29,674 research outputs found
Extensions of q-Chu–Vandermonde's identity
No full textAbstractIn this paper, we apply q-exponential operator to get some general q-Chu–Vandermonde's identities
q-Differential equations for q-classical polynomials and q-Jacobi-Stirling numbers
No full textWe introduce, characterise and provide a combinatorial interpretation for the so-called q-Jacobi–Stirling numbers.
This study is motivated by their key role in the (reciprocal) expansion of any power of a second order
q-differential operator having the q-classical polynomials as eigenfunctions in terms of other even order operators,
which we explicitly construct in this work. The results here obtained can be viewed as the q-version of
those given by Everitt et al. and by the first author, whilst the combinatorics of this new set of numbers is a
q-version of the Jacobi–Stirling numbers given by Gelineau and the second author
Generalizations of Milne's -Chu-Vandermonde summation
No full textsummary:We derive two identities for multiple basic hyper-geometric series associated with the unitary group. In order to get the two identities, we first present two known -exponential operator identities which were established in our earlier paper. From the two identities and combining them with the two -Chu-Vandermonde summations established by Milne, we arrive at our results. Using the identities obtained in this paper, we give two interesting identities involving binomial coefficients. In addition, we also derive two nontrivial summation equations from the two multiple extensions
Continuous Stieltjes-Wigert Limiting Behaviour of a Family of Confluent q-Chu-Vandermonde Distributions
No full textFrom Kemp [1], we have a family of confluent q-Chu- Vandermonde distributions, consisted by three members I, II and III, interpreted as a family of q-steady-state distributions from Markov chains. In this article, we provide the moments of the distributions of this family and we establish a continuous limiting behavior for the members I and II, in the sense of pointwise convergence, by applying a q-analogue of the usual Stirling asymptotic formula for the factorial number of order n. Specifically, we initially give the q-factorial moments and the usual moments for the family of confluent q-Chu- Vandermonde distributions and then we designate as a main theorem the conditions under which the confluent q-Chu-Vandermonde distributions I and II converge to a continuous Stieltjes-Wigert distribution. For the member III we give a continuous analogue. Moreover, as applications of this study we present a modified q-Bessel distribution, a generalized q-negative Binomial distribution and a generalized over/underdispersed (O/U) distribution. Note that in this article we prove the convergence of a family of discrete distributions to a continuous distribution which is not of a Gaussian type
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