Abstract

Wt apply inverse scattering theory to a Schrodinger operator with a regular reflectionless Poschl-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,H- 1 =SO0(2N+3, 1; R)/SO0(2N+2. 1; R), H-1,H- 2N+3 =SO0(2N+3, 1; R)/SO(2N+3; R) and the sphere S2N+3 = SO(2N+4; R)/SO(2N + 3; R) are presented. (C) 2000 Academic Press.