Related papers: Heine's method and $A_n$ to $A_m$ transformation f…
Early 17th-century mathematical publications of Johann Faulhaber contain some remarkable theorems, such as the fact that the $r$-fold summation of $1^m,2^m,...,n^m$ is a polynomial in $n(n+r)$ when $m$ is a positive odd number. The present…
The classical summation and transformation theorems for very well-poised hypergeometric functions, namely, $_{5}F_4(1)$ summation, Dougall's $_{7}F_6(1)$ summation, Whipple's $_{7}F_6(1)$ to $_{4}F_3(1)$ transformation and Bailey's…
In the previous work we introduced the higher order $q$-Painlev\'{e} system $q$-$P_{(n+1,n+1)}$ as a generalization of the Jimbo-Sakai's $q$-Painlev\'{e} VI equation. It is derived from a $q$-analogue of the Drinfeld-Sokolov hierarchy of…
Factorization -- a simple form of standardization -- is concerned with reduction strategies, i.e. how a result is computed. We present a new technique for proving factorization theorems for compound rewriting systems in a modular way, which…
We extend the well-known Rainwater-Simons convergence theorem to various generalized convergence methods such as strong matrix summability, statistical convergence and almost convergence. In fact we prove these theorems not only for…
The method of double extension, introduced by A.~Medina and Ph.~Revoy, is a procedure which decomposes a Lie algebra with an invariant symmetric form into elementary pieces. Such decompositions were developed for other algebras, for…
We present generalizations of the Newton-Raphson-Simpson method. Specifically, for a positive integer $m$ and the sequence of coefficients of a Taylor series of a function $f(z)$, we define an algorithm we denote by NRS($m$) which is a way…
An operatorial method, already employed to formulate a generalization of the Ramanujan master theorem, is applied to the evaluation of integrals of various type. This technique provide a very flexible and powerful tool yielding new results…
Using the framework relating hypergeometric motives to modular forms, we define an explicit family of weight 2 Hecke eigenforms with complex multiplication. We use the theory of ${}_2F_1(1)$ hypergeometric series and Ramanujan's theory of…
We study a one-parameter family of binomial-convolution operators acting on sequences. These operators form an additive semigroup with an explicit inverse, and they subsume iterated classical binomial transforms as a special case. We…
The divergent series for a function defined through Lapalce integral and the ground state energy of the quartic anharmonic oscillator to large orders are studied to test the generalized binomial transform which is the renamed version of…
In terms of the difference operators, we establish several curious transformation and summation formulas for basic hypergeometric series. When the parameters are specified, they produce $q$-analogues of Ramanujan's three series for 1/$\pi$…
Ingham studied two types of convolution sums of the divisor function, namely the shifted convolution sum $\sum_{n \le N} d(n) d(n+h)$ and the additive convolution sum $\sum_{n < N} d(n) d(N-n)$ for integers $N, h$ and derived their…
In this paper, transformation formulas for the function \[ A_{1}\left(z,s:\chi\right)=\sum\limits_{n=1}^{\infty}\sum\limits_{m=1}^{\infty}\chi\left(n\right)\chi\left(m\right)\left(-1\right)^{m}n^{s-1}e^{2\pi imnz/k} \] are obtained. Sums…
We show how the formulas in paper Variae considerationes circa series hypergeometricas written by Euler imply the duplication formula for the Gamma-function. This paper can be seen as an Addendum to a previous paper by the author.
The aim of this research paper is to obtain explicit expressions of (i) $ {}_1F_1 \left[\begin{array}{c} \alpha \\ 2\alpha + i \end{array} ; x \right]. {}_1F_1\left[ \begin{array}{c} \beta \\ 2\beta + j \end{array} ; x \right]$ (ii)…
A rapid transformation is derived between spherical harmonic expansions and their analogues in a bivariate Fourier series. The change of basis is described in two steps: firstly, expansions in normalized associated Legendre functions of all…
We consider the derivatives of Horn hypergeometric functions of any number variables with respect to their parameters. The derivative of the function in $n$ variables is expressed as a Horn hypergeometric series of $n+1$ infinite summations…
The Factorial Basis method, initially designed for quasi-triangular, shift-compatible factorial bases, provides solutions to linear recurrence equations in the form of definite-sums. This paper extends the Factorial Basis method to its…
Given a permutation, there is a well-developed literature studying the number of ways one can factor it into a product of other permutations subject to certain conditions. We initiate the analogous theory for the type A Iwahori-Hecke…