Summing a family of generalized Pell numbers

2020/10/27 by Helmut Prodinger, Prodinger, Helmut
Mathematics · #FOS: Mathematics #Number Theory (math.NT) #math.NT

paper · pdf · doi:10.48550/arxiv.2010.14321

arxiv created 2020/10/27 · arxiv updated 2020/10/28

Abstract

A new family of generalized Pell numbers was recently introduced and studied by Bród \citeDorota. 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.

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