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

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 $t^{\alpha }$, $0<\alpha \le 1$, and it expresses the sum of a certain series in terms of a bibasic integral.

arXiv comment14 pages