Abstract
This paper aims to obtain extorial type solutions of Fibonacci difference equations having shift value. A higher order Fibonacci summation formula of product of polynomial and extorial functions is obtained by higher order Fibonacci nabla difference operator, its inverse and the higher order Fibonacci numbers. Extorial function is a function obtained by replacing polynomial into polynomial factorials in the exponential function. Suitable examples with numerical verification are inserted to illustrate our main results.
Highlights
Difference equations usually describe the evolution of certain phenomena over the course of time
A standard approach in numerical integration of differential equations is to replace it by a suitable difference equation whose solutions can be obtained in a stable manner
The qualitative properties of solutions of the difference equations are quite different from the solutions of the corresponding differential equations
Summary
Difference equations usually describe the evolution of certain phenomena over the course of time. A standard approach in numerical integration of differential equations is to replace it by a suitable difference equation whose solutions can be obtained in a stable manner. The qualitative properties of solutions of the difference equations are quite different from the solutions of the corresponding differential equations. Solutions of several well known difference equations like Clairaut’s, Euler’s, Riccati’s, Bernoulli’s, Verhulst’s, Duffing’s, Mathieu’s and Volterra’s difference equations retain most of the properties of the corresponding differential equations. Call that sequence a solution of the equation. The sequence Fn of usual Fibonacci numbers is obtained by the recurrence relation Fn = Fn−1 + Fn−2, F0 = 1, F1 = 1, n ≥ 2 In [7], for an m-tuple a = (a1, a2, a3, · · · am) ∈ Rm, a-Fibonacci number is defined as n.
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