Generating Function for Bernoulli Numbers and Its Generalizations

This paper explores certain generalizations of the generating function of Bernoulli numbers, the computation of integrals, and the investigation of the convergence of integrals from these functions. The primary tools employed in the research include the use of Taylor series, theorems on uniform continuity (such as Weierstrass's and Dini's theorems), as well as special functions such as the gamma function, incomplete gamma function, Riemann zeta function, and Lambert function. Various examples for specific parameter values are considered in the article. The obtained results can be strengthened in subsequent works and generalized to a broader class of functions. The derived estimates can be applied in various tasks related to the assessment of similar integrals.