On the m-extension of Fibonacci p-functions with period k

Abstract

Let fp;m be a real valued function on R, p be nonnegative integer, k be a positive integer and m be a nonnegative real number. For all x 2 R, fp;m(x + (p + 1)k) = mfp;m(x + pk) + fp;m(x), we call this function mð€€€ï¿½ï¿½ï¿½ï¿½ï¿½extension of Fibonacci pð€€€ï¿½ï¿½ï¿½ï¿½ï¿½function with period k. In this paper, we present basic properties of mð€€€ï¿½ï¿½ï¿½ï¿½ï¿½extension of Fibonacci pð€€€ï¿½ï¿½ï¿½ï¿½ï¿½functions with period k. Specifying p and m, we obtain Fibonacci (p = 1, m = 1) and Pell (p = 1, m = 2) functions. Furthermore, we de ne mð€€€ï¿½ï¿½ï¿½ï¿½ï¿½extension of odd Fibonacci pð€€€ï¿½ï¿½ï¿½ï¿½ï¿½functions with period k. Moreover, we analyze some properties by using notion of fð€€€ï¿½ï¿½ï¿½ï¿½ï¿½even and fð€€€ï¿½ï¿½ï¿½ï¿½ï¿½odd functions with period k. We also demonstrate the products and quotients of these functions and provide new results in the development of Fibonacci functions with period k.