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Some continuous analogs of expansion in Jacobi polynomials and vector valued hypergeometric orthogonal bases
, 2003
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Fourier transforms related to a root system of rank 1
"... Abstract. We introduce an algebra H consisting of differencereflection 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 (C ∨ 1, C1). We study eigenfunctions of a DunklCheredniktype ope ..."
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Abstract. We introduce an algebra H consisting of differencereflection 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 (C ∨ 1, C1). We study eigenfunctions of a DunklCheredniktype operator in the algebra H, and the corresponding Fourier transforms. These eigenfunctions are nonsymmetric versions of the Wilson polynomials and the Wilson functions. 1.
PROPERTIES OF GENERALIZED UNIVARIATE HYPERGEOMETRIC FUNCTIONS
, 2006
"... Abstract. Based on Spiridonov’s analysis of elliptic generalizations of the Gauss hypergeometric function, we develop a common framework for 7parameter families of generalized elliptic, hyperbolic and trigonometric univariate hypergeometric functions. In each case we derive the symmetries of the ge ..."
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Abstract. Based on Spiridonov’s analysis of elliptic generalizations of the Gauss hypergeometric function, we develop a common framework for 7parameter families of generalized elliptic, hyperbolic and trigonometric univariate hypergeometric functions. In each case we derive the symmetries of the generalized hypergeometric function under the Weyl group of type E7 (elliptic, hyperbolic) and of type E6 (trigonometric) using the appropriate versions of the NassrallahRahman beta integral, and we derive contiguous relations using fundamental addition formulas for theta and sine functions. The top level degenerations of the hyperbolic and trigonometric hypergeometric functions are identified with Ruijsenaars ’ relativistic hypergeometric function and the AskeyWilson function, respectively. We show that the degeneration process yields various new and known identities for hyperbolic and trigonometric special functions. We also describe an intimate connection between the hyperbolic and trigonometric theory, which yields an expression of the hyperbolic hypergeometric function as an explicit bilinear sum in trigonometric hypergeometric functions. 1.
WILSON FUNCTION TRANSFORMS RELATED TO RACAH COEFFICIENTS
, 2005
"... Abstract. The irreducible ∗representations of the Lie algebra su(1,1) consist of discrete series representations, principal unitary series and complementary series. We calculate Racah coefficients for tensor product representations that consist of at least two discrete series representations. We us ..."
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Abstract. The irreducible ∗representations of the Lie algebra su(1,1) consist of discrete series representations, principal unitary series and complementary series. We calculate Racah coefficients for tensor product representations that consist of at least two discrete series representations. We use the explicit expressions for the ClebschGordan coefficients as hypergeometric functions to find explicit expressions for the Racah coefficients. The Racah coefficients are Wilson polynomials and Wilson functions. This leads to natural interpretations of the Wilson function transforms. As an application several sum and integral identities are obtained involving Wilson polynomials and Wilson functions. We also compute Racah coefficients for Uq(su(1, 1)), which turn out to be AskeyWilson functions and AskeyWilson polynomials. 1.
THE JMATRIX METHOD: A SURVEY OF TRIDIAGONALIZATION
, 810
"... Abstract. Given an operator L acting on a function space, the Jmatrix method consists of finding a sequence yn of functions such that the operator L acts tridiagonally on yn. Once such a tridiagonalization is obtained, a number of characteristics of such an operator L can be obtained. In particular ..."
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Abstract. Given an operator L acting on a function space, the Jmatrix method consists of finding a sequence yn of functions such that the operator L acts tridiagonally on yn. Once such a tridiagonalization is obtained, a number of characteristics of such an operator L can be obtained. In particular, information on eigenvalues and eigenfunctions, bound states, spectral decompositions, etc. can be obtained in this way. We review the general setup, and we discuss two examples in detail; the Schrödinger operator with Morse potential and the Lamé equation. 1.