Some extensions of Ramanujan's 1-psi-1 summation formula
  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. 
N. M. Vildanov
math.CA