On polynomials of binomial type, Ramanujan-Soldner constant and inverse logarithmic derivative operator

Some binomial series obtained by the WZ-method

Using the WZ-method we find some of the easiest Ramanujan's formulae and also some new interesting Ramanujan-like sums.

关于第11级Ramanujan-Sato级数的一个问题

On a problem on Ramanujan-Sato series of level 11

本文解决了关于第11级Ramanujan-Sato级数系数$c_{11}(n)$的长期未解问题，通过引入并证明一个双二项式求和公式，直接类比较低级系数公式，成功给出了所有非负整数$n$的显式表达式。

A rapidly converging Ramanujan-type series for Catalan's constant

In this note, by making use of a known hypergeometric series identity, I prove two Ramanujan-type series for the Catalan's constant. The convergence rate of these central binomial series surpasses those of all known similar series, including a classical formula by Ramanujan and a recent formula by Lupas. Interestingly, this suggests that an Apéry-like irrationality proof could be found for this constant.

Type-(I,II) Interpolations and some asymptotic expansions using Ramanujan's master theorem

The theory of Mellin transform is an incredibly useful tool in evaluating some of the well known results for the zeta function. Ramanujan in his quarterly reports \cite{1} gave a theorem for Mellin transform which is now known as Ramanujan's master theorem \cite{2}. In this paper, we have derived some extended versions of Ramanujan's master theorem based on our previous results \cite{3} and applied them to some special functions such as known as the Riesz function $R(z)$ and generalized binomial function. Some asymptotic expansions using extended Ramanujan's master theorem are also derived.

On the q-binomial identities involving the Legendre symbol modulo 3

I use polynomial analogue of the Jacobi triple product identity together with the Eisenstein formula for the Legendre symbol modulo 3 . to prove six identities involving the $q$-binomial coefficients. These identities are then extended to the new infinite hierarchies of q-series identities by means of the special case of Bailey's lemma. Some of the identities of Ramanujan, Slater, McLaughlin and Sills are obtained this way.

Tiling proofs of Jacobi triple product and Rogers-Ramanujan identities

We use the method of tiling to give elementary combinatorial proofs of some celebrated $q$-series identities, such as Jacobi triple product identity, Rogers-Ramanujan identities, and some identities of Rogers. We give a tiling proof of the q-binomial theorem and a tiling interpretation of the q-binomial coefficients. A new generalized $ k $-product $q$-series identity is also obtained by employing the `tiling-method', wherein the generating function of the set of all possible tilings of a rectangular board is computed in two different ways to obtain the desired $q$-series identity. Several new…