Some results on integrals involving generalized Jacobi and related functions (Q1903763)

The partial derivatives of the integral \[ I_{a,b}^{\nu, m} (z; \lambda, \mu)= \int^1_{-1} (1-x)^a (1+x)^b S^m_\nu (z(1- x)^\lambda (1+ x)^\mu )dx, \] where \[ S^m_\nu (z)= \sum^\infty_{k=0} {{(-\nu)_{mk}} \over {k!}} A_{\nu,k} z^k, \] with respect to \(a\) and \(b\) are established, and formulae for linear combinations of these derivatives are also given. Many known integral formulae would follow from these results by suitably specializing the parameters and the \(A\)-sequence. A few such cases are indicated, e.g., the Gould-Hopper polynomials.