# An application of hypergeometric shift operators to the chi-spherical Fourier transform

@article{Ho2015AnAO, title={An application of hypergeometric shift operators to the chi-spherical Fourier transform}, author={Vivian M. Ho and Gestur {\'O}lafsson}, journal={arXiv: Classical Analysis and ODEs}, year={2015} }

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 $\chi$-spherical functions on Hermitian symmetric spaces $U/K$ where $\chi$ is a nontrivial character of $K$. We apply shift operators to the hypergeometric functions to move negative multiplicities to positive ones. This allows us to use many well-known results of the… Expand

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#### References

SHOWING 1-10 OF 15 REFERENCES

Harmonic Analysis and Special Functions on Symmetric Spaces

- Mathematics
- 1995

Part 1 The hypergeometric differential operators: the periodic Calogero-Moser system the hypergeometric shift operators the hypergeometric function spherical functions of type x. Part 2 Structure… Expand

Paley-Wiener Theorem for Line Bundles over Compact Symmetric Spaces

- Mathematics
- 2012

Paley-Wiener type theorems describe the image of a given space of func- tions, often compactly supported functions, under an integral transform, usually a Fourier transform on a group or homogeneous… Expand

Root systems and hypergeometric functions. I

- Mathematics
- 1988

On considere les systemes dynamiques d'hamiltoniens: H=∑ i=1 n p i 2 /2+g∑ i<j (q i −q j ) −2 et H=∑ i=1 n p i 2 /2+g∑ i<j sin −2 (q i −q j ). On etudie la resolution spectrale simultanee de… Expand

The Plancherel Formula for Spherical Functions with a One-Dimensional K-Type on a Simply Connected Simple Lie Group of Hermitian Type

- Mathematics
- 1994

Abstract We generalize the Harish-Chandra inversion formula for the spherical transform on a Riemannian symmetric space to homogeneous line bundles on a Hermitian symmetric space. We determine the… Expand

Differential Geometry, Lie Groups, and Symmetric Spaces

- Mathematics
- 1978

Elementary differential geometry Lie groups and Lie algebras Structure of semisimple Lie algebras Symmetric spaces Decomposition of symmetric spaces Symmetric spaces of the noncompact type Symmetric… Expand

Some applications of hypergeometric shift operators

- Mathematics
- 1989

Disclaimer/Complaints regulations If you believe that digital publication of certain material infringes any of your rights or (privacy) interests, please let the Library know, stating your reasons.… Expand

Orthogonal polynomials in two variables which are eigenfunctions of two algebraically independent partial differential operators. III

- Mathematics
- 1974

Abstract Let the region S={(x, y)∥μ(x+iy, x−iy)>0} be the interior of Steiner's hypocycloid, where μ(z, z )=−z 2 z 2 +4z 3 +4 z 3 −18z z +27 . For each real α>− 5 6 an orthogonal system of… Expand

Groups and geometric analysis

- Mathematics
- 1984

Geometric Fourier analysis on spaces of constant curvature Integral geometry and Radon transforms Invariant differential operators Invariants and harmonic polynomials Spherical functions and… Expand

Harmonic analysis for certain representations of graded Hecke algebras

- Mathematics
- 1995

0. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 1. Notations and preliminaries . . . . . . . . . . . . . . . . . . . . . . . . 78 2. Cherednik's operators . . .… Expand

Root systems and hypergeometric functions III

- Medicine
- 1988

A hand held, motor driven, wire joining tool is disclosed which includes two wire gripping jaws adapted to twist wires gripped therein by counter-rotating with respect to each other and a heater for… Expand