DDNSDiscrete Dynamics in Nature and Society1607-887X1026-0226Hindawi Publishing Corporation68428010.1155/2012/684280684280Research ArticleOn Reciprocal Series of Generalized Fibonacci Numbers with Subscripts in Arithmetic ProgressionÖmürNeşeBravermanElena1Department of MathematicsKocaeli University41380 IzmitTurkeykocaeli.edu.tr2012512012201224102011161120112012Copyright © 2012 Neşe Ömür.This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

We investigate formulas for closely related series of the forms: n=01/(Uan+b+c), n=0(-1)nUan+b/(Uan+b+c)2, n=0U2(an+b)/(Uan+b2+c)2 for certain values of a, b, and c.

1. Introduction

Let p be a nonzero integer such that Δ=p2+40. The generalized Fibonacci and Lucas sequences are defined by the following recurrences:Un+1=pUn+Un-1,Vn+1=pVn+Vn-1,   where U0=0,  U1=1 and V0=2,  V1=p, respectively. When p=1,  Un=Fn (nth Fibonacci number) and Vn=Ln (nth Lucas number).

If α and β are the roots of equation x2-px-1=0, the Binet formulas of the sequences {Un} and {Vn} have the forms:Un=αn-βnα-β,Vn=αn+βn, respectively.

In , Backstrom developed formulas for closely related series of the form:n=01Fan+b+c, for certain values of a, b, and c. For example, he obtained the following series:n=01F2n+1+FK=K52LK,n=01F(2n+1)K+2t+FK={(5-5Ft/Lt)2LK,t  even,(5-Lt/Ft)2LK,t  odd, where K represents an odd integer and t is an integer in the range -(K-1)/2 to (K-1)/2 inclusive. Also, he gave the similar results for Lucas numbers.

In , Popov found in explicit form series of the form:n=01Fan+b±c,n=01Fan+bFcn+d,  n=01Fan+b2±Fcn+d2, for certain values of a, b, c, and d.

In , Popov generalized some formulas of Backstrom  related to sums of reciprocal series of Fibonacci and Lucas numbers. For example, Δn=0(-q)s+nrV(2n+1)r+2s-(-q)s+nrVr={βsUrUs,|βα|<1,αsUrUs,|αβ|<1, where s and r are integers.

In , Gauthier found the closed form expressions for the following sums:k=1m(-1)knf(2k+1)nf(k+1)n2fkn2,m,n1,k=0m(-1)knf(2k+1)nl(k+1)n2lkn2,m,n0, where for x0 an indeterminate, the generalized Fibonacci and Lucas polynomials {fn}n and {ln}n are given by the following recurrences:fn+2=xfn+1+fn,f0=0,  f1=1,  n0,ln+2=xln+1+ln,l0=2,  l1=x,  n0, respectively.

In this paper, we investigate formulas for closely related series of the forms:n=01Uan+b+c,  n=0(-1)nUan+b(Uan+b+c)2,  n=0U2(an+b)(Uan+b2+c)2, for certain values of a, b and c.

2. On Some Series of Reciprocals of Generalized Fibonacci Numbers

In this section, firstly, we will give the following lemmas for further use.

Lemma 2.1.

Let n be an arbitrary nonzero integer. For integer m1, k=1m(-1)knU(2k+1)nU(k+1)n2Ukn2=14Un(Vn2Un2-V(m+1)n2U(m+1)n2), and for integer m0, k=0m(-1)knU(2k+1)nV(k+1)n2Vkn2=U(m+1)n24UnV(m+1)n2.

Proof.

We give the proof of Lemma 2.1 as the proofs of the sums in , using the following equalities: U(2k+1)nU(k+1)nUkn=12(VknUkn+V(k+1)nU(k+1)n),U(2k+1)nV(k+1)nVkn=12(U(k+1)nV(k+1)n+UknVkn),(-1)knUnU(k+1)nUkn=12(VknUkn-V(k+1)nU(k+1)n),(-1)knUnV(k+1)nVkn=12(U(k+1)nV(k+1)n-UknVkn).

Lemma 2.2.

For arbitrary integers n and t, V2n-(-1)n-tV2t=ΔUn-tUn+t,V2n+(-1)n-tV2t=Vn-tVn+t,Un2-(-1)n-tUt2=Un-tUn+t,Vn2-(-1)n-tVt2=ΔUn-tUn+t.

Proof.

From Binet formulas of sequences {Un} and {Vn}, the desired results are obtained.

Theorem 2.3.

For an odd integer t, n=1m1U(2n+1)t+Ut=12Vt(2-V2tU2t-2-V2(m+1)tU2(m+1)t),n=1m1U(2n+1)t-Ut=12Vt(2+V2tU2t-2+V2(m+1)tU2(m+1)t).

Proof.

By replacing n with (2n+1)t in (2.5), we have U(2n+1)t2-Ut2=U2ntU2(n+1)t, or 1U(2n+1)t+Ut=U(2n+1)t-UtU2ntU2(n+1)t. Taking r=(2n+1)t and s=t in the equality VsUr=Ur+s+(-1)sUr-s , the equality (2.8) is rewritten as follows: 1U(2n+1)t+Ut=1Vt(1U2nt+(-1)tU2(n+1)t)-UtU2ntU2(n+1)t. We have the sum n=1m1U(2n+1)t+Ut=1Vtn=1m(1U2nt+(-1)tU2(n+1)t)-Utn=1m1U2ntU2(n+1)t. For an odd integer t, we have n=1m(1U2nt-1U2(n+1)t)=1U2t-1U2(m+1)t, and taking s=2nt and r=2t in identity : Us+rVs-UsVs+r=2(-1)sUr, we get n=1m1U2ntU2(n+1)t=12U2tn=1m(V2ntU2nt-V2(n+1)tU2(n+1)t)=12U2t(V2tU2t-V2(m+1)tU2(m+1)t).

