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

A new multivariable 6-psi-6 summation formula

By multidimensional matrix inversion, combined with an A_r extension of Jackson's 8-phi-7 summation formula by Milne, a new multivariable 8-phi-7 summation is derived. By a polynomial argument this 8-phi-7 summation is transformed to another multivariable 8-phi-7 summation which, by taking a suitable limit, is reduced to a new multivariable extension of the nonterminating 6-phi-5 summation. The latter is then extended, by analytic continuation, to a new multivariable extension of Bailey's very-well-poised 6-psi-6 summation formula.

Ramanujan Summation and the Exponential Generating Function $ \sum_{k=0}^{\infty}\frac{z^{k}}{k!}ζ^{\prime}(-k)$

In the sixth chapter of his notebooks Ramanujan introduced a method of summing divergent series which assigns to the series the value of the associated Euler-MacLaurin constant that arises by applying the Euler-MacLaurin summation formula to the partial sums of the series. This method is now called the Ramanujan summation process. In this paper we calculate the Ramanujan sum of the exponential generating functions $\sum_{n\geq 1}\log n e^{nz}$ and $\sum_{n\geq 1}H_n^{(j)} e^{-nz}$ where $H_n^{(j)}=\sum_{m=1}^n \frac{1}{m^j}$. We find a surprising relation between the two sums when $j=1$ from w…

Another proof of Bailey's 6-psi-6 summation

Adapting a method used by Cauchy, Bailey, Slater, and more recently, the second author, we give a new proof of Bailey's celebrated 6-psi-6 summation formula.

Nombres de Bernoulli et une formule de Ramanujan

In the first part we establish a connection between the Euler-Maclaurin summation formula and the Rota-Baxter functional equation. In the second part we give a simple proof of a formula, due to Ramanujan, on the summation of certain exponential series.

Ramanujan and Koshliakov Meet Abel and Plana

The neglected Russian mathematician, N.~S.~Koshliakov, derived beautiful generalizations of the classical Abel--Plana summation formula through a setting arising from a boundary value problem in heat conduction. When we let the parameter $p$ in this setting tend to infinity, his formulas reduce to the classical Abel--Plana summation formula. Rigorous formulations and proofs of these summation formulas are given. In his notebooks, Ramanujan derived different analogues of the Abel--Plana summation formula. One particular example provides a vast new generalization of the classical transformation …

Elementary derivations of identities for bilateral basic hypergeometric series

We give elementary derivations of several classical and some new summation and transformation formulae for bilateral basic hypergeometric series. For purpose of motivation, we review our previous simple proof ("A simple proof of Bailey's very-well-poised 6-psi-6 summation", Proc. Amer. Math. Soc., to appear) of Bailey's very-well-poised 6-psi-6 summation. Using a similar but different method, we now give elementary derivations of some transformations for bilateral basic hypergeometric series. In particular, these include M. Jackson's very-well-poised 8-psi-8 transformation, a very-well-poised …

A Purely Algebraic Summation Method

It is mathematical folklore that 1 + 2 + 3 + 4 + ... = --1/12. This result is usually achieved using elaborate analytical methods, such as zeta function regularization or Ramanujan summation. However, in its notebooks, Ramanujan has also provided a very simple derivation which relied instead on algebraic manipulations. Recently, a video from Numberphile has presented a similar derivation of the result (provoking lots of discussions and debates about the meaning of such an equality). But this derivation, simple as it is, is usually considered as less rigorous than those using more elaborate ana…

Voronoi求和公式、Riesz和的振荡，以及Ramanujan-Guinand型与Cohen型恒等式

Voronoi summation formulas, oscillations of Riesz sums, and Ramanujan-Guinand and Cohen type identities

本研究推导了刘维尔函数、莫比乌斯函数及除数函数平方的Voronoi求和公式，得到多类恒等式特例，并在相关数论猜想假设下证明了对应Riesz和及其误差项的振荡性质。

Summation formulae for the bilateral basic hypergeometric series ${}_1ψ_1 ( a; b; q, z )$

We give summation formulae for the bilateral basic hypergeometric series ${}_1ψ_1( a; b; q, z )$ through Ramanujan's summation formula, which are generalizations of nontrivial identities found in the physics of three-dimensional Abelian mirror symmetry on $\mathbf{R}P^2 \times S^1$. We also show the $q \to 1 - 0$ limit of our summation formulae.

Multilateral inversion of A_r, C_r and D_r basic hypergeometric series

In [Electron. J. Combin. 10 (2003), #R10], the author presented a new basic hypergeometric matrix inverse with applications to bilateral basic hypergeometric series. This matrix inversion result was directly extracted from an instance of Bailey's very-well-poised 6-psi-6 summation theorem, and involves two infinite matrices which are not lower-triangular. The present paper features three different multivariable generalizations of the above result. These are extracted from Gustafson's A_r and C_r extensions and of the author's recent A_r extension of Bailey's 6-psi-6 summation formula. By combi…

Applications of Lipschitz summation formula and a generalization of Raabe's cosine transform

General summation formulas have been proved to be very useful in analysis, number theory and other branches of mathematics. The Lipschitz summation formula is one of them. In this paper, we give its application by providing a new transformation formula which generalizes that of Ramanujan. Ramanujan's result, in turn, is a generalization of the modular transformation of Eisenstein series $E_k(z)$ on SL$_2(\mathbb{Z})$, where $z\to-1/z, z\in\mathbb{H}$. The proof of our result involves delicate analysis containing Cauchy Principal Value integrals. A simpler proof of a recent result of ours with …

广义Zwegers $μ$-函数的退化形式根据Ramanujan差分方程

A degeneration of the generalized Zwegers' $μ$-function according to the Ramanujan difference equation

本文引入了小$μ$-函数，它是广义$μ$-函数的退化极限。通过$q$-Borel求和得到该函数，用于处理Ramanujan方程的发散解。同时给出了小$μ$-函数的对称性和连接公式，以及与$q,t$-斐波那契序列和罗杰斯-拉马努金连分数相关的连续关系。

Abel-Rothe type generalizations of Jacobi's triple product identity

Using a simple classical method we derive bilateral series identities from terminating ones. In particular, we show how to deduce Ramanujan's 1-psi-1 summation from the q-Pfaff-Saalschuetz summation. Further, we apply the same method to our previous q-Abel-Rothe summation to obtain, for the first time, Abel-Rothe type generalizations of Jacobi's triple product identity. We also give some results for multiple series.