Mersenne-Horadam identities using generating functions

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 wn=pw_{n−1}+qw_{n−2}, n≥2, with w_0=a and w_1=b, where a, b, p >0 and q≠0 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.