scholarly journals Lattice paths and multiple basic hypergeometric series

1989 ◽  
Vol 136 (2) ◽  
pp. 209-228 ◽  
Author(s):  
Ashok Agarwal ◽  
David Bressoud
Keyword(s):  
Hypergeometric Series ◽  
Basic Hypergeometric Series ◽  
Lattice Paths ◽  
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 An extension to overpartitions of Rogers-Ramanujan identities for even moduli

2006 ◽  
Vol DMTCS Proceedings vol. AG,... (Proceedings) ◽  
Author(s):  
Sylvie Corteel ◽  
Jeremy Lovejoy ◽  
Olivier Mallet
Keyword(s):  
Generating Functions ◽  
Hypergeometric Series ◽  
Basic Hypergeometric Series ◽  
Lattice Paths ◽  
Infinite Products ◽  
International Audience

International audience We investigate class of well-poised basic hypergeometric series $\tilde{J}_{k,i}(a;x;q)$, interpreting these series as generating functions for overpartitions defined by multiplicity conditions. We also show how to interpret the $\tilde{J}_{k,i}(a;1;q)$ as generating functions for overpartitions whose successive ranks are bounded, for overpartitions that are invariant under a certain class of conjugations, and for special restricted lattice paths. We highlight the cases $(a,q) \to (1/q,q)$, $(1/q,q^2)$, and $(0,q)$, where some of the functions $\tilde{J}_{k,i}(a;x;q)$ become infinite products. The latter case corresponds to Bressoud's family of Rogers-Ramanujan identities for even moduli.

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 From Kajihara’s transformation formula to deformed Macdonald–Ruijsenaars and Noumi–Sano operators

2021 ◽  
Vol 28 (2) ◽  
Author(s):  
Martin Hallnäs ◽  
Edwin Langmann ◽  
Masatoshi Noumi ◽  
Hjalmar Rosengren
Keyword(s):  
Hypergeometric Series ◽  
Basic Hypergeometric Series ◽  
Root Systems ◽  
Higher Order ◽  
Explicit Description ◽  
Transformation Formula ◽  
Inverse Limits ◽  
First Order ◽  
Order Case ◽  
Multiple Basic Hypergeometric Series

AbstractKajihara obtained in 2004 a remarkable transformation formula connecting multiple basic hypergeometric series associated with A-type root systems of different ranks. By specialisations of his formula, we deduce kernel identities for deformed Macdonald–Ruijsenaars (MR) and Noumi–Sano (NS) operators. The deformed MR operators were introduced by Sergeev and Veselov in the first order case and by Feigin and Silantyev in the higher order cases. As applications of our kernel identities, we prove that all of these operators pairwise commute and are simultaneously diagonalised by the super-Macdonald polynomials. We also provide an explicit description of the algebra generated by the deformed MR and/or NS operators by a Harish-Chandra type isomorphism and show that the deformed MR (NS) operators can be viewed as restrictions of inverse limits of ordinary MR (NS) operators.

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