scholarly journals Kernel identities for van Diejen’s q-difference operators and transformation formulas for multiple basic hypergeometric series

2013 ◽  
Vol 32 (2) ◽  
pp. 281-314 ◽  
Author(s):  
Yasuho Masuda
Keyword(s):  
Hypergeometric Series ◽  
Basic Hypergeometric Series ◽  
Difference Operators ◽  
Transformation Formulas ◽  
Multiple Basic Hypergeometric Series

scholarly journals $n$-color overpartitions, lattice paths, and multiple basic hypergeometric series

2008 ◽  
Vol DMTCS Proceedings vol. AJ,... (Proceedings) ◽  
Author(s):  
Olivier Mallet
Keyword(s):  
Generating Functions ◽  
Hypergeometric Series ◽  
Basic Hypergeometric Series ◽  
Combinatorial Identities ◽  
Lattice Paths ◽  
Multiple Series ◽  
Infinite Products ◽  
Nous Montrons ◽  
International Audience ◽  
Multiple Basic Hypergeometric Series

International audience We define two classes of multiple basic hypergeometric series $V_{k,t}(a,q)$ and $W_{k,t}(a,q)$ which generalize multiple series studied by Agarwal, Andrews, and Bressoud. We show how to interpret these series as generating functions for special restricted lattice paths and for $n$-color overpartitions with weighted difference conditions. We also point out that some specializations of our series can be written as infinite products, which leads to combinatorial identities linking $n$-color overpartitions with ordinary partitions or overpartitions. Nous définissons deux classes de séries hypergéométriques basiques multiples $V_{k,t}(a,q)$ et $W_{k,t}(a,q)$ qui généralisent des séries multiples étudiées par Agarwal, Andrews et Bressoud. Nous montrons comment interpréter ces séries comme les fonctions génératrices de chemins avec certaines restrictions et de surpartitions $n$-colorées vérifiant des conditions de différences pondérées. Nous remarquons aussi que certaines spécialisations de nos séries peuvent s'écrire comme des produits infinis, ce qui conduit à des identités combinatoires reliant les surpartitions $n$-colorées aux partitions ou surpartitions ordinaires.

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.

scholarly journals A Partial Theta Function Borwein Conjecture

2019 ◽  
Vol 23 (3-4) ◽  
pp. 561-572
Author(s):  
Gaurav Bhatnagar ◽  
Michael J. Schlosser
Keyword(s):  
Theta Function ◽  
Hypergeometric Series ◽  
Basic Hypergeometric Series ◽  
Infinite Family ◽  
Macdonald Polynomial ◽  
Partial Theta Function ◽  
Multiple Basic Hypergeometric Series

Abstract We present an infinite family of Borwein type $$+ - - $$+-- conjectures. The expressions in the conjecture are related to multiple basic hypergeometric series with Macdonald polynomial argument.

scholarly journals Some q-Supercongruences from Transformation Formulas for Basic Hypergeometric Series

2020 ◽  
Author(s):  
Victor J. W. Guo ◽  
Michael J. Schlosser
Keyword(s):  
Hypergeometric Series ◽  
Basic Hypergeometric Series ◽  
Double Series ◽  
Summation Formula ◽  
Transformation Formula ◽  
Quadratic Transformation ◽  
Ultraspherical Polynomials ◽  
Higher Power ◽  
Series Transformation ◽  
Transformation Formulas

AbstractSeveral new q-supercongruences are obtained using transformation formulas for basic hypergeometric series, together with various techniques such as suitably combining terms, and creative microscoping, a method recently developed by the first author in collaboration with Zudilin. More concretely, the results in this paper include q-analogues of supercongruences (referring to p-adic identities remaining valid for some higher power of p) established by Long, by Long and Ramakrishna, and several other q-supercongruences. The six basic hypergeometric transformation formulas which are made use of are Watson’s transformation, a quadratic transformation of Rahman, a cubic transformation of Gasper and Rahman, a quartic transformation of Gasper and Rahman, a double series transformation of Ismail, Rahman and Suslov, and a new transformation formula for a nonterminating very-well-poised $${}_{12}\phi _{11}$$ 12 ϕ 11 series. Also, the nonterminating q-Dixon summation formula is used. A special case of the new $${}_{12}\phi _{11}$$ 12 ϕ 11 transformation formula is further utilized to obtain a generalization of Rogers’ linearization formula for the continuous q-ultraspherical polynomials.

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