Related papers: Askey--Wilson Integral and its Generalizations
We consider a quaternionic analogue of the univariate complex Hermite polynomials and study some of their analytic properties in some detail. We obtain their integral representation as well as the operational formulas of exponential and…
Transformation formulas for four-parameter refinements of the q-trinomial coefficients are proven. The iterative nature of these transformations allows for the easy derivation of several infinite series of q-trinomial identities, and can be…
Based on the theory of Dunkl operators, this paper presents a general concept of multivariable Hermite polynomials and Hermite functions which are associated with finite reflection groups on $\b R^N$. The definition and properties of these…
We use recent results proved by Berrick and the author (math.KT/0509404) to improve the periodicity theorem in hermitian K-theory. We define also a new filtration of the classical Witt ring W(A), built from non degenerate quadratic forms…
By providing a variant of Weyl's inequality for general systems of forms we establish the Hardy-Littlewood asymptotic formula for the density of integer zeros of systems of quadratic or cubics forms under weaker rank conditions than…
We obtain new explicit formulas for the recurrence coefficients of the q-orthogonal polynomial sequences in a class that extends the q-Askey scheme. Our formulas express the recurrence coefficients in terms of four parameters that determine…
Small perturbations of the Jacobi matrix with weights \sqrt n and zero diagonal are considered. A formula relating the asymptotics of polynomials of the first kind to the spectral density is obtained, which is analogue of the classical…
The problem of barycentric Hermite interpolation is highly susceptible to overflows or underflows. In this paper, based on Sturm-Liouville equations for Jacobi orthogonal polynomials, we consider the fast implementation on the second…
The aim of this paper is to give a new approach to modified $q$-Bernstein polynomials for functions of several variables. By using these polynomials, the recurrence formulas and some new interesting identities related to the second Stirling…
A hyperbinary expansion of a positive integer n is a partition of n into powers of 2 in which each part appears at most twice. In this paper, we consider a generalization of this concept and a certain statistic on the corresponding set of…
We give expansions for the distribution, density, and quantiles of an estimate, building on results of Cornish, Fisher, Hill, Davis and the authors. The estimate is assumed to be non-lattice with the standard expansions for its cumulants.…
In this paper, we first get a criterion formula for whether a differential form is holomorphic with respect to the generalized complex structure induced by $\epsilon$. Next, we get the local extensions of $\overline\partial$-closed forms on…
The Askey-Wilson polynomials are a four-parameter family of orthogonal symmetric Laurent polynomials $R_n[z]$ which are eigenfunctions of a second-order $q$-difference operator $L$, and of a second-order difference operator in the variable…
We present an asymmetric $q$-Painlev\'e equation. We will derive this using $q$-orthogonal polynomials with respect to generalized Freud weights: their recurrence coefficients will obey this $q$-Painlev\'e equation (up to a simple…
We list the so-called Askey-scheme of hypergeometric orthogonal polynomials. In chapter 1 we give the definition, the orthogonality relation, the three term recurrence relation and generating functions of all classes of orthogonal…
Let $d_S$ denote the arithmetic density of a subset $S \subseteq \mathbb N$. We derive a power series in $q\in \mathbb C$, $|q|<1$, with co\"efficients related to integer partitions and integer compositions, that yields $1/d_S$ in the limit…
The aim of this paper is to present a new simple recurrence for Appell and Sheffer sequences in terms of the linear functional that defines them, and to explain how this is equivalent to several well-known characterizations appearing in the…
We construct a set $S$ such that every translate of $S$ is a set of recurrence and a set of rigidity for a weak mixing measure preserving system. This construction generalizes or strengthens results of Katznelson, Saeki, Forrest, and Fayad…
A q-generalization of the product densities in stochastic point processes is developed. The properties of these functions are studied and a q-generalization of the usual $C^r_s$ coefficients is obtained. This for fixed q-number of particles…
Using the algebraic method of Gardner's deformations for completely integrable systems, we construct the recurrence relations for densities of the Hamiltonians for the Boussinesq and the Kaup-Boussinesq equations. By extending the Magri…