On Two New Classes of Fibonacci and Lucas Reciprocal Sums with Subscripts in Arithmetic Progression

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1 Applied Mathematical Sciences Vol. 207 no HIKARI Ltd On Two New Classes of Fibonacci Lucas Reciprocal Sums with Subscripts in Arithmetic Progression Robert Frontczak Lesbank Baden-Wuerttemberg (LBBW Am Hauptbahnhof Stuttgart Germany Copyright c 207 Robert Frontczak. This article is distributed under the Creative Commons Attribution License which permits unrestricted use distribution reproduction in any medium provided the original work is properly cited. Abstract In this paper expressions are derived for two new classes of Fibonacci Lucas number reciprocal sums with subscripts in arithmetic progression. We give expressions for both finite infinite sums. The series that are studied are similar to that of Backstrom [3] Popov [0] Melham [9] the results complete the discoveries from [5]. Mathematics Subject Classification: B39 Y60 Keywords: Fibonacci number Lucas number Reciprocal sum Introduction Let F n L n (n 0 denote the Fibonacci Lucas numbers respectively. In his paper from 98 Backstrom [3] develops formulas for series of the form i=0 F ai+b + c i=0 L ai+b + c for certain integer values of a b c. His initial study was carried on by several authors results of similar kind were established among others by Disclaimer: Statements conclusions made in this article are entirely those of the author. They do not necessarily reflect the views of LBBW.

2 22 Robert Frontczak Popov ([0] [] André-Jeannin ([] Zhao ([2]. Reciprocal Fibonacci- Lucas series of a more general nature are studied in [2] [6] [7] [8]. Two other special forms of these series have been considered recently in papers by Melham [9] Frontczak [5]. Melham derives results for i=0 L 2ai+b (F 2ai+b + c 2 i=0 F 2ai+b (L 2ai+b + c 2 for certain values of a b c whereas Frontczak examines series of the form F 2pi+m (F 2pi+q + n(f 2p(i++q + n L 2pi+m (F 2pi+q + n(f 2p(i++q + n F 2pi+m (L 2pi+q + n(l 2p(i++q + n L 2pi+m (L 2pi+q + n(l 2p(i++q + n for integer values of p m q reals n. The aim of the current paper is to evaluate in closed-form two new classes of reciprocal Fibonacci-Lucas sums. The first class consists of the following types of finite infinite sums: (F p(i++q + n(f p(i+q + n (F p(i++q + n(f p(i+q + n (L p(i++q + n(l p(i+q + n (L p(i++q + n(l p(i+q + n for integer values of p q non-negative reals n. We show that these sums are special cases of even more general series which we solve in closed-form here. In the second part of the paper we also solve the problem for another class of reciprocal series which is closely related to the above one. 2 Main Result The following Lemma will be used later. Lemma 2. Let u v be integers such that u + v u v have the same parity. Then { Lu F F u+v F uv = v if v is even F u L v if v is odd { 5Fu F L u+v L uv = v if v is even L u L v if v is odd. ( (2

3 Two new classes of Fibonacci Lucas reciprocal sums 23 PROOF: Both identities are well-known. They can be proved easily using the Binet forms (see for instance [4]. Our main results for the first class of sums are contained in two theorems. Theorem 2.2 Let p q be integers n a 0 be real numbers. For p even define the series S (N p q n a as S (N p q n a = ((F p(i++q + n(f p(i+q + n a 2. (3 ((F p(i++q + n(f p(i+q + n + a a 2 Fp 2 L 2 pi+q For p odd define the series S 2 (N p q n a as S 2 (N p q n a = ((F p(i++q + n(f p(i+q + n a 2. (4 ((F p(i++q + n(f p(i+q + n + a a 2 L 2 pfpi+q 2 Then S (N p q n a = ( F q + n F p (F q + n 2 + a + F p+q + n 2 (F p+q + n 2 + a 2 F p(n++q + n (F p(n++q + n 2 + a F pn+q + n (5 2 (F pn+q + n 2 + a 2 S 2 (N p q n a = ( F q + n L p (F q + n 2 + a + F p+q + n 2 (F p+q + n 2 + a 2 F p(n++q + n (F p(n++q + n 2 + a F pn+q + n. (6 2 (F pn+q + n 2 + a 2 Especially for a = 0 we have the evaluations (F p(i++q + n(f p(i+q + n = ( F p F q + n + F p+q + n F p(n++q + n F pn+q + n (7 (F p(i++q + n(f p(i+q + n = ( F p F q + n + (8 F p+q + n (F p(i++q + n(f p(i+q + n = ( L p (F p(i++q + n(f p(i+q + n = ( L p F q + n + F p+q + n F p(n++q + n F pn+q + n (9 F q + n + F p+q + n. (0 For the last four equations to hold we need the assumption on q n that (q n (0 0.

