Evaluation of certain families of log-cosine integrals using hypergeometric function approach and applications

In this paper, we provide the analytical solutions of the families of certain definite integrals: $\int_0^\pi x^{m}\{\ln(2\cos\frac{x}{2})\}^{n}dx$ $(m\in\mathbb{N}_{0}$ and $n\in\mathbb{N}),$ in terms of multiple hypergeometric functions of Kampé de Fériet having the arguments $\pm1$ and Riemann zeta functions. As applications, we obtain some mixed summation formulas (19), (35) and (46) involving generalized hypergeometric functions $_3F_2,$ $_5F_4$ and $_7F_6$ having the arguments $\pm 1$ and other (possibly) new summation formulas (38) and (40) for multiple hypergeometric functions of Kampé de Fériet having the arguments $\pm 1$ also mixed relations (36) and (47) involving Riemann zeta functions.