Ramanujan provided several results involving the modified Bessel function [Formula: see text] in his Lost Notebook. One of them is famous Ramanujan–Guinand formula, equivalent to functional equation non-holomorphic Eisenstein series on SL[Formula: text]. Recently, this formula was generalized by Dixit, Kesarwani, and Moll. In paper, we first obtain a generalization theorem Watson and, as an app...

For ν an unrestricted real (or complex) number, let Iν be the modified Bessel function of the first kind of order ν, defined by [1, p. 77] Iν(x) = ∑ n≥0 (x/2)2n+ν n!0(ν + n+ 1) , which occurs frequently in problems of electrical engineering, finite elasticity, quantum billiards, wave mechanics, mathematical physics and chemistry, etc. Here x is an arbitrary real (or complex) number, and as usua...

Abstract: We define q-analogues of Fourier-Bessel series, by means of complete qorthogonal systems constructed with the third Jackson q-Bessel function. Sufficient conditions for pointwise convergence of these series are obtained, in terms of a general convergence principle valid for other Fourier series on grids defined over numerable sets. The results are illustrated with specific examples of...

The discrete Painlevé III equation is investigated based on the bilinear formalism. It is shown that it admits the solutions expressed by the Casorati determinant whose entries are given by the discrete Bessel function. Moreover, based on the observation that these discrete Bessel functions are transformed to the q-Bessel functions by a simple variable transformation, we present a q-difference ...

the theory of c-frames and c-bessel mappings are the generalizationsof the theory of frames and bessel sequences. in this paper, weobtain several equivalent conditions for dual of c-bessel mappings.we show that for a c-bessel mapping $f$, a retrievalformula with respect to a c-bessel mapping $g$ is satisfied if andonly if $g$ is sum of the canonical dual of $f$ with a c-besselmapping which  wea...