Related papers: Hypergeometric-Type Sequences
A constant term sequence is a sequence of rational numbers whose $n$-th term is the constant term of $P^n(\boldsymbol{x}) Q(\boldsymbol{x})$, where $P(\boldsymbol{x})$ and $Q(\boldsymbol{x})$ are multivariate Laurent polynomials. While the…
We give a systematic and unified discussion of various classes of hypergeometric type equations: the hypergeometric equation, the confluent equation, the F_1 equation (equivalent to the Bessel equation), the Gegenbauer equation and the…
A hypergeometric type equation satisfying certain conditions defines either a finite or an infinite system of orthogonal polynomials. We present in a unified and explicit way all these systems of orthogonal polynomials, the associated…
We prove a summation formula for a bilateral series whose terms are products of two basic hypergeometric functions. In special cases, series of this type arise as matrix elements of quantum group representations.
We construct a set $H$ of orthogonal polynomial sequences that contains all the families in the Askey scheme and the $q$-Askey scheme. The polynomial sequences in $H$ are solutions of a generalized first-order difference equation which is…
A sequence is difference algebraic (or D-algebraic) if finitely many shifts of its general term satisfy a polynomial relationship; that is, they are the coordinates of a generic point on an affine hypersurface. The corresponding equations…
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…
A supercongruence is a congruence between rational numbers modulo a power of a prime. In this paper, we give a technique for finding and algorithmically proving supercongruences by expressing terms as infinite series involving certain…
We generalize the known constructions of A-hypergeometric functions. In particular, we show that periods of middle dimension on affine or projective complex algebraic varieties are A-hypergeometric functions of coefficients of polynomial…
This paper describes a method to find a connection between combinatorial identities and hypergeometric series with a number of examples. Combinatorial identities can often be written as hypergeometric series with unit argument. In a number…
Pseudorandom number generators are required to generate pseudorandom numbers which have good statistical properties as well as unpredictability in cryptography. An m-sequence is a linear feedback shift register sequence with maximal period…
A hypergeometric type equation satisfying certain conditions defines either a finite or an infinite system of orthogonal polynomials. We present in a unified and explicit way all these systems of orthogonal polynomials, the associated…
The origin of this study is based on not only explicit formulas of finite sums involving higher powers of binomial coefficients, but also explicit evaluations of generating functions for this sums. It should be emphasized that this study…
In this paper, we investigate the relationships among hypergeometric series, truncated hypergeometric series, and Gaussian hypergeometric functions through some families of `hypergeometric' algebraic varieties that are higher dimensional…
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…
By dividing hypergeometric series representations of the inverse sine by sin^-1 (x) and integrating, new double series representations of integers and constants arise. Binomial coefficients and the sine integral are thus combined in double…
Sequences that are defined by multisums of hypergeometric terms with compact support occur frequently in enumeration problems of combinatorics, algebraic geometry and perturbative quantum field theory. The standard recipe to study the…
We study a new type of sequences whose elements are defined in terms of the position, sign and magnitude of another element of the sequence. The name ultra-recursive comes from the fact that these sequences possess terms that are generated…
Feynman integrals that have been evaluated in dimensional regularization can be written in terms of generalized hypergeometric functions. It is well known that properties of these functions are revealed in the framework of intersection…
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…