Some explicit and asymptotic formulas related to the generalized arithmetic triangles

This talk first introduces the m-arithmetic triangle — a generalization of Pascal’s triangle, and then presents an explicit formula for the numbers of the m-arithmetic triangle; we have found some asymptotic expansions for these numbers. With a new formula for inversion of analytic functions, which we introduced recently, we get inversion formulas, in which new special numbers An appear. We present a recurrence formula for An using an analogue of the Pascal’s triangle, and give a relation formula of An with Bernoulli numbers. We also derive a new explicit formula for Bernoulli numbers. In conclusion, we present the power series expansions for some elementary functions as well as asymptotic expansions of certain special functions that involve An.