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

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 $α=1/N$, $N$ 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 $t^α$, $0<α\leq 1$, and it expresses the sum of a certain series in terms of a bibasic integral.

Noncommutative extensions of Ramanujan's 1-psi-1 summation

Using functional equations, we derive noncommutative extensions of Ramanujan's 1-psi-1 summation.

A new A_n extension of Ramanujan's 1-psi-1 summation with applications to multilateral A_n series

In this article, we derive some identities for multilateral basic hypergeometric series associated to the root system A_n. First, we apply Ismail's argument to an A_n q-binomial theorem of Milne and derive a new A_n generalization of Ramanujan's 1-psi-1 summation theorem. From this new A_n 1-psi-1 summation and from an A_n 1-psi-1 summation of Gustafson we deduce two lemmas for deriving simple A_n generalizations of bilateral basic hypergeometric series identities. These lemmas are closely related to the Macdonald identities for A_n. As samples for possible applications of these lemmas, we pro…

$q$-类比的Wilf-Zeilberger种子与Ramanujan $1/π^k$-公式

$q$-analogues of Wilf-Zeilberger seeds and Ramanujan $1/π^k$-formulas

本文提出Wilf-Zeilberger种子的概念，用于生成$q$-类比的Ramanujan型$1/π^k$公式，并揭示了部分非模形式的模伪模实例。

A simple proof of Bailey's very-well-poised 6-psi-6 summation

We give elementary derivations of some classical summation formulae for bilateral (basic) hypergeometric series. In particular, we apply Gauss' 2-F-1 summation and elementary series manipulations to give a simple proof of Dougall's 2-H-2 summation. Similarly, we apply Rogers' nonterminating 6-phi-5 summation and elementary series manipulations to give a simple proof of Bailey's very-well-poised 6-psi-6 summation. Our method of proof extends M. Jackson's first elementary proof of Ramanujan's 1-psi-1 summation.

New summation and transformation formulas of the Poisson, Müntz, Möbius and Voronoi type

Basing on properties of the Mellin transform and Ramanujan's identities, which represent a ratio of products of Riemann's zeta- functions of different arguments in terms of the Dirichlet series of arithmetic functions, we obtain a number of the Poisson, Müntz, Möbius and Voronoi type summation formulas. The corresponding analogs of the Müntz operators are investigated. Interesting and curious particular cases of summation formulas involving arithmetic functions are exhibited. Necessary and sufficient conditions for the validity of the Riemann hypothesis are derived.