4 24 Robert Frontczak When p = the special results in (9 (0 are trivial since the sums are telescoping sums. The same is true for (7 (8 when p = 2. To reveal the telescoping nature one uses the relation L i = F i+2 F i2 i 2. The second theorem has an analogous structure: Theorem 2.3 Let p q be integers n a 0 be real numbers. For p even define the series S 3 (N p q n a as ((L p(i++q + n(l p(i+q + n a 2 S 3 (N p q n a =. ((L p(i++q + n(l p(i+q + n + a a 2 Fp 2 Fpi+q 2 ( For p odd define the series S 4 (N p q n a as ((L p(i++q + n(l p(i+q + n a 2 S 4 (N p q n a =. ((L p(i++q + n(l p(i+q + n + a a 2 L 2 pl 2 pi+q (2 Then S 3 (N p q n a = ( L q + n 5F p (L q + n 2 + a + L p+q + n 2 (L p+q + n 2 + a 2 L p(n++q + n (L p(n++q + n 2 + a L pn+q + n (3 2 (L pn+q + n 2 + a 2 S 4 (N p q n a = ( L q + n L p (L q + n 2 + a + L p+q + n 2 (L p+q + n 2 + a 2 L p(n++q + n (L p(n++q + n 2 + a L pn+q + n. (4 2 (L pn+q + n 2 + a 2 Especially for a = 0 we have the evaluations (L p(i++q + n(l p(i+q + n = ( 5F p L q + n + L p+q + n L p(n++q + n L pn+q + n (5 (L p(i++q + n(l p(i+q + n = ( 5F p L q + n + (6 L p+q + n (L p(i++q + n(l p(i+q + n = ( L p (L p(i++q + n(l p(i+q + n = ( L p L q + n + L p+q + n L p(n++q + n L pn+q + n (7 L q + n + L p+q + n. (8

5 Two new classes of Fibonacci Lucas reciprocal sums 25 Again when p = then (7 (8 reduce to trivial telescoping sums. We prove both theorems simultaneously. PROOF: The proof is an application of the following result from [4]: Theorem 2.4 Let f(x g(x be two real functions. integer consider the new function h(x defined as For k an h(x = f(g(x + k f(g(x k + f(g(x + kf(g(x k. (9 Let tan (x denote the principal branch of the inverse tangent function. If f(g(i + kf(g(i k > for all i then n tan h(i = k m=k+ tan f(g(n + m k m=k+ tan f(g(m (20 k tan h(i = 2k tan f(g( tan f(g(m. (2 m=k+ We are going to apply this result in case k =. To ensure a concise presentation we define the generalized Fibonacci numbers G i through the relation G i+ = G i + G i with initial terms G 0 G. For a 0 define f(x as f(x = a/x. Furthermore for integers p q 0 n 0 a real number define g(i = G pi+q + n. Then obviously f(g(0 = a/g q + n f(g( = a/g p+q + n lim i f(g(i = 0. It follows that ( tan axi x 2i + a 2 ( = tan a ( + tan a G q + n G p+q + n ( tan a ( tan G p(n++q + n a G pn+q + n where we have set x i = G p(i++q G p(i+q x 2i = (G p(i++q + n(g p(i+q + n. (22 The first theorem follows by setting G = F using { Lpi+q F F p(i++q F p(i+q = p if p is even L p if p is odd (23

6 26 Robert Frontczak differentiating w.r.t. the parameter a. The second theorem is obtained upon setting G = L using { 5Fpi+q F L p(i++q L p(i+q = p if p is even L p if p is odd (24 differentiating w.r.t. the parameter a. It is obvious from the proof that the identities also hold for negative values of n. However some caution is necessary to avoid singularities. We conclude this section with a short list of examples of infinite series evaluations that were discovered in this paper: L 4i (F 4i4 + (F 4i+4 + = 5 2 L 4i+2 = 3 F 4i2 F 4i+6 8 F 3i+ = F 3i2 F 3i+4 3 = F 3i (F 3i3 + (F 3i+3 + F 2i+2 (L 2i + 2(L 2i = L 3i+3 (L 3i (L 3i+6 = A second class of sums Using the same approach as in the previous section it is possible to derive formulas for another class of reciprocal series. For p q a a real number we define the four series S 5 S 8 according to S 5 (N p q a = p even S 6 (N p q a = p odd S 7 (N p q a = ( + L pi ((F p(i++q + F p(i+ (F p(i+q + F p(i a 2 ((F p(i++q + F p(i+ (F p(i+q + F p(i + a a 2 F 2 p ( + L pi 2 (25 ( + F pi ((F p(i++q + F p(i+ (F p(i+q + F p(i a 2 ((F p(i++q + F p(i+ (F p(i+q + F p(i + a a 2 L 2 p( + F pi 2 (26 ( + F pi ((L p(i++q + L p(i+ (L p(i+q + L p(i a 2 ((L p(i++q + L p(i+ (L p(i+q + L p(i + a a 2 F 2 p ( + F pi 2 (27

