On some series involving the binomial coefficients (cid:0) 3 n

: Using a simple transformation, we obtain much simpler forms for some series involving binomial coefficients (cid:0) 3 nn (cid:1) derived by Necdet Batır. New evaluations are given and connections with Fibonacci numbers and the golden ratio are established. Finally we derive some Fibonacci and Lucas series involving the reciprocals of (cid:0) 3 nn (cid:1) .


Introduction
In an article published in the year 2005, Batır [1], inspired by the results of Lehmer [6], studied the series k n , giving particular attention to the special cases n ∈ N ∪ {0}, for which he derived explicit closed formulas.He obtained many interesting formulas by evaluating the closed forms at appropriate arguments.Some of his results had earlier been obtained experimentally by Borwein and Girgensohn [2].In [4], D'Aurizio and Di Trani studied this kind of series using hypergeometric functions.In the recent paper [3], Chu evaluated many series having the form where a ∈ {0; ±1; ±2} and b ∈ {0; 1; ±2}.
The purpose of this note is to derive equivalent but much simpler expressions for the special cases and thereby obtain new evaluations.

Evaluations at selected arguments
In this section we will evaluate identities (A), (B) and (C) at carefully selected values of x and y.Some of the resulting summation identities will involve Fibonacci and Lucas numbers in the summand and possibly in the evaluations.Let F n and L n denote the n-th Fibonacci and Lucas numbers, both satisfying the recurrence relation Extending Fibonacci and Lucas numbers to negative subscripts gives F −j = (−1) j−1 F j and L −j = (−1) j L j .Throughout this paper, we denote the golden ratio α = 1+ 2 , so that αβ = −1 and α + β = 1.For any integer j, the explicit formulas (Binet formulas) for Fibonacci and Lucas numbers are We will often require the following identities, valid for any integer r, which are straightforward consequences of (6): We also require the following well-known identities [5,7]: 2.1 Results from identity (A) Proof.Identity (17) is proved by setting x = α 2r , y = (−1) r+1 in (A) and making use of identity (7).Identity (18) follows from setting x = α 2r , y = (−1) r in (A) and using (8).
Theorem 2.2.Let m and n be positive integers such that n ≥ m unless stated otherwise.Then Proof.Straightforward using identities (11) to (16) and identity (A).
Example 2.4.1.Evaluation of (26) at r = 1, 2, 3 and (27) at r = 2, 3, 6, respectively, gives 3 Fibonacci and Lucas series involving inverses of the binomial coefficients 3n n In this section we will derive Fibonacci and Lucas identities which contain reciprocals of the binomial coefficients 3n n .
Lemma 3.1.[5] If p and q are integers, then 3.1 Fibonacci series associated with identity (A) Theorem 3.2.Let p and q be integers such that p ≤ −2, q ≥ 4 with q > |p| + 1. and Proof.Set (x, y) = (F p α q , −F p+q ) in identity (A) and use (28) to obtain Similarly, (x, y) = (F p+q , −β q F p ) in identity (A) and the use of (29) gives Identities ( 30) and (31) follow from the subtraction and addition of (32) and (33) with the use of the Binet formulas (6).
Example 3.2.1.At p = −2 and q = 5 from (30) and (31) we have the following series: 3.2 Fibonacci series associated with identity (B) Theorem 3.3.Let p and q be integers such that p ≤ −2, q ≥ 4, and q > |p| + 1.Then Proof.The proof is similar to that one given for Theorem 3.2 and omitted.

3
, where Proof.The proof is similar to the previous two proofs.