Evaluation of integrals involving powers of (1-x2) and two associated Legendre functions or Gegenbauer polynomials

The author considers the integral integral -1 1 P l+ alpha - alpha (x)P k+ beta - beta (x)(1-x 2 ) -p-1 dx where l, k are non-negative integers and Re( 1 / 2 ( alpha + beta )-p)>0 for convergence as a multiple of a terminating Saalchutzian hypergeometric functions 4 F 3 of unit argument and utilises the results to compute the following special integrals: (i) integral -1 1 P l m (x)P k n (x)(1-x 2 ) -p-1 dx where l, k, m, n are non-negative integers and l>or=m>or=0, k>or=n>or=0, Re( 1 / 2 (m+n)-p)>0; (ii) integral -1 1 C l alpha (x)C k beta (x)(1-x 2 ) ( alpha + beta -3) 2-p/dx where l, k are non-negative integers and Re( 1 / 2 ( alpha + beta -1)-p)>0. The symbols P alpha beta (x) and C l alpha (x) represent the associated Legendre functions and Gegenbauer polynomials respectively. Both of the above integrals are expressed as multiples of a terminating Saalchutzian hypergeometric functions 4 F 3 of unit argument. For the special case m=n( alpha = beta ) and integral p, the 4 F 3 function in (i) and (ii) has min(p+1, mod 1 / 2 l mod +1) terms for p>or=0 and min(-p, mod 1 / 2 l mod +1) for p<0. The work generalises the known results for diagonal integrals to off-diagonal integrals of the type given in (i) and (ii).