On the m-extension of the Fibonacci and Lucas p-numbers

Abstract

In this article, we define the m-extension of the Fibonacci and Lucas p-numbers ( p ⩾ 0 is integer and m > 0 is real number) from which, specifying p and m, classic Fibonacci and Lucas numbers ( p = 1, m = 1), Pell and Pell–Lucas numbers ( p = 1, m = 2), Fibonacci and Lucas p-numbers ( m = 1), Fibonacci m-numbers ( p = 1), Pell and Pell–Lucas p-numbers ( m = 2) are obtained. Afterwards, we obtain the continuous functions for the m-extension of the Fibonacci and Lucas p-numbers using the generalized Binet formulas. Also we introduce in the article a new class of mathematical constants – the Golden ( p, m)-Proportions, which are a wide generalization of the classical golden mean, the golden p-proportions and the golden m-proportions. The article is of fundamental interest for theoretical physics where Fibonacci numbers and the golden mean are used widely.