7 Two new classes of Fibonacci Lucas reciprocal sums 27 p even ( + L pi ((L p(i++q + L p(i+ (L p(i+q + L p(i a 2 S 8 (N p q a = ((L p(i++q + L p(i+ (L p(i+q + L p(i + a a 2 L 2 p( + L pi 2 (28 p odd. Then each of the series allows an evaluation which is very similar to the one from the previous section. We state the results in one theorem a proof of which we leave as an exercise: Theorem 3. We have S 5 (N p q a = ( F q F p Fq 2 + a + F p+q + F p (29 2 (F p+q + F p 2 + a 2 F p(n++q + F p(n+ (F p(n++q + F p(n+ 2 + a F pn+q + F pn 2 (F pn+q + F pn 2 + a 2 S 6 (N p q a = ( F q L p Fq 2 + a + F p+q + F p (30 2 (F p+q + F p 2 + a 2 F p(n++q + F p(n+ (F p(n++q + F p(n+ 2 + a F pn+q + F pn 2 (F pn+q + F pn 2 + a 2 S 7 (N p q a = ( L q + 2 5F p (L q a + L p+q + L p (3 2 (L p+q + L p 2 + a 2 L p(n++q + L p(n+ (L p(n++q + L p(n+ 2 + a L pn+q + L pn 2 (L pn+q + L pn 2 + a 2 S 8 (N p q a = ( L q + 2 L p (L q a + L p+q + L p (32 2 (L p+q + L p 2 + a 2 L p(n++q + L p(n+ (L p(n++q + L p(n+ 2 + a L pn+q + L pn. 2 (L pn+q + L pn 2 + a 2 Upon setting a = 0 letting N we get + L pi (F p(i++q + F p(i+ (F p(i+q + F p(i = ( + (33 F p F q F p+q + F p + F pi (F p(i++q + F p(i+ (F p(i+q + F p(i = ( + (34 L p F q F p+q + F p

8 28 Robert Frontczak + F pi (L p(i++q + L p(i+ (L p(i+q + L p(i = ( 5F p L q L p+q + L p (35 + L pi (L p(i++q + L p(i+ (L p(i+q + L p(i = ( L p L q L p+q + L p (36 The interesting reader is invited to explore the relationships between the sums S S 4 S 5 S 8. References [] R. André-Jeannin Summation of Certain Reciprocal Series Related to Fibonacci Lucas Numbers The Fibonacci Quarterly 29 ( [2] R. André-Jeannin Summation of Reciprocals in Certain Second-Order Recurring Sequences The Fibonacci Quarterly 35 ( [3] R.P. Backstrom On Reciprocal Series Related to Fibonacci Numbers with Subscripts in Arithmetic Progression The Fibonacci Quarterly 9 ( [4] R. Frontczak Further Results on Arctangent Sums with Applications to Generalized Fibonacci Numbers Notes on Number Theory Discrete Mathematics 23 (207 no. to appear. [5] R. Frontczak New Results on Reciprocal Series Related to Fibonacci Lucas Numbers with Subscripts in Arithmetic Progression International Journal of Contemporary Mathematical Sciences (206 no [6] H. Hu Z.-W. Sun J.-X. Liu Reciprocal Sums of Second-Order Recurrent Sequences The Fibonacci Quarterly 39 ( [7] R. S. Melham Finite sums that involve reciprocals of products of generalized Fibonacci numbers Integers 3 (203 #A40. [8] R. S. Melham More on finite sums that involve reciprocals of products of generalized Fibonacci numbers Integers 4 (204 #A4.

9 Two new classes of Fibonacci Lucas reciprocal sums 29 [9] R. S. Melham Reciprocal Series of Squares of Fibonacci Related Sequences with Subscripts in Arithmetic Progression Journal of Integer Sequences 8 (205 Article [0] B. S. Popov On certain series of reciprocals of Fibonacci numbers The Fibonacci Quarterly 22 ( [] B. S. Popov Summation of reciprocal series of numerical functions of second order The Fibonacci Quarterly 24 ( [2] F.-Z. Zhao Notes on Reciprocal Series Related to Fibonacci Lucas Numbers The Fibonacci Quarterly 37 ( Received: March 9 207; Published: April