Extended Hadamard products, trigonometric integrals and associated sums

The notion of the Hadamard product is extended to evaluate an extensive number of trigonometric integrals in terms of sums. These sums are taken over index sets defined by a Diophantine equation which can lie simplified in certain circumstances. The results obtained include generalizations of integrals defining sums of products of Bessel functions and integrals of powers of cosines evaluated in terms of sums of products of binomial coefficients. Generating functions for special polynomials are also called upon in some of the developments.