A RECURRENCE RELATION WITH COMBINATORIAL IDENTITIES

George Grossman1 §, Aklilu Zeleke2, Xinyun Zhu3, Tomas Zdrahal4 1Department of Mathematics Central Michigan University Mount Pleasant, MI 48859, USA 2Lyman Briggs College and Department of Statistics and Probability Michigan State University East Lansing, MI 48824, USA 3Department of Mathematics University of Texas of the Permian Basin Odessa, TX 79762, USA 4Katedra matematiky Palacký University Olomouc, CZECH REPUBLIC

The objective of this paper is to investigate a pair of (nonhomogeneous) second order recurrence relations in one unknown w k,n involving two subscripts n and k.We then outline a methodology to derive new representations of Fibonacci numbers and give two results.Recurrence relations and generating functions are essentially the focus of the present paper.The paper is organized as follows.In Section 2 we find the generating function for w k,n .Several other formulas for w k,n are given, including the general solution using two approaches.In Section 3 we will present some representations of the Fibonacci numbers by applying the main results.

Main Results
The generating function of the standard linear recurrence relation −w n − w n+1 + w n+2 = 0 that generates the Fibonacci numbers with F 0 = 0, F 1 = 1, is known to be (2.1) Consider now the linear recurrence relation with initial conditions Then, by using standard methods [11], [4], one can show that its generating function has the form Similarly the generating function of the nonhomogeneous linear recurrence equation .
The following theorems show how to generalize this result for a double indexed recurrence relation.
This result first appeared in [5] and later in [6].The generating function g(x) for (2.3) is given by Theorem 2. Let w k,n be as in (2.3).Then the generating function g(x) is given by After employing (2.5) in (2.3), multiplying by x n and summing, we obtain from which the result follows.

Consider now the polynomials p
is the characteristic polynomial of (2.3).The zeros of p 1 (x), p 2 (x) and the relations among them reveal some interesting facts.Let the zeors of p 1 (x) be β 1 and β 2 that of p 2 (x) be α 1 and α 2 .Then, and (2.9) This leads then to Proposition 1. (2.10) Proof.The proof follows from (2.6-2.9).
Using Proposition 1 and Theorem 2 we get, Proposition 2.
Proof.In Proposition 1, divide by x, sum from n = 1 and use the definiton of p 1 (x).
From Theorem 1 and Proposition 2 it follows that, Proof.This follows from Propostion 1 and the definition of the n-th Fibonacci number Next, a closed form for w k,n is found using two approaches: analytic and inductive approach.Theorem 3.
From Theorem 3 one gets Corollary 2.

.22)
With w k,n given by 2.11 and employing Proposition 2 it follows that, Theorem 4. (2.23) Proof.In Proposition 2, multiply both sides by (1 − x) k+1 , use definition of g(x) and expand to find power of x n .Corollary 3. (2.24) The case b = 0 is also considered to get the following,
Several examples involving the golden section, the Fibonacci and the Lucas numbers are presented in next section.

Combinatorial Identities
In this section we present some examples for the representation of a linear combination of the Fibonacci and Lucas numbers that involve a polynomial and a double sum combinatorial term.These representations are derived by employing the method of undetermined coefficients for specific values of k in (2.3).