27 research outputs found
Elliptic Hypergeometric Summations by Taylor Series Expansion and Interpolation
We use elliptic Taylor series expansions and interpolation to deduce a number
of summations for elliptic hypergeometric series. We extend to the well-poised
elliptic case results that in the -case have previously been obtained by
Cooper and by Ismail and Stanton. We also provide identities involving S.
Bhargava's cubic theta functions
Elliptic rook and file numbers
Utilizing elliptic weights, we construct an elliptic analogue of rook numbers
for Ferrers boards. Our elliptic rook numbers generalize Garsia and Remmel's
q-rook numbers by two additional independent parameters a and b, and a nome p.
These are shown to satisfy an elliptic extension of a factorization theorem
which in the classical case was established by Goldman, Joichi and White and
later was extended to the q-case by Garsia and Remmel. We obtain similar
results for our elliptic analogues of Garsia and Remmel's q-file numbers for
skyline boards. We also provide an elliptic extension of the j-attacking model
introduced by Remmel and Wachs. Various applications of our results include
elliptic analogues of (generalized) Stirling numbers of the first and second
kind, Lah numbers, Abel numbers, and r-restricted versions thereof.Comment: 45 pages; 3rd version shortened (elliptic rook theory for matchings
has been taken out to keep the length of this paper reasonable
Some combinatorial identities involving noncommuting variables
International audienceWe derive combinatorial identities for variables satisfying specific sets of commutation relations. The identities thus obtained extend corresponding ones for -commuting variables and satisfying . In particular, we obtain weight-dependent binomial theorems, functional equations for generalized exponential functions, we propose a derivative of noncommuting variables, and finally utilize one of the considered weight functions to extend rook theory. This leads us to an extension of the -Stirling numbers of the second kind, and of the -Lah numbers.Nous obtenons des identités combinatoires pour des variables satisfaisant des ensembles spécifiques de relations de commutation. Ces identités ainsi obtenues généralisent leurs analogues pour des variables -commutantes et satisfaisant . En particulier, nous obtenons des théorèmes binomiaux dépendant du poids, des équations fonctionnelles pour les fonctions exponentielles généralisées, nous proposons une dérivée des variables non-commutatives, et finalement nous utilisons l’une des fonctions de poids considérées pour étendre la théorie des tours. Nous en déduisons une généralisation des -nombres de Stirling de seconde espèce et des -nombres de Lah