Substituting (2.11) and (2.13) in (2.10), we have the desired result.

For example, if we take t=1 and p=1 in (2.6), we have n=1m1F2n+1+1=F2m+1-1F2(m+1). Note that F2(m+1)n=1m1F2n+1+1=n=1mF2n.

Corollary 2.4.

For an odd integer t, n=1m1U(2n+1)t+Ut={12Vt(V(m+1)tU(m+1)t-VtUt),m  is  even,12Vt(ΔU(m+1)tV(m+1)t-VtUt),m  is  odd,n=1m1U(2n+1)t-Ut={Δ2Vt(UtVt-U(m+1)tV(m+1)t),m  is  even,12Vt(ΔUtVt-V(m+1)tU(m+1)t),m  is  odd.

Proof.

Using the equalities V2n=Vn2-2(-1)n=ΔUn2+2(-1)n and U2n=UnVn in Theorem 2.3, the results are obtained.

Corollary 2.5.

Let t be an odd integer. For |β/α|<1,t>0 and |α/β|<1,t<0, n=11U(2n+1)t+Ut=12Vt(Δ-VtUt),n=11U(2n+1)t-Ut=Δ2Vt(UtVt-1Δ), and for |β/α|<1,t<0 and |α/β|<1,t>0, n=11U(2n+1)t+Ut=-12Vt(Δ+VtUt),n=11U(2n+1)t-Ut=Δ2Vt(UtVt+1Δ).

Proof.

Since limn(Van+bUan+b)={Δ|βα|<1,a>0,|αβ|<1,a<0,-Δ|βα|<1,a<0,|αβ|<1,a>0, the results are easily seen by equalities (2.16).

Theorem 2.6.

For an integer m1 and an arbitrary nonzero integer t, n=1m(-1)ntU(2n+1)t(V(2n+1)t-(-1)ntVt)2=14Δ2Ut(Vt2Ut2-V(m+1)t2U(m+1)t2).

Proof.

By replacing n with (2n+1)t/2 and t with t/2 in (2.4), we have V(2n+1)t-(-1)ntVt=ΔUntU(n+1)t, or 1V(2n+1)t-(-1)ntVt=1ΔUntU(n+1)t. Multiplying equality (2.22) by (-1)ntU(2n+1)t/UntU(n+1)t, we get (-1)ntU(2n+1)tUntU(n+1)t(V(2n+1)t-(-1)ntVt)=(-1)ntU(2n+1)tΔUnt2U(n+1)t2. We have the sum: n=1m(-1)ntU(2n+1)tUntU(n+1)t(V(2n+1)t-(-1)ntVt)=1Δn=1m(-1)ntU(2n+1)tUtn2Ut(n+1)2. Using the equalities (2.1) and (2.21), the proof is obtained.

Corollary 2.7.

For an arbitrary nonzero integer t, n=1(-1)ntU(2n+1)t(V(2n+1)t-(-1)ntVt)2=14Δ2Ut(Vt2Ut2-Δ).

Proof.

Taking m in Theorem 2.6 and using (2.19), the result is easily obtained.

Theorem 2.8.

For an integer m0 and an arbitrary nonzero integer t, n=0m(-1)ntU(2n+1)t(V(2n+1)t+(-1)ntVt)2=U(m+1)t24UtV(m+1)t2.

Proof.

The proof of the theorem is similar to the proof of Theorem 2.6.

Corollary 2.9.

For an arbitrary nonzero integer t, n=0(-1)ntU(2n+1)t(V(2n+1)t+(-1)ntVt)2=14ΔUt.

Proof.

Taking m in Theorem 2.8 and using (2.19), the result is easily obtained.

For example, if we take t=3 and p=1 in (2.27), we have n=0(-1)nF3(2n+1)(L3(2n+1)+(-1)n4)2=140.

Theorem 2.10.

For an integer m1 and an arbitrary nonzero integer t, n=1mU2(2n+1)t(U(2n+1)t2-Ut2)2=14U2t(V2t2U2t2-V2(m+1)t2U2(m+1)t2),n=1mU2(2n+1)t(V(2n+1)t2-Vt2)2=14Δ2U2t(V2t2U2t2-V2(m+1)t2U2(m+1)t2).

Proof.

The proof of theorem is similar to the proof of Theorem 2.6.

Corollary 2.11.

For an arbitrary nonzero integer t, n=1U2(2n+1)t(U(2n+1)t2-Ut2)2=14U2t(V2t2U2t2-Δ),n=1U2(2n+1)t(V(2n+1)t2-Vt2)2=14Δ2U2t(V2t2U2t2-Δ).

Proof.

Taking m in Theorem 2.10 and using (2.19), the result is easily obtained.

For example, if we take t=2 in the equality (2.30), we have n=1U4(2n+1)(V2(2n+1)2-V22)2=1Δ2U43.

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