Related papers: A quadratic formula for basic hypergeometric serie…
Orthogonal polynomials of a continuous variable in the Askey scheme satisfying second order difference equations, such as the Askey-Wilson polynomial, can be studied by the quantum mechanical formulation, idQM (discrete quantum mechanics…
We derive inversion formulas involving orthogonal polynomials which can be used to find coefficients of differential equations satisfied by certain generalizations of the classical orthogonal polynomials. As an example we consider special…
We prove a transformation formula relating two determinants involving elliptic shifted factorials. Similar determinants have been applied to multiple elliptic hypergeometric series.
We give two general transformations that allows certain quite general basic hypergeometric multi-sums of arbitrary depth (sums that involve an arbitrary sequence $\{g(k)\}$), to be reduced to an infinite $q$-product times a single basic…
Based on Spiridonov's analysis of elliptic generalizations of the Gauss hypergeometric function, we develop a common framework for 7-parameter families of generalized elliptic, hyperbolic and trigonometric univariate hypergeometric…
In this paper we present an addition to Askey's scheme of q-hypergeometric orthogonal polynomials involving classes of q-special functions which do not consist of polynomials only. The special functions are q-analogues of the Jacobi and…
In this paper, we compute the contributions from double cover maps to genus 0 degree 2 Gromov-Witten invariants of general type projective hypersurfaces. Our results correspond to a generalization of Aspinwall-Morrison formula to general…
The central idea of this article is to present a systematic approach to construct some recurrence relations for the solutions of the second-order linear difference equation of hypergeometric-type defined on the quadratic-type lattices. We…
Several new identities for elliptic hypergeometric series are proved. Remarkably, some of these are elliptic analogues of identities for basic hypergeometric series that are balanced but not very-well-poised.
The Askey--Wilson integral is very important in the theory of orthogonal polynomials. Liu's integral is a generalization of the Askey--Wilson integral with many parameters. With the help of the series rearrangement method, we give the…
A proof of an unusual summation formula for a basic hypergeometric series associated to the affine root system $\tilde A_n$ that was conjectured by Warnaar is given. It makes use of Milne's $A_n$ extension of Watson's transformation,…
We generalize the known constructions of A-hypergeometric functions. In particular, we show that periods of middle dimension on affine or projective complex algebraic varieties are A-hypergeometric functions of coefficients of polynomial…
Motivated by the Gaussian symplectic ensemble, Mehta and Wang evaluated the $n$ by $n$ determinant $\det((a+j-i)\Gamma(b+j+i))$ in 2000. When $a=0$, Ciucu and Krattenthaler computed the associated Pfaffian $\Pf((j-i)\Gamma(b+j+i))$ with an…
A general class of Newton algorithms on Gra{\ss}mann and Lagrange-Gra{\ss}mann manifolds is introduced, that depends on an arbitrary pair of local coordinates. Local quadratic convergence of the algorithm is shown under a suitable condition…
In this article, we solve the connection problem of the Hermite polynomials with the classical continuous orthogonal polynomials belonging to Askey scheme, using the hypergeometric functions method combined is with the work the Fields and…
We apply a symbolic approach of the general quadratic decomposition of polynomial sequences - presented in a previous article referenced herein - to polynomial sequences fulfilling specific orthogonal conditions towards two given…
We develop the relation between hyperbolic geometry and arithmetic equidistribution problems that arises from the action of arithmetic groups on real hyperbolic spaces, especially in dimension up to 5. We prove generalisations of Mertens'…
For quadratic spaces which represent 1 there is a characterization of hermitian compositions in the language of algebras-with-involutions using the even Clifford algebra. We extend this notion to define a generalized composition based on…
This paper gives an explicit formula for the Ehrhart quasi-polynomial of certain 2-dimensional polyhedra in terms of invariants of surface quotient singularities. Also, a formula for the dimension of the space of quasi-homogeneous…
By systematically applying ten inequivalent two-part relations between hypergeometric sums 3F2(1) to the published database of all such sums, 66 new sums are obtained. Many results extracted from the literature are shown to be special cases…