On a class of generalized elliptic-type integrals (Q2764613)

The integrals are of the form NEWLINE\[NEWLINE R_\gamma^{\omega}(x,l,\beta)=\int_0^\pi { \cos^{2l/\omega-1}(\theta/2)(1-\cos^{2/\omega} (\theta/2))^{\beta-l-1}\sin(\theta/2) \over (1-x^2\cos\theta)^{\gamma+1/2} } d\theta, NEWLINE\]NEWLINE where \(0\leq x<1\), \(\omega>0\), \(\text{Re} \beta>\text{Re} l\geq 0\), \(\text{Re} \gamma>-1/2\). Series expansions and recurrence relations are obtained, and the relation with Wright's hypergeometric function is established. Several earlier results follow as special cases of the results in this paper.