Evaluation of Integrals Involving Powers of (1-x exp 2 ) and Two Associated Legendre Functions or Gegenbauer Polynomials

摘要

Integrals involving powers of (1-x exp 2 ) and two associated Legendre functions or two Gegenbauer polynomials are evaluated as finite sums which can be expressed in terms of terminating hypergeometric function sub 4 F sub 3 . The integrals which are evaluated are integral sub(-1)sup(1)(Psub(l)sup(m)(x)Psub(k)sup(n)(x))/((1-x exp 2 )sup(p+1))dx and integral sub(-1)sup(1)Csub(l)sup( alpha )(x)Csub(k)sup( beta )(x)((1-x exp 2 )sup(( alpha + beta -3)/2-p))dx where k,l,m,n,p are integers. The answers are presented in two forms: one in which the symmetry with respect to the interchange between l,m and k,n (l, alpha and k, beta ) is evident, and the other, in which the number of terms in the diagonal cases is at most equal to p+1 for p>=0 and -p for p<0. (Atomindex citation 16:027838)