On a Family of Infinite Series with Reciprocal Catalan Numbers

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.

Sums of Pell/Lucas Polynomials and Fibonacci/Lucas Numbers

“…In particular, when x = 1 2 , we confirm the following identity about Lucas numbers, which was proposed recently by Ohtsuka [16] as a problem in "Fibonacci Quarterly". More identities of a similar type were derived recently by Adegoke-Frontczak-Goy [4].…”

“…Now, by letting y → u u+v and y → v u+v , and then by adding the two resulting equations, we derive the following formula. Its equivalent forms can be found in two different papers by Adegoke-Frontczak-Goy [4], and Lehmer [20].…”

“…We begin with the following infinite series identity. An equivalent form of it can be found in Adegoke-Frontczak-Goy [4], Lehmer [20], and Zucker [21].…”

“…As polynomial extensions of the Fibonacci/Lucas numbers, the Pell/Lucas polynomials have many remarkable properties and wide applications in mathematics, physics, and computer sciences (see Koshy [1,2] and Grimaldi [3]). There exist numerous summation formulae involving Fibonacci/Lucas numbers [4][5][6][7][8] and the Pell/Lucas polynomials [9][10][11][12][13][14][15]. Motivated by a beautiful formula (see Corollary 3) due to Ohtsuka [16], we shall investigate in this paper further summation formulae involving inverse central binomial coefficients and Pell/Lucas polynomials, as well as their applications to identities concerning Fibonacci/Lucas numbers.…”

Fibonacci-Catalan Series

“…Other recently published works on infinite sums with (reciprocal) Catalan numbers and central binomial coefficients include the articles by Zhao and Wang [32], Sprugnoli [24], Chu and Zheng [12], Boyadzhiev [8,9], Chen [10], Frontczak et al [17] and Uhl [27]. Some Fibonacci-Catalan series were also derived, among other results, in a recent article by Adegoke et al [2,Section 2].…”