Generating Functions for the Values of Some Dirichlet Functions at Positive Integer Points

This paper is devoted to representations of generating functions for the values of the Dirichlet beta function at even points, the Riemann zeta function, and related functions at odd points, i.e., the numbers $$\beta(2m)$$ , $$\zeta(2m+1)$$ , $$\lambda(2m+1)$$ , $$\eta(2m-1)$$ for $$m=1,2, \dots$$ , and some of their linear combinations, in the form of definite integrals of elementary functions and in the form of generalized hypergeometric and similar series. Some of the obtained results are known, while others are new. The paper also discusses consequences of the obtained theorems for the Catalan constant $$\beta(2)$$ , the Apéry constant $$\zeta(3)$$ , and the numbers $$\beta(4)$$ and $$\zeta(5)$$ .