Related papers: Summations of Linear Recurrent Sequences
We systematically exploit a new generalized hypergeometric identity to obtain new hypergeometric summation formulas. As a consistency test, alternative proofs for some special cases are also provided. As a byproduct new summation formulas…
We present efficient methods for calculating linear recurrences of hypergeometric double sums and, more generally, of multiple sums. In particular, we supplement this approach with the algorithmic theory of contiguous relations, which…
The article addresses the problem whether indefinite double sums involving a generic sequence can be simplified in terms of indefinite single sums. Depending on the structure of the double sum, the proposed summation machinery may provide…
Several new identities for elliptic hypergeometric series are proved. Remarkably, some of these are elliptic analogues of identities for basic hypergeometric series that are balanced but not very-well-poised.
We introduce several new identities combining basic hypergeometric sums and integrals. Such identities appear in the context of superconformal index computations for three-dimensional supersymmetric dual theories. We give both analytic…
We perform certain alternating binomial summations with parameters that occur in the analysis of algorithms. A combination of integral and special function and special number representations is used. The results are sufficiently general to…
Motivated by the substantial development of the special functions, we contribute to establish some rigorous results on the general series identities with bounded sequences and hypergeometric functions with different arguments, which are…
We prove hypergeometric type summation identities for a function defined in terms of quotients of the $p$-adic gamma function by counting points on certain families of hyperelliptic curves over $\mathbb{F}_{q}$. We also find certain special…
By means of inversion techniques and several known hypergeometric series identities, summation formulas for Fox-Wright function are explored. They give some new hypergeometric series identities when the parameters are specified.
In a recent paper (Appl. Math. Comput. 215, 1622--1645, 2009), the authors proposed a method of summation of some slowly convergent series. The purpose of this note is to give more theoretical analysis for this transformation, including the…
Polynomial reduction, designed first for hypergeometric terms, can be used to automatically prove and generate new hypergeometric identities from old ones. In this paper, we extend the reduction method to holonomic sequences. As…
Recursive formulas extending some known $_{2}F_{1}$ and $_{3}F_{2}$ summation formulas by using contiguous relations have been obtained. On the one hand, these recursive equations are quite suitable for symbolic and numerical evaluation by…
Power series in which the summand satisfies a linear recurrence relation with polynomial coefficients are shown to be the solution of a linear differential or algebraic equation. Solving the associated differential or algebraic equation…
By employing certain extended classical summation theorems, several surprising \pi and other formulae are displayed.
In this paper we extend the notion of Melham sum to the Pell and Pell-Lucas sequences. While the proofs of general statements rely on the binomial theorem, we prove some spacial cases by the known Pell identities. We also give extensions of…
In this paper we give a systematized treatment to some coincidence situations for multiple summing multilinear mappings which extend, generalize and simplify the methods and results obtained thus far. The application of our general results…
We state and prove a general summation identity. The identity is then applied to derive various summation formulas involving the generalized harmonic numbers and related quantities. Interesting results, mostly new, are obtained for both…
The considered problem is uniform convergence of sequences of hypergeometric series. We give necessary and sufficient conditions for uniformly dominated convergence of infinite sums of proper bivariate hypergeometric terms. These conditions…
Some examples of naturally arising multisum $q$-series which turn out to have representations as fermionic single sums are presented. The resulting identities are proved using transformation formulas from the theory of basic hypergeometric…
The classical hypergeometric summation theorems are exploited to derive several striking identities on harmonic numbers including those discovered recently by Paule and Schneider (2003).