Successive derivatives of Fibonacci type polynomials of higher order in two variables

Abstract

The purpose of this work is to compute the successive derivatives of Fibonacci type polynomials in two variables; polynomials these introduced by G. Ozdemir,Y. Simsek in [3] and generalized by G. Ozdemir, Y. Simsek and G. Milovanović in [2] to a higher order. In addition we construct their recursive formula different of that given in Theorem 2.2 [3] p.6. Finally we define a novel generalized class of those polynomials similar to that given in [1] and found its recursive formula.

An affirmative answer to two questions concerning special case of Simsek numbers and open problems

“…Taking f 0 (x, t) = 1 1−(e t +1)x and g(x, t) = 1 − x − xe t . By using some techniques illustrated in the work [4] we deduce for n ≥ 1 that n j=0 n j ∂ j ∂t j f 0 (x, t) ∂ n−j ∂t n−j g(x, t) = 0 and then…”

“…The following theorem explain how the numbers B(n, k, λ) are connected to first kind Apostol-Euler numbers E From the well-known Cauchy product of two generating function [4] we deduce that Since e t + 1 = n≥0 B(n, 1) t n n! and comparing with the identity e t + 1 = 2 + n≥1 t n n!…”

“…Using Leibnitz formula (see [4]) for computing derivative at any order of product of two functions we conclude that…”

On the Cotangent Sums Related to Estermann Zeta Function and Arithmetic Properties of their Arguments

“…It's interesting to remember the proof of the explicit formula by using Cauchy product of series (see [7]) and associated arithmetical properties. Taking |x| < 1, it is well known that…”

On the recursion formula of polynomials generated by rational functions

“…In this section we revisit the work [2,3] and illustrate how to get a new recursion formula of generalized Catalan polynomials P λ,A i,m (x) defined by 1 + A(x)t…”