We show that the three quantum states (Pólya states, the generalized non-classical states related to Hahn polynomials and negative hypergeometric states) introduced recently as intermediates states which interpolate between the binomial states and negative binomial states are essentially identical. By using the Hermitial-phase-operator formalism, the phase properties of the hypergeometric state...

Exton introduced 20 distinct triple hypergeometric functions whose names are Xi (i = 1, . . . , 20) to investigate their twenty Laplace integral representations whose kernels include the confluent hypergeometric functions 0F1, 1F1, a Humbert function Ψ1, and a Humbert function Φ2. The object of this paper is to present 18 new integral representations of Euler type for the Exton hypergeometric f...

In [7], Greene introduced the notion of general hypergeometric series over finite fields or Gaussian hypergeometric series, which are analogous to classical hypergeometric series. The motivation for his work was to develop the area of character sums and their evaluations through parallels with the theory of hypergeometric functions. The basis for this parallel was the analogy between Gauss sums...

In this paper, we propose a matrix-variate generalization of the Gauss hypergeometric distribution and study several of its properties. We also derive probability density functions of the product of two independent random matrices when one of them is Gauss hypergeometric. These densities are expressed in terms of Appell’s first hypergeometric function F1 and Humbert’s confluent hypergeometric f...

In 2012, H. M. Srivastava et al. [37] introduced and studied a number of interesting fundamental properties and characteristics of a family of potentially useful incomplete hypergeometric functions. The definitions of these incomplete hypergeometric functions were based essentially upon some generalization of the Pochhammer symbol by mean of the incomplete gamma functions γ(s,x) and Γ (s,x). Ou...

It is suggested that the (p, q)-hypergeometric series studied by Burban and Klimyk (in Integral Transforms and Special Functions 2 (1994) 15-36) can be considered as a special case of a more general (P, Q)-hypergeometric series. 1. Introduction I propose a general (P, Q)-hypergeometric series. In this, I am inspired mainly by the paper of Burban and Klimyk [1] titled " P, Q-differentiation, P, ...

Using functional equations, we derive noncommutative extensions of Ramanujan's 1 ψ 1 summation. 1. Introduction. Hypergeometric series with noncommutative parameters and argument, in the special case involving square matrices, have been the subject of recent study, see e.g. the papers by Duval and Ovsienko [DO], Grünbaum [G], Tirao [T], and some of the references mentioned therein. Of course, t...

We consider the following problem: Given a nested sum expression, find a sum representation such that the nested depth is minimal. We obtain a symbolic summation framework that solves this problem for sums defined, e.g., over hypergeometric, q-hypergeometric or mixed hypergeometric expressions. Recently, our methods have found applications in quantum field theory.

We express the real period of a family of elliptic curves in terms of classical hypergeometric series. This expression is analogous to a result of Ono which relates the trace of Frobenius of the same family of elliptic curves to a Gaussian hypergeometric series. This analogy provides further evidence of the interplay between classical and Gaussian hypergeometric series.

Elliptic hypergeometric series form a natural generalization of hypergeometric and basic hypergeometric (or q-) series. It is surprising that they were introduced only very recently, by Frenkel and Turaev [FT], who expressed the 6j-symbols corresponding to certain elliptic solutions of the Yang–Baxter equation, cf. [DJ], in terms of the 10ω9-sums defined below. It is expected that elliptic hype...