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

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 in, 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 in-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. (c) 2007 Elsevier Ltd. All rights reserved.