Hindawi Publishing Corporation Discrete Dynamics in Nature and Society Volume 2012, Article ID 684280, 7 pages doi:10.1155/2012/684280 Research Article On Reciprocal Series of Generalized Fibonacci Numbers with Subscripts in Arithmetic Progression Neşe Ömür Department of Mathematics, Kocaeli University, 41380 Izmit, Turkey Correspondence should be addressed to Neşe Ömür, neseomur@kocaeli.edu.tr Received 24 October 2011; Accepted 16 November 2011 Academic Editor: Elena Braverman Copyright q 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: n0 1/Uanb c, ∞ 2 n 2 ∞ 2 −1 U /U c , U /U c for certain values of a, b, and c. anb anb 2anb n0 n0 anb 1. Introduction Let p be a nonzero integer such that Δ p2 4 / 0. The generalized Fibonacci and Lucas sequences are defined by the following recurrences: Un1 pUn Un−1 , Vn1 pVn Vn−1 , 1.1 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 − βn , α−β Vn αn βn , 1.2 respectively. In 1, Backstrom developed formulas for closely related series of the form: ∞ 1 n0 Fanb c , 1.3 2 Discrete Dynamics in Nature and Society for certain values of a, b, and c. For example, he obtained the following series: √ ∞ 1 K 5 , F FK 2LK n0 2n1 ⎧ √ ⎪ 5 − 5F /L ⎪ t t ⎪ ⎪ ∞ ⎨ , t even, 1 √ 2LK F FK ⎪ ⎪ n0 2n1K2t ⎪ 5 − Lt /Ft ⎪ ⎩ , t odd, 2LK 1.4 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 2, Popov found in explicit form series of the form: ∞ 1 n0 Fanb ± c ∞ , n0 ∞ 1 , Fanb Fcnd 2 n0 Fanb 1 , 2 ± Fcnd 1.5 for certain values of a, b, c, and d. In 3, Popov generalized some formulas of Backstrom 1 related to sums of reciprocal series of Fibonacci and Lucas numbers. For example, ⎧ s β β ⎪ ⎪ < 1, , ⎪ snr ∞ ⎨ −q Ur Us α 1.6 Δ snr α αs Vr ⎪ ⎪ n0 V2n1r2s − −q ⎪ , < 1, ⎩ Ur Us β where s and r are integers. In 4, Gauthier found the closed form expressions for the following sums: m −1kn f2k1n k1 2 2 fk1n fkn m −1kn f2k1n k0 2 2 lk1n lkn m, n ≥ 1, , 1.7 m, n ≥ 0, , where for x / 0 an indeterminate, the generalized Fibonacci and Lucas polynomials {fn }n and {ln }n are given by the following recurrences: fn2 xfn1 fn , ln2 xln1 ln , f0 0, f1 1, n ≥ 0, 1.8 l0 2, l1 x, n ≥ 0, respectively. In this paper, we investigate formulas for closely related series of the forms: ∞ 1 n0 Uanb c , for certain values of a, b and c. ∞ −1n Uanb n0 2 Uanb c , ∞ n0 U2anb 2 c Uanb 2 , 1.9 Discrete Dynamics in Nature and Society 3 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 m ≥ 1, m −1kn U2k1n k1 2 2 Uk1n Ukn 1 4Un Vn2 Un2 − 2 Vm1n 2 Um1n , 2.1 and for integer m ≥ 0, m −1kn U2k1n 2 2 Vk1n Vkn k0 2 Um1n 2 4Un Vm1n . 2.2 Proof. We give the proof of Lemma 2.1 as the proofs of the sums in 4, using the following equalities: Vk1n U2k1n 1 Vkn , Uk1n Ukn 2 Ukn Uk1n U2k1n 1 Uk1n Ukn , Vk1n Vkn 2 Vk1n Vkn 2.3 Vk1n 1 Vkn −1kn Un − , Uk1n Ukn 2 Ukn Uk1n 1 Uk1n Ukn −1kn Un − . Vk1n Vkn 2 Vk1n Vkn Lemma 2.2. For arbitrary integers n and t, V2n − −1n−t V2t ΔUn−t Unt , V2n −1n−t V2t Vn−t Vnt , Un2 − −1n−t Ut2 Un−t Unt , Vn2 − −1n−t Vt2 ΔUn−t Unt . 2.4 2.5 Proof. From Binet formulas of sequences {Un } and {Vn }, the desired results are obtained. Theorem 2.3. For an odd integer t, m 1 n1 U2n1t Ut m 1 n1 1 2Vt 1 U2n1t − Ut 2Vt 2 − V2t 2 − V2m1t − , U2t U2m1t 2 V2t 2 V2m1t − . U2t U2m1t 2.6 4 Discrete Dynamics in Nature and Society Proof. By replacing n with 2n 1t in 2.5, we have 2 − Ut2 U2nt U2n1t , U2n1t or 1 U2n1t Ut 2.7 U2n1t − Ut . U2nt U2n1t 2.8 Taking r 2n 1t and s t in the equality Vs Ur Urs −1s Ur−s 5, the equality 2.8 is rewritten as follows: 1 Ut 1 1 −1t − . 