Summing a family of generalized Pell numbers

TL;DR

This paper derives explicit formulas for summing powers of a new family of generalized Pell numbers using Binet formulas and generating functions, despite complex algebraic manipulations.

Contribution

It introduces a method to explicitly sum powers of generalized Pell numbers with a parameter, expanding the understanding of their algebraic properties.

Findings

Explicit formulas for sums of powers of generalized Pell numbers.

Use of Binet formulas and generating functions for summation.

Handling complex algebraic expressions with computer algebra.

Abstract

A new family of generalized Pell numbers was recently introduced and studied by Br\'od \cite{Dorota}. These number possess, as Fibonacci numbers, a Binet formula. Using this, partial sums of arbitrary powers of generalized Pell numbers can be summed explicitly. For this, as a first step, a power $P_n^l$ is expressed as a linear combination of $P_{mn}$. The summation of such expressions is then manageable using generating functions. Since the new family contains a parameter $R=2^r$, the relevant manipulations are quite involved, and computer algebra produced huge expressions that where not trivial to handle at times.