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: ∫₀^π xᵐln(2/2)\ⁿdx (m₀ and n), in terms of multiple hypergeometric functions of Kampé de Fériet having the arguments ±1 and Riemann zeta functions. As applications, we obtain some mixed summation formulas (19), (35) and (46) involving generalized hypergeometric functions ₃F₂, ₅F₄ and ₇F₆ having the arguments ± 1 and other (possibly) new summation formulas (38) and (40) for multiple hypergeometric functions of Kampé de Fériet having the arguments ± 1 also mixed relations (36) and (47) involving Riemann zeta functions.