383 research outputs found
Mellin Transforms of the Generalized Fractional Integrals and Derivatives
We obtain the Mellin transforms of the generalized fractional integrals and
derivatives that generalize the Riemann-Liouville and the Hadamard fractional
integrals and derivatives. We also obtain interesting results, which combine
generalized operators with generalized Stirling numbers and Lah
numbers. For example, we show that corresponds to the Stirling
numbers of the kind and corresponds to the unsigned Lah
numbers. Further, we show that the two operators and
, , generate the same sequence given by the
recurrence relation
with and for and or
. Finally, we define a new class of sequences for and in turn show
that corresponds to the generalized Laguerre
polynomials.Comment: 17 pages, 1 figure, 9 tables, Accepted for publication in Applied
Mathematics and Computatio
Identities via Bell matrix and Fibonacci matrix
AbstractIn this paper, we study the relations between the Bell matrix and the Fibonacci matrix, which provide a unified approach to some lower triangular matrices, such as the Stirling matrices of both kinds, the Lah matrix, and the generalized Pascal matrix. To make the results more general, the discussion is also extended to the generalized Fibonacci numbers and the corresponding matrix. Moreover, based on the matrix representations, various identities are derived
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 observations on the Lah and Laguerre transforms of integer sequences
We study the Lah and related Laguerre transforms
within the context of exponential Riordan arrays. Links to the
Stirling numbers are explored. Results for finite matrices are
generalized, leading to a number of useful matrix factorizations
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
Some combinatorial identities involving noncommuting variables
We 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
- …