2.9 U2n1t Ut Vt U2nt U2n1t U2nt U2n1t We have the sum m n1 m 1 U2n1t Ut Vt n1 1 1 −1t U2nt U2n1t − Ut m n1 1 . U2nt U2n1t 2.10 For an odd integer t, we have m 1 1 1 1 − − , U2nt U2n1t U2t U2m1t n1 2.11 and taking s 2nt and r 2t in identity 5: Usr Vs − Us Vsr 2−1s Ur , we get m n1 m V2n1t V2m1t 1 1 V2nt 1 V2t − − . U2nt U2n1t 2U2t n1 U2nt U2n1t 2U2t U2t U2m1t 2.12 2.13 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 m n1 1 F2m1 − 1 . F2n1 1 F2m1 2.14 Note that F2m1 m 1 n1 F2n1 1 m n1 F2n . 2.15 Discrete Dynamics in Nature and Society 5 Corollary 2.4. For an odd integer t, ⎧ Vm1t Vt 1 ⎪ ⎪ − , m is ⎪ ⎪ m ⎨ 2Vt Um1t Ut 1 U2n1t Ut ⎪ ⎪ 1 ΔUm1t Vt n1 ⎪ ⎪ − , m is ⎩ 2Vt Vm1t Ut ⎧ Δ Ut Um1t ⎪ ⎪ − , m is ⎪ ⎪ ⎨ 2Vt Vt Vm1t m 1 ⎪ n1 U2n1t − Ut ⎪ 1 ΔUt Vm1t ⎪ ⎪ − , m is ⎩ 2Vt Vt Um1t even, odd, 2.16 even, odd. Proof. Using the equalities V2n Vn2 − 2−1n ΔUn2 2−1n and U2n Un Vn 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, ∞ 1 n1 U2n1t Ut ∞ n1 1 √ Vt Δ− , 2Vt Ut Δ U2n1t − Ut 2Vt 1 2.17 1 Ut −√ , Vt Δ and for |β/α| < 1, t < 0 and |α/β| < 1, t > 0, ∞ 1 n1 U2n1t Ut ∞ n1 −1 √ Vt , Δ 2Vt Ut Δ U2n1t − Ut 2Vt 1 2.18 1 Ut √ . Vt Δ Proof. Since lim n→∞ Vanb Uanb ⎧ √ ⎪ ⎪ ⎪ Δ ⎪ ⎨ ⎪ √ ⎪ ⎪ ⎪ ⎩− Δ β < 1, a > 0, α β < 1, a < 0, α α < 1, a < 0, β α < 1, a > 0, β 2.19 the results are easily seen by equalities 2.16. Theorem 2.6. For an integer m ≥ 1 and an arbitrary nonzero integer t, m n1 −1nt U2n1t V2n1t − −1nt Vt 1 2 4Δ2 Ut Vt2 Ut2 − 2 Vm1t 2 Um1t . 2.20 6 Discrete Dynamics in Nature and Society Proof. By replacing n with 2n 1t/2 and t with t/2 in 2.4, we have V2n1t − −1nt Vt ΔUnt Un1t , or 1 nt V2n1t − −1 Vt 2.21 1 . ΔUnt Un1t 2.22 Multiplying equality 2.22 by −1nt U2n1t /Unt Un1t , we get −1nt U2n1t −1nt U2n1t . 2 2 ΔUnt Un1t Unt Un1t V2n1t − −1nt Vt 2.23 We have the sum: m n1 m −1nt U2n1t −1nt U2n1t 1 . 2 2 Δ n1 Utn Utn1 Unt Un1t V2n1t − −1nt Vt 2.24 Using the equalities 2.1 and 2.21, the proof is obtained. Corollary 2.7. For an arbitrary nonzero integer t, ∞ n1 −1nt U2n1t V2n1t − −1nt Vt 1 2 4Δ2 Ut Vt2 Ut2 −Δ . 2.25 Proof. Taking m → ∞ in Theorem 2.6 and using 2.19, the result is easily obtained. Theorem 2.8. For an integer m ≥ 0 and an arbitrary nonzero integer t, m n0 −1nt U2n1t V2n1t −1nt Vt 2 2 Um1t 2 4Ut Vm1t . 2.26 Proof. The proof of the theorem is similar to the proof of Theorem 2.6. Corollary 2.9. For an arbitrary nonzero integer t, ∞ n0 −1nt U2n1t nt V2n1t −1 Vt 1 2 4ΔU . t 2.27 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 ∞ n0 −1n F32n1 1 2 40 . L32n1 −1 4 n 2.28 Discrete Dynamics in Nature and Society 7 Theorem 2.10. For an integer m ≥ 1 and an arbitrary nonzero integer t, m n1 m n1 U22n1t 1 2 4U 2t 2 2 U2n1t − Ut U22n1t 1 2 2U 4Δ 2t 2 V2t2 − 2 U2t 2 V2n1t − Vt V2t2 2 V2m1t 2 U2m1t 2 V2m1t − 2 U2t 2 U2m1t , 2.29 . Proof. The proof of theorem is similar to the proof of Theorem 2.6. Corollary 2.11. For an arbitrary nonzero integer t, ∞ n1 ∞ n1 U22n1t 1 2 4U 2t 2 1 2 2U 4Δ 2t 2 2 V2n1t − Vt −Δ , 2 U2t 2 U2n1t − Ut U22n1t V2t2 V2t2 2 U2t 2.30 −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 ∞ n1 U42n1 2 V22n1 − V22 2 1 Δ2 U43 . 2.31 References 1 R. P. Backstrom, “On reciprocal series related to Fibonacci numbers with subscripts in arithmetic progression,” The Fibonacci Quarterly, vol. 19, pp. 14–21, 1981. 2 B. S. Popov, “On certain series of reciprocals of Fibonacci numbers,” The Fibonacci Quarterly, vol. 22, no. 3, pp. 261–265, 1984. 3 B. S. Popov, “Summation of reciprocal series of numerical functions of second order,” The Fibonacci Quarterly, vol. 24, no. 1, pp. 17–21, 1986. 4 N. Gauthier, “Solution to problem H-680,” The Fibonacci Quarterly, vol. 49, no. 1, pp. 90–92, 2011. 5 S. Vajda, Fibonacci & Lucas Numbers, and the Golden Section. Theory and Applications, John Wiley & Sons, New York, NY, USA, 1989. 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