Some extensions of Ramanujan's 1-psi-1 summation formula

TL;DR

This paper extends Ramanujan's 1-psi-1 summation formula by introducing parameterized summations and a q-analogue of the binomial theorem, connecting to q-beta integrals and bibasic integrals.

Contribution

It introduces new summation formulas extending Ramanujan's psi sum, including a parameterized form and a bibasic integral representation, advancing q-series theory.

Findings

Derived a summation formula with parameter ta=1/N, N positive integer.

Connected the summation to a q-beta integral as N.

Established a q-analogue of the generalized binomial theorem.

Abstract

We have found several summation formulas that extend Ramanujan's psi sum. First contains a parameter $\alpha=1/N$, $N$ is a positive integer, and transforms to $q$-beta integral in the limit $N\to\infty$. The other is a $q$-analogue of generalized binomial theorem, expansion of $(1+t)^a$ in powers of $t^\alpha$, $0<\alpha\leq 1$, and it expresses the sum of a certain series in terms of a bibasic integral.