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A Combinatorial Formula for the Associated Legendre Functions of Integer Degree

✍ Scribed by J.F. van Diejen; A.N. Kirillov

Publisher
Elsevier Science
Year
2000
Tongue
English
Weight
552 KB
Volume
149
Category
Article
ISSN
0001-8708

No coin nor oath required. For personal study only.

✦ Synopsis

dedicated to professor richard a. askey on the occasion of his 65th birthday

We apply inverse scattering theory to a Schro dinger operator with a regular reflectionless Po schl Teller potential on the line, to arrive at a combinatorial formula for the associated Legendre functions of integer degree. The expansion coefficients in the combinatorial formula are identified as dimensions of irreducible representations of gl(N), where N corresponds to the degree of the associated Legendre function. As an application, combinatorial formulas for the zonal spherical functions on the real hyperboloids H 2N+3, 1 =SO 0 (2N+3, 1; R)Γ‚SO 0 (2N+ 2, 1; R), H 1, 2N+3 + =SO 0 (2N+3, 1; R)Γ‚SO(2N+3; R) and the sphere S 2N+3 = SO(2N+4; R)Γ‚SO(2N+3; R) are presented.

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