Generating functions for reciprocal Catalan-type sums: approach to linear differentiation equation and ($p$-adic) integral equations

Abstract

This article is inspired by the reciprocal Catalan sums associated with problem 11765, proposed by David Beckwith and Sag Harbor. For this reason, partial derivative equations, the first-order linear differentiation equation and integral representations for series and generating functions for reciprocal Catalan-type sums containing the Catalan-type numbers are constructed. Some special values of these series and generating functions, which are given solutions of problem 11765, are found. Partial derivative equations of the generating function for the Catalan-type numbers are given. By using these equations, recurrence relations and derivative formulas involving these numbers are found. Finally, applying the $p$-adic Volkenborn integral to the Catalan-type polynomials, some combinatorial sums and identities involving the Bernoulli numbers, the Stirling numbers and the Catalan-type numbers are derived.