with integer exponents—and determine whether the second term of Heine’s integral representation of the Bessel function Jν(kx cos(θ)) given in equation (36) can be integrated over both cos(θ) and t so as to remove the integer restriction on the parameters p and v in the Fourier–Legendre series derivation.

Background

The paper’s derivation of Fourier–Legendre series for Bessel functions, which underpins the main results on expressing certain 3F4 hypergeometric functions as sums of pair-products of 2F3 functions, relies on identities that are restricted to integer parameters. In particular, equation (47) provides an integral formula for ∫_0^π cos^m(θ) cos(nθ) dθ that the authors use with integer m and n to facilitate the truncation and orthogonality steps.

The authors note that while Heine’s integral representation (equation (36)) offers a more general framework, the present derivation remains limited by the lack of a non-integer analogue of equation (47). They further indicate that, if such an extension exists, one must investigate the integrability of the second term in Heine’s representation over both cos(θ) and t to remove the integer restrictions on the parameters p and v, thereby generalizing the series expansions to non-integer values.