Some integrals and series involving the Gegenbauer polynomials and the Legendre functions on the cut (-1,1)

In this study, we use the recent findings of Cohl [On a generalization of the generating function for Gegenbauer polynomials, arXiv: 1105.2735v1] and evaluate two integrals involving the Gegenbauer polynomials: integral(x)(-1) dt(1-t(2))(lambda-1/2)(x-t)C--kappa-1/2(n)lambda(t) and integral(1)(x) dt(1-t(2))(lambda-1/2)(t-x)C--kappa-1/2(n)lambda (t), both with Re lambda > -1/2, Re kappa < 1/2, -1 < x < 1. The results are expressed in terms of the on-the-cut associated Legendre functions P-n+lambda-1/2(kappa-lambda)(+/- x) and Q(n+lambda-1/2)(kappa-lambda)(x). In addition, we find closed-form representations of the series Sigma(infinity)(n=0)(+/-)(n)[(n+lambda)/lambda]P-n+lambda-1/2(kappa-lambda)(+/- x)C-n(lambda)(t) and Sigma(infinity)(n=0) (+/-)(n) [(n+lambda)/lambda]Q(n+lambda-1/2)(kappa-lambda)(+/- x)C-n(lambda)(t), both with Re lambda > -1/2, Re kappa < 1/2, -1 < t < 1, -1 < x < 1.