A Property of Legendre Polynomials

Abstract

Remark I: The Legendre polynomials are the spherical functions for the symmetric space SO(3)/SO(2). A property analogous to that stated in the above theorem holds for other symmetric spaces. In this fashion we get also new properties for Gegenbauer polynomials, Bessel, and Legendre functions. The methods of proof presented in this note extend to these cases. Remark II: The case of Euclidean symmetric spaces can be easily handled once we have the property analogous to (2) for those of compact type. Remark III: If r and s are even and n is odd, (2) cannot hold close to 0 = r. But this is the only case in which the validity of (2) cannot be extended to the whole interval [0, 7r]. This is also the situation in all the cases SO(2p + 1)/ SO(2p). In the cases SO(2p)/SO(2p-1) the property analogous to (2) holds in [0, r] and thus it truly becomes an inequality for spherical functions. The latter is also the case for Euclidean and noncompact type of symmetric spaces. Proof of the Theorem: Assume 1 5, a restriction that will prove to be quite useful. Let f(x) stand for cos(x1/2). If one rewrites (1) as