Iteratioal Joural of Cotemporary Mathematical Scieces Vol. 10, 015, o. 8, 363-379 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.1988/ijcms.015.594 O Ifiite Series Ivolvig Fiboacci Numbers Robert Frotczak 1 Lesbak Bade-Wuerttemberg (LBBW Am Hauptbahhof, 70173 Stuttgart, Germay Copyright c 015 Robert Frotczak. This article is distributed uder the Creative Commos Attributio Licese, which permits urestricted use, distributio, reproductio i ay medium, provided the origial work is properly cited. Abstract The aim of the paper is to derive closed-form expressios for some ew types of ifiite series ivolvig Fiboacci umbers. For each type umerous examples are preseted. Some classical results are rediscovered as particular cases of the more geeral idetities preseted here. Mathematics Subject Classificatio: 11B39, 11Y60 Keywords: Fiboacci umber, Iverse taget fuctio, Ifiite series 1 Itroductio The Fiboacci sequece (or Fiboacci umbers is oe of the most popular fasciatig liear sequeces i mathematics. It is defied recursively as F 0 0, F 1 1, F + F +1 + F for 0. Sice its itroductio by Leoardo of Pisa at the begiig of the thirteeth cetury the sequece attracted the iterest of umerous mathematicias. As a result, hudreds of formulas idetities ivolvig Fiboacci umbers were developed (see [3] [5] for a excellet itroductio. The Biet form for the Fiboacci sequece is F α β, 0, (1 α β 1 Disclaimer: Statemets coclusios made i this article are etirely those of the author. They do ot ecessarily reflect the views of LBBW.
364 Robert Frotczak where α β are roots of the quadratic equatio x x 1 0. We have α 1 + 5, β 1 5. ( It follows that α α + 1, β β + 1, αβ 1, α + β 1 α + β 3. The study of ifiite series ivolvig Fiboacci umbers starts with the followig iverse taget (i.e. arctaget idetity which was discovered i 1936 by Lehmer (Lehmer s idetity, see [3]: ta 1 ( 1 F +1 π 4. (3 Motivated by this strikig result, may extesios geeralizatios were obtaied i [4], [6], [7] [8]. For istace, i the very recet paper [8] the above iverse taget formula was exteded to ( ta 1 1 ( 1 ta 1, k 1. (4 F +k 1 F k O the other h a well-kow summatio techique for ifiite series is employed by Brousseau i [] to derive closed-form evaluatios for a rage of ifiite series ivolvig Fiboacci umbers. As a classical example he proves that F 1. (5 F +1 F + The above iverse taget evaluatios are based o a telescopig property of the iverse taget fuctio. The required trigoometric formula is ( x y ta 1 (x ta 1 (y ta 1, xy > 1. (6 1 + xy However, i may situatios the telescopig structure is hidde. I this paper extedig a theorem foud i [1] ew closed-form evaluatios of ifiite series ivolvig Fiboacci umbers are derived by elemetary methods. Mai Result Let ta 1 (x deote the pricipal value of the iverse taget fuctio. Our mai result is the followig statemet: Theorem.1 Let g(x be a real fuctio of oe variable. Let h(x be of fixed sig composite, h(x h(g(x. Defie H(x by H(x h(g(x h(g(x + 1 1 + h(g(xh(g(x + 1. (7
O ifiite series ivolvig Fiboacci umbers 365 The we have k ta 1 H( ta 1 h(g(1 ta 1 h(g(k + 1, (8 ta 1 H( ta 1 h(g(1 ta 1 ( lim k h(g(k + 1. (9 PROOF: By (6 we have that k ta 1 H( k ( ta 1 h(g( ta 1 h(g( + 1. The statemets follow from telescopig. Remark. If g(x x the the Theorem reduces to Theorem.1 i [1]. Also, as poited out i [1] the coditio o h(x to be of fixed sig is icluded i order to avoid a jump i Eq. (6 by π multiplied by the sig of x whe xy < 1. I the followig we are goig to preset several applicatios of the previous theorem. The use the term type for a specific form of the fuctio h(x. To provide a compact treatmet, the presetatio will be