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Secant tree calculus

Dominique Foata, Guo-Niu Han (2014)

Open Mathematics

A true Tree Calculus is being developed to make a joint study of the two statistics “eoc” (end of minimal chain) and “pom” (parent of maximum leaf) on the set of secant trees. Their joint distribution restricted to the set {eoc-pom ≤ 1} is shown to satisfy two partial difference equation systems, to be symmetric and to be expressed in the form of an explicit three-variable generating function.

Several q -series identities from the Euler expansions of ( a ; q ) and 1 ( a ; q )

Zhizheng Zhang, Yang, Jizhen (2009)

Archivum Mathematicum

In this paper, we first give several operator identities which extend the results of Chen and Liu, then make use of them to two q -series identities obtained by the Euler expansions of ( a ; q ) and 1 ( a ; q ) . Several q -series identities are obtained involving a q -series identity in Ramanujan’s Lost Notebook.

Some q-supercongruences for truncated basic hypergeometric series

Victor J. W. Guo, Jiang Zeng (2015)

Acta Arithmetica

For any odd prime p we obtain q-analogues of van Hamme’s and Rodriguez-Villegas’ supercongruences involving products of three binomial coefficients such as k = 0 ( p - 1 ) / 2 [ 2 k k ] q ² 3 ( q 2 k ) / ( ( - q ² ; q ² ) ² k ( - q ; q ) ² 2 k ² ) 0 ( m o d [ p ] ² ) for p≡ 3 (mod 4), k = 0 ( p - 1 ) / 2 [ 2 k k ] q ³ ( ( q ; q ³ ) k ( q ² ; q ³ ) k q 3 k ) ( ( q ; q ) k ² ) 0 ( m o d [ p ] ² ) for p≡ 2 (mod 3), where [ p ] = 1 + q + + q p - 1 and ( a ; q ) = ( 1 - a ) ( 1 - a q ) ( 1 - a q n - 1 ) . We also prove q-analogues of the Sun brothers’ generalizations of the above supercongruences. Our proofs are elementary in nature and use the theory of basic hypergeometric series and combinatorial q-binomial identities including a new q-Clausen type summation formula.

Systèmes aux q -différences singuliers réguliers : classification, matrice de connexion et monodromie

Jacques Sauloy (2000)

Annales de l'institut Fourier

G.D. Birkhoff a posé, par analogie avec le cas classique des équations différentielles, le problème de Riemann-Hilbert pour les systèmes “fuchsiens” aux q -différences linéaires, à coefficients rationnels. Il l’a résolu dans le cas générique: l’objet classifiant qu’il introduit est constitué de la matrice de connexion P et des exposants en 0 et . Nous reprenons sa méthode dans le cas général, mais en traitant symétriquement 0 et et sans recours à des solutions à croissance “sauvage”. Lorsque q ...

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