Integral and series representations of q-polynomials and functions: Part III theta, Ramanujan and q-Bessel functions

Abstract

In this paper, we use an identity connecting a modified [Formula: see text]-Bessel function and a [Formula: see text] function to give [Formula: see text]-versions of entries in the Lost Notebook of Ramanujan. We also establish an identity which gives an [Formula: see text]-version of a partition identity. We prove new relations and identities involving theta functions, the Ramanujan function, the Stieltjes–Wigert, [Formula: see text]-Lommel and [Formula: see text]-Bessel polynomials. We introduce and study [Formula: see text]-analogues of the spherical Bessel functions.