Abstract
Abstract A new family of generalized Pell numbers was recently introduced and studied by Bród ([2]). These numbers 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 𝓁 n 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.
Highlights
The Pell numbers Pn are defined via the recursion Pn = 2Pn−1 + Pn−2 and P0 = 0, P1 = 1; they are sequence A000129 in [4], they possess a Binet formula
The goal of the present note is to compute the sums 0≤k≤n efficiently, for positive parameters m and. We presented such a treatment in [8] for Fibonacci numbers, and, more recently, in [9] for balancing numbers studied by Komatsu ([5]), but the current instance is definitely the most challenging, as it contains the parameter R, and that makes computations difficult
We used Maple’s package gfun for that. Let us rewrite this in terms of generating functions: Set
Summary
The Pell numbers Pn are defined via the recursion Pn = 2Pn−1 + Pn−2 and P0 = 0, P1 = 1; they are sequence A000129 in [4], they possess a Binet formula. Key words and phrases: Pell numbers, Binet formula, generating functions. A non-complete list of references about Pell numbers and generalizations is here: [1, 3, 10, 6]. 0≤k≤n efficiently, for positive parameters m and We presented such a treatment in [8] for Fibonacci numbers, and, more recently, in [9] for balancing numbers studied by Komatsu ([5]), but the current instance is definitely the most challenging, as it contains the parameter R, and that makes computations difficult (even for a computer)
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