Open AccessMersenne-Horadam identities using generating functions
Author(s) -
Robert Frontczak,
Taras Goy
Publication year - 2020
Publication title -
karpatsʹkì matematičnì publìkacìï
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.63
H-Index - 4
eISSN - 2313-0210
pISSN - 2075-9827
DOI - 10.15330/cmp.12.1.34-45
Subject(s) - fibonacci number , mersenne prime , mathematics , lucas number , combinatorics , order (exchange) , exponential function , lucas sequence , arithmetic , fibonacci polynomials , object (grammar) , discrete mathematics , mathematical analysis , computer science , finance , economics , orthogonal polynomials , difference polynomials , artificial intelligence
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.