Summation Formulas, Generating Functions, and Polynomial Division

We describe a general method that finds closed forms for partial sums of power series whose coefficients arise from linear recurrence relations. These closed forms allow one to derive a vast collection of identities involving the Fibonacci numbers and other related sequences. Although motivated by a...

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Bibliographic Details
Published in:Mathematics magazine 2022-12, Vol.95 (5), p.509-519
Main Authors: Berkove, Ethan, Brilleslyper, Michael A.
Format: Magazinearticle
Language:English
Subjects:
Algorithms
Fibonacci numbers
Functions (mathematics)
Identities
Mathematical analysis
Mathematical functions
Mathematical problems
Polynomials
Power series
Primary 05A15
Secondary 11B39
Sequences
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Summary:We describe a general method that finds closed forms for partial sums of power series whose coefficients arise from linear recurrence relations. These closed forms allow one to derive a vast collection of identities involving the Fibonacci numbers and other related sequences. Although motivated by a polynomial long division problem, the method fits naturally into a standard generating function framework. We also describe an explicit way to calculate the generating function of the Hadamard product of two generating functions, a construction on power series which resembles the dot product. This allows one to use the method for many examples where the recurrence relation for the coefficients is not initially known.
ISSN:0025-570X
1930-0980
DOI:10.1080/0025570X.2022.2127302