scholarly journals The $q$-Binomial Theorem and two Symmetric $q$-Identities

10.37236/1727 ◽  
2003 ◽  
Vol 10 (1) ◽  
Author(s):  
Victor J. W. Guo
Keyword(s):  
Hypergeometric Series ◽  
Basic Hypergeometric Series ◽  
Binomial Theorem ◽  
Cambridge University ◽  
Special Cases

We notice two symmetric $q$-identities, which are special cases of the transformations of $_2\phi_1$ series in Gasper and Rahman's book (Basic Hypergeometric Series, Cambridge University Press, 1990, p. 241). In this paper, we give combinatorial proofs of these two identities and the $q$-binomial theorem by using conjugation of $2$-modular diagrams.

scholarly journals In Praise of an Elementary Identity of Euler

10.37236/2009 ◽  
2011 ◽  
Vol 18 (2) ◽  
Author(s):  
Gaurav Bhatnagar
Keyword(s):  
Hypergeometric Series ◽  
Fibonacci Numbers ◽  
Basic Hypergeometric Series ◽  
Binomial Theorem ◽  
Pell Numbers ◽  
Generalized Hypergeometric Series ◽  
Pentagonal Number Theorem ◽  
Wz Method ◽  
Elementary Identity

We survey the applications of an elementary identity used by Euler in one of his proofs of the Pentagonal Number Theorem. Using a suitably reformulated version of this identity that we call Euler's Telescoping Lemma, we give alternate proofs of all the key summation theorems for terminating Hypergeometric Series and Basic Hypergeometric Series, including the terminating Binomial Theorem, the Chu–Vandermonde sum, the Pfaff–Saalschütz sum, and their $q$-analogues. We also give a proof of Jackson's $q$-analog of Dougall's sum, the sum of a terminating, balanced, very-well-poised $_8\phi_7$ sum. Our proofs are conceptually the same as those obtained by the WZ method, but done without using a computer. We survey identities for Generalized Hypergeometric Series given by Macdonald, and prove several identities for $q$-analogs of Fibonacci numbers and polynomials and Pell numbers that have appeared in combinatorial contexts. Some of these identities appear to be new.

scholarly journals Bisymmetric functions, Macdonald polynomials and basic hypergeometric series

2008 ◽  
Vol 144 (2) ◽  
pp. 271-303 ◽  
Author(s):  
S. Ole Warnaar
Keyword(s):  
Hypergeometric Series ◽  
Basic Hypergeometric Series ◽  
Macdonald Polynomials ◽  
Difference Operators ◽  
Jack Polynomials ◽  
Selberg Integral ◽  
Binomial Theorem ◽  
Alternating Sign Matrices ◽  
Selberg Integrals ◽  
New Type

AbstractA new type of $\mathfrak {sl}_3$ basic hypergeometric series based on Macdonald polynomials is introduced. Besides a pair of Macdonald polynomials attached to two different sets of variables, a key ingredient in the $\mathfrak {sl}_3$ basic hypergeometric series is a bisymmetric function related to Macdonald’s commuting family of q-difference operators, to the $\mathfrak {sl}_3$ Selberg integrals of Tarasov and Varchenko, and to alternating sign matrices. Our main result for $\mathfrak {sl}_3$ series is a multivariable generalization of the celebrated q-binomial theorem. In the limit this q-binomial sum yields a new $\mathfrak {sl}_3$ Selberg integral for Jack polynomials.

THE BRENKE TYPE GENERATING FUNCTIONS AND EXPLICIT FORMS OF MRM-TRIPLES BY MEANS OF Q-HYPERGEOMETRIC SERIES

2013 ◽  
Vol 16 (02) ◽  
pp. 1350010 ◽  
Author(s):  
NOBUHIRO ASAI ◽  
IZUMI KUBO ◽  
HUI-HSIUNG KUO
Keyword(s):  
Orthogonal Polynomials ◽  
Generating Function ◽  
Generating Functions ◽  
Hypergeometric Series ◽  
Basic Hypergeometric Series ◽  
Special Cases ◽  
Triple H

An MRM-triple (h(x), ρ(t), B(t)) gives a generating function B(t)h(ρ(t)x) of some orthogonal polynomials on ℝ. In particular, B(t)h(tx) is called the Brenke type.19 In this paper, we shall determine all MRM-triples and associated Jacobi-Szegö parameters of this type with showing very careful computations in detail. (h(x), t, B(t)) is classified into four categories. In any case, h(x) and B(t) can be expressed in terms of two kinds of q-hypergeometric series, old basic and basic hypergeometric series, rΦs and rϕs, respectively. As examples, our results contain generating functions of the Al-Salam-Carlitz (I and II), little q-Laguerre, q-Laguerre, and discrete q-Hermite (I and II) polynomials. Our results are more complete and general than those of Refs. 20 and 21 by Chihara. The following are special cases of our results in each class. Here {αn, ωn} are the Jacobi-Szegö parameters. [Formula: see text]

scholarly journals Approach of q-Derivative Operators to Terminating q-Series Formulae

2018 ◽  
Vol 26 (2) ◽  
pp. 99-111
Author(s):  
Xiaoyuan Wang ◽  
Wenchang Chu
Keyword(s):  
Hypergeometric Series ◽  
Basic Hypergeometric Series ◽  
Operator Approach ◽  
Derivative Operator ◽  
Summation Formulae

AbstractThe q-derivative operator approach is illustrated by reviewing several typical summation formulae of terminating basic hypergeometric series.

scholarly journals Inversion of Bilateral Basic Hypergeometric Series

10.37236/1703 ◽  
2003 ◽  
Vol 10 (1) ◽  
Author(s):  
Michael Schlosser
Keyword(s):  
Hypergeometric Series ◽  
Basic Hypergeometric Series ◽  
Matrix Inversion ◽  
Infinite Matrices ◽  
Inversion Result ◽  
Matrix Inverse ◽  
Bilateral Basic Hypergeometric Series ◽  
Summation Theorem ◽  
Lower Triangular

We present a new matrix inverse with applications in the theory of bilateral basic hypergeometric series. Our matrix inversion result is directly extracted from an instance of Bailey's very-well-poised ${}_6\psi_6$ summation theorem, and involves two infinite matrices which are not lower-triangular. We combine our bilateral matrix inverse with known basic hypergeometric summation theorems to derive, via inverse relations, several new identities for bilateral basic hypergeometric series.

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