Mersenne-Horadam identities using generating functions

  • R. Frontczak Landesbank Baden-Württemberg, 70173, Stuttgart, Germany
  • T.P. Goy Vasyl Stefanyk Precarpathian National University, 57 Shevchenka str., 76018, Ivano-Frankivsk, Ukraine https://orcid.org/0000-0002-6212-3095
Keywords: Mersenne numbers, Horadam sequence, Fibonacci sequence, Lucas sequence, Pell sequence, generating function, binomial transform
Published online: 2020-06-12

Abstract


The main object of the present paper is to reveal connections between Mersenne numbers $M_n=2^n-1$ and generalized Fibonacci (i.e., Horadam) numbers $w_n$ defined by a second order linear recurrence $w_n=pw_{n-1}+qw_{n-2}$, $n\geq 2$, with $w_0=a$ and $w_1=b$, where $a$, $b$, $p>0$ and $q\ne0$ are integers. This is achieved by relating the respective (ordinary and exponential) generating functions to each other. Several explicit examples involving Fibonacci, Lucas, Pell, Jacobsthal and balancing numbers are stated to highlight the results.

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How to Cite
(1)
Frontczak R., Goy T. Mersenne-Horadam Identities Using Generating Functions. Carpathian Math. Publ. 2020, 12 (1), 34-45.