mathematics, basic hypergeometric series, or q-hypergeometric series, are q-analogue generalizations of generalized hypergeometric series, and are in turn...

the Gaussian or ordinary hypergeometric function 2F1(a,b;c;z) is a special function represented by the hypergeometric series, that includes many other...

generalized hypergeometric series is a power series in which the ratio of successive coefficients indexed by n is a rational function of n. The series, if convergent...

hypergeometric series where the ratio is a rational function of n, and basic hypergeometric series where the ratio is a periodic function of the complex number...

functions (Handbuch der Kugelfunctionen). He also investigated basic hypergeometric series. He introduced the Mehler–Heine formula. Heinrich Eduard Heine...

theory of basic hypergeometric series, it plays the role that the ordinary Pochhammer symbol plays in the theory of generalized hypergeometric series. Unlike...

}}z^{n}} and their generalizations (such as basic hypergeometric series and elliptic hypergeometric series) frequently appear in integrable systems and...

scientific skepticism, freethinking and rationalism. He co-authored Basic Hypergeometric Series with George Gasper. This book is widely considered as the standard...

known results. The earliest q-analog studied in detail is the basic hypergeometric series, which was introduced in the 19th century. q-analogs are most...

William Barnes (1908, 1910). They are closely related to generalized hypergeometric series. The integral is usually taken along a contour which is a deformation...

119–120. doi:10.1093/imamat/34.1.119. Gasper; Rahman (2004). Basic Hypergeometric Series. Cambridge University Press. p.20-22. ISBN 978-0-521-83357-8...

1960) was an English clergyman and mathematician who worked on basic hypergeometric series. He introduced several q-analogs such as the Jackson–Bessel functions...

Bringmann and Ken Ono showed that certain q-series arising from the Rogers–Fine basic hypergeometric series are related to holomorphic parts of weight...

distribution q-Weibull diribution Tsallis q-Gaussian Tsallis entropy Basic hypergeometric series Elliptic gamma function Hahn–Exton q-Bessel function Jackson...

In mathematics, a bilateral hypergeometric series is a series Σan summed over all integers n, and such that the ratio an/an+1 of two terms is a rational...

Chu, Wenchang (2007). "Abel's lemma on summation by parts and basic hypergeometric series". Advances in Applied Mathematics. 39 (4): 490–514. doi:10.1016/j...

Cambridge University Press. Gasper, George; Rahman, Mizan (2004). Basic Hypergeometric Series. Encyclopedia of Mathematics and Its Applications. Vol. 96 (2nd ed...

z ) . {\displaystyle E_{q}(z).} It is a special case of the basic hypergeometric series, E q ( z ) = 1 ϕ 1 ( 0 0 ; z ) = ∑ n = 0 ∞ q ( n 2 ) ( − z )...

the Rogers–Ramanujan identities are two identities related to basic hypergeometric series and integer partitions. The identities were first discovered...

geometry Quantum differential calculus Time scale calculus q-analog Basic hypergeometric series Quantum dilogarithm Abreu, Luis Daniel (2006). "Functions q-Orthogonal...

ISSN 0950-1207, JSTOR 92601 Gasper, George; Rahman, Mizan (2004), Basic hypergeometric series, Encyclopedia of Mathematics and its Applications, vol. 96 (2nd ed...

mathematics, the q-Bessel polynomials are a family of basic hypergeometric orthogonal polynomials in the basic Askey scheme. Roelof Koekoek, Peter A. Lesky, and...

functions for distinct and any parts respectively. (See also Basic hypergeometric series.) With the ordinary binomial coefficients, we have: ∑ k = 0 n...

In mathematics, Appell series are a set of four hypergeometric series F1, F2, F3, F4 of two variables that were introduced by Paul Appell (1880) and that...

Beach, Florida) was an American mathematician who worked on basic hypergeometric series. He is best known for his lecture notes on the subject which...

polynomials and basic hypergeometric series, who introduced the Askey–Gasper inequality. Gasper, George; Rahman, Mizan (2004), Basic hypergeometric series, Encyclopedia...

continuous q-Hermite polynomials are a family of basic hypergeometric orthogonal polynomials in the basic Askey scheme. Roelof Koekoek, Peter A. Lesky, and...

orthogonal family of polynomials defined in terms of Heine's basic hypergeometric series as P n ( x ; c ; q ) = 3 ϕ 2 ( q − n , q n + 1 , x ; q , c q...

scheme is a way of organizing orthogonal polynomials of hypergeometric or basic hypergeometric type into a hierarchy. For the classical orthogonal polynomials...

functions are given in terms of the q-Pochhammer symbol and the basic hypergeometric function ϕ {\displaystyle \phi } by J ν ( 1 ) ( x ; q ) = ( q ν +...