On a family of infinite series with reciprocal Catalan numbers

TL;DR

This paper investigates a family of infinite series involving reciprocal Catalan numbers, providing explicit formulas, special case evaluations, and integral representations, revealing new relations with Fibonacci numbers.

Contribution

It introduces new explicit formulas and integral representations for a family of series with reciprocal Catalan numbers, including special case evaluations and Catalan-Fibonacci relations.

Findings

Closed-form evaluations for two special series

Explicit formulas and integral representations for the general family

Discovery of Catalan-Fibonacci relations

Abstract

We study a certain family of infinite series with reciprocal Catalan numbers. We first evaluate two special candidates of the family in closed form, where we also present some Catalan-Fibonacci relations. Then we focus on the general properties of the family and prove explicit formulas, including two types of integral representations.