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Register a new userMoments of q-Jacobi Polynomials and q-Zeta Values
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Authors
Frederic Chapoton
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We explore some connections between moments of rescaled little q-Jacobi polynomials, q-analogues of values at negative integers for some Dirichlet series, and the q-Eulerian polynomials of wreath products of symmetric groups.
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In this paper we give the q-extension of Euler numbers which can be viewed as interpolating of the q-analogue of Euler zeta function ay negative integers, in the same way that Riemann zeta function interpolates Bernoulli numbers at negative integers. Finally we woll treat some identities of the q-extension of the euler numbers by using fermionic p-adic q-integration on Z_p.
A polynomial $A(q)=sum_{i=0}^n a_iq^i$ is said to be unimodal if $a_0le a_1le cdots le a_kge a_{k+1} ge cdots ge a_n$. We investigate the unimodality of rational $q$-Catalan polynomials, which is defined to be $C_{m,n}(q)= frac{1}{[n+m]} left[ m+n atop nright]$ for a coprime pair of positive integers $(m,n)$. We conjecture that they are unimodal with respect to parity, or equivalently, $(1+q)C_{m+n}(q)$ is unimodal. By using generating functions and the constant term method, we verify our conjecture for $mle 5$ in a straightforward way.
A two-parameter deformation of the Touchard polynomials, based on the NEXT $q$-exponential function of Tsallis, defines two statistics on set partitions. The generating function of classical Touchard polynomials is a composition of two exponential functions. By applying analysis of a combinatorial structure of the deformed exponential function, we establish explicit formulae for both statistics. Moreover, the explicit formulae for the deformed Touchard polynomials makes possible to evaluate coefficients of Taylor series expansion for wide variety of functions with different values of parameters $p$ and $q$.
We describe various aspects of the Al-Salam-Chihara $q$-Laguerre polynomials. These include combinatorial descriptions of the polynomials, the moments, the orthogonality relation and a combinatorial interpretation of the linearization coefficients. It is remarkable that the corresponding moment sequence appears also in the recent work of Postnikov and Williams on enumeration of totally positive Grassmann cells.
We introduce a notion of $q$-deformed rational numbers and $q$-deformed continued fractions. A $q$-deformed rational is encoded by a triangulation of a polygon and can be computed recursively. The recursive formula is analogous to the $q$-deformed Pascal identitiy for the Gaussian binomial coefficients, but the Pascal triangle is replaced by the Farey graph. The coefficients of the polynomials defining the $q$-rational count quiver subrepresentations of the maximal indecomposable representation of the graph dual to the triangulation. Several other properties, such as total positivity properties, $q$-deformation of the Farey graph, matrix presentations and $q$-continuants are given, as well as a relation to the Jones polynomial of rational knots.
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