Related papers: Transformation and summation formulas for basic hy…
We deduce several curious q-series expansions by applying inverse relations to certain identities for basic hypergeometric series. After rewriting some of these expansions in terms of q-integrals, we obtain, in the limit q -> 1, some…
We study the quotient of hypergeometric functions \begin{equation*} \mu_{a}^*(r)=\frac{\pi}{2\sin{(\pi a)}}\frac{F(a,1-a;1;1-r^3)}{F(a,1-a;1;r^3)} \quad (r\in(0,1)) \end{equation*} in the theory of Ramanujan's generalized modular equation…
In this paper we prove some Ramanujan-type formulas for $1/\pi$ but without using the theory of modular forms. Instead we use the WZ-method created by H. Wilf and D. Zeilberger and find some hypergeometric functions in two variables which…
Recently, the authors with Lea Beneish established a recipe for constructing Ramanujan-Sato series for $1/\pi$, and used this to construct 11 explicit examples of Ramanujan-Sato series arising from modular forms for arithmetic triangle…
Using the WZ-method we find some of the easiest Ramanujan's formulae and also some new interesting Ramanujan-like sums.
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…
Each of Ramanujan's series for $\frac{1}{\pi}$ is of the form $$ \sum_{n=0}^{\infty} z^n \frac{ (a_{1})_{n} (a_{2})_{n} (a_{3})_{n} }{ (b_{1})_{n} (b_{2})_{n} (b_{3})_{n} } (c_{1} n + c_2) $$ for rational parameters such that the difference…
Several terminating generalizations of Ramanujan's formula for $\frac{1}{\pi}$ with complete WZ proofs are given.
Via symbolic computation we deduce 97 new type series for powers of $\pi$ related to Ramanujan-type series. Here are three typical examples: $$\sum_{k=0}^\infty \frac{P(k) \binom{2k}k\binom{3k}k…
By means of the extended Gould-Hsu inverse series relations, we find that the dual relation of Dougall's summation theorem for the well--poised $_7F_6$-series can be utilized to construct numerous interesting Ramanujan--like infinite series…
In this paper, we propose a general method to express explicitly the inversion and the connection coefficients between two basic hypergeometric polynomial sets. As application, we consider some $d$-orthogonal basic hypergeometric…
We state and prove a number of unilateral and bilateral $q$-series identities and explore some of their consequences. Those include certain generalizations of the $q$-binomial sum which also generalize the $q$-Airy function introduced by…
In this note, we shall provide several properties of hypergeometric Bernoulli numbers and polynomials, including sums of products identity, differential equations and recurrence formulas.
A new class of integrals involving the confluent hypergeometric function ${}_1F_{1}(a;c;z)$ and the Riemann $\Xi$-function is considered. It generalizes a class containing some integrals of S. Ramanujan, G.H. Hardy and W.L. Ferrar and gives…
This paper demonstrates the power of the calculus developed in the two previous parts of the series for all real forms of the almost Hermitian symmetric structures on smooth manifolds, including e.g. conformal Riemannian and almost…
The bilateral series corresponding to many of the third-, fifth-, sixth- and eighth order mock theta functions may be derived as special cases of $_2\psi_2$ series \[ \sum_{n=-\infty}^{\infty}\frac{(a,c;q)_n}{(b,d;q)_n}z^n. \] Three…
We find new hypergeometric identities which, in a certain aspect, are stron-ger than others of the same style found by the author in a previous paper. The identities in Section \ref{section-pi} are related to some Ramanujan-type series for…
Some rapidly convergent formulae for special values of the Riemann zeta function are given. We obtain a generating function formula for zeta(4n+3) which generalizes Apery's series for zeta(3), and appears to give the best possible series…
In the lecture notes we start off with an introduction to the $q$-hypergeometric series, or basic hypergeometric series, and we derive some elementary summation and transformation results. Then the $q$-hypergeometric difference equation is…
Symmetry analysis of Ramanujan's system of differential equations is performed by representing it as a third-order equation. A new system consisting of a second-order and a first-order equation is derived from Ramanujan's system. The Lie…