The infinite sum of the cubes of reciprocal Pell numbers

Abstract

Given the sequence of Pell numbers {P n }, we evaluate the integral part of the reciprocal of the sum ∞ k=n 1 P 3 n explicitly in terms of the Pell numbers themselves. MSC: Primary 11B39

The Reciprocal Sums of the Pell Numbers

“…Wang AYZ, Zhang F. [3] The Pell numbers also provide boundless opportunities to experiment, explore, and conjecture, they are a lot of fun for inquisitive amateurs and professionals alike. The authors [7] and [8] studied the infinite sums derived from the Pell numbers and proved the following identities: k=n∞1Pk−1=Pn−1+Pn−2, if n is even and n≥2,Pn−1+Pn−2−1 , if n is odd and n≥1. (1.12) k=n∞1Pk2−1=2Pn−1+Pn−1, if n is even and n≥2,2Pn−1+Pn , if n is odd and n≥1. (1.13) Xu and Wang [9] proofed the following interesting identities for the Pell numbers:…”

Partial reciprocal sums of the Mathieu series

“…Xu and Wang [ 8 ] studied a similar problem. They considered the infinite sum of cubes of reciprocal and then obtained a complex computational formula for …”

On the reciprocal sums of generalized Fibonacci numbers

“…where x ≥ 2. For other interesting results, see [5], [6], [7], [8], [9] and references cited therein.…”