Related papers: Inversion formula for the hypergeometric Fourier t…
Many problems of applied mathematics are reduced to the solution of integral equations with special functions in kernels, therefore the inversion formulas for such equations play an important role in solving boundary value problems for…
By means of inversion techniques and several known hypergeometric series identities, summation formulas for Fox-Wright function are explored. They give some new hypergeometric series identities when the parameters are specified.
We consider different pentagon identities realized by the hyperbolic hypergeometric functions and investigate their degenerations to the level of complex hypergeometric functions. In particular, we show that one of the degenerations yields…
A new Hilbert-type integral inequality in the whole plane with the non-homogeneous kernel and parameters is given. The constant factor related to the hypergeometric function and the beta function is proved to be the best possible. As…
We give conceptual proofs of certain basic properties of the arrangement of shifted root hyperplanes associated to a root system and a Weyl group invariant real valued parameter function on the root system. The method is based on the role…
A factorization formula for wave functions, which is basic in the inverse spectral transform approach to initial-boundary value problems, is proved in greater generality than before. Applications follow. Related compatibility questions for…
We study the action of hypergeometric shift operators on the Heckman-Opdam hypergeometric functions associated with the $BC_n$ type root system and some negative multiplicities. Those hypergeometric functions are connected to the…
In this note we state (with minor corrections) and give an alternative proof of a very general hypergeometric transformation formula due to Slater. As an application, we obtain a new hypergeometric transformation formula for a ${}_5F_4(-1)$…
A master formula of transformation formulas for bilinear sums of basic hypergeometric series is proposed. It is obtained from the author's previous results on a transformation formula for Milne's multivariate generalization of basic…
We introduce an algebra $\mathcal H$ consisting of difference-reflection operators and multiplication operators that can be considered as a $q=1$ analogue of Sahi's double affine Hecke algebra related to the affine root system of type…
Using matrix inversion and determinant evaluation techniques we prove several summation and transformation formulas for terminating, balanced, very-well-poised, elliptic hypergeometric series.
We examine the problem of the Fourier transform mapping one weighted Lebesgue space into another, by studying necessary conditions and sufficient conditions which expose an underlying geometry. In the necessary conditions, this geometry is…
Multidimensional matrix inversions provide a powerful tool for studying multiple hypergeometric series. In order to extend this technique to elliptic hypergeometric series, we present three new multidimensional matrix inversions. As…
We study multivariable (bilateral) basic hypergeometric series associated with (type $A$) Macdonald polynomials. We derive several transformation and summation properties for such series including analogues of Heine's ${}_2\phi_1$…
In this paper, we prove several transformation formulas for the very-well-poised bilateral basic hypergeometric $_5\psi_5$ series by using the relationship between the bilateral basic hypergeometric $_5\psi_5$ series and basic…
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 express explicitly the Heckman-Opdam hypergeometric function for the root system of type A with a certain degenerate parameter in terms of the Lauricella hypergeometric function.
For the super-hyperbolic space in any dimension, we introduce the non-Euclidean Helgason--Fourier transform. We prove an inversion formula exhibiting residue contributions at the poles of the Harish-Chandra c-function, signalling discrete…
This is a discussion of miscellaneous summation, integration and transformation formulas obtained using Fourier analysis. The topics covered are: Series of the form $\sum_{n\in\mathbb{Z}} c_ne^{\pi i \gamma n^2}$; Fusion of integrals, and…
The inversion problem for rational B\'ezier curves is addressed by using resultant matrices for polynomials expressed in the Bernstein basis. The aim of the work is not to construct an inversion formula but finding the corresponding value…