Related papers: Some summation theorems for truncated Clausen seri…
The Watson-Harkins sum involving the product of the cosine and cosecant functions is extended to derive the finite sum of generalized Hurwitz-Lerch Zeta functions is derived in terms of the Hurwitz-Lerch Zeta function. A transformation…
Following a previous article we continue our study on non-terminating hypergeometric series with one free parameter, which aims to find arithmetical constraints for a given hypergeometric series to admit a gamma product formula. In this…
This paper presents a family of rapidly convergent summation formulas for various finite sums of analytic functions. These summation formulas are obtained by applying a series acceleration transformation involving Stirling numbers of the…
We provide numerical procedures for possibly best evaluating the sum of positive series. Our procedures are based on the application of a generalized version of Kummer's test.
It is known that Plotkin's reduction theorem is very important for his theory of universal algebraic geometry [arXiv:math. GM/0210187], [arXiv:math. GM/0210194]. It turns out that this theorem can be generalized to arbitrary categories…
We present some elementary derivations of summation and transformation formulas for q-series, which are different from, and in several cases simpler or shorter than, those presented in the Gasper and Bahman [1990] "Basic Hypergeometric…
We study the arithmetic property which allows to sharpen number-theoretic estimates. Previous results on this property are, as a rule, quantitive. The application of our general qualitive theorems to generalized hypergeometric functions…
In recent years, Z.-W. Sun proposed several sophisticated conjectures on congruences for finite sums with terms involving combinatorial sequences such as central trinomial coefficients, Domb numbers and Franel numbers. These sums are double…
In this paper, we establish three new and general transformations with sixteen parameters and bases via Abel's lemma on summation by parts. As applications, we set up a lot of new transformations of basic hypergeometric series. Among…
Employing a quadratic transformation formula of Rahman and the method of `creative microscoping' (introduced by the author and Zudilin in 2019), we provide some new $q$-supercongruences for truncated basic hypergeometric series. In…
We define a truncated Euler polynomial $E_{m,n}(x)$ as a generalization of the classical Euler polynomial $E_n(x)$. In this paper we give its some properties and relations with the hypergeometric Bernoulli polynomial.
In this document, some elements of the theory and algorithmics corresponding to the existence and computability of approximate joint eigenpairs for finite collections of matrices with applications to model order reduction, are presented.…
In this paper, we obtain analytical solutions of Laplace transform based some generalized class of the hyperbolic integrals in terms of hypergeometric functions ${}_3F_2 (\pm1)$, ${}_4F_3 (\pm1)$, ${}_5F_4(\pm1)$, ${}_6F_5(\pm1)$,…
A summation is a shift-invariant ${\rm R}$-module homomorphism from a submodule of ${\rm R}[[\sigma]]$ to ${\rm R}$ or another ring. [11] formalized a method for extending a summation to a larger domain by telescoping. In this paper, we…
We consider the ratio of two Gauss hypergeometric functions, in which the parameters of the numerator function differ from the respective parameters of the denominator function by integers. We derive explicit integral representations for…
The main result of the paper gives criteria for extendibility of sesquilinear form-valued mappings defined on symmetric subsets of *-semigroups to positive definite ones. By specifying this we obtain new solutions of: * the truncated…
We principally present reductions of certain generalized hypergeometric functions $_3F_2(\pm 1)$ in terms of products of elementary functions. Most of these results have been known for some time, but one of the methods, wherein we…
Slowly convergent or divergent sequences and series occur abundantly in quantum physics and quantum chemistry. These convergence problems can be overcome with the help of nonlinear sequence transformations (Wynn's epsilon or rho algorithm,…
The predictions of the geometric collective model (GCM) for different sets of Hamiltonian parameter values are related by analytic scaling relations. For the quartic truncated form of the GCM -- which describes harmonic oscillator, rotor,…
Given a Feynman parameter integral, depending on a single discrete variable $N$ and a real parameter $\epsilon$, we discuss a new algorithmic framework to compute the first coefficients of its Laurent series expansion in $\epsilon$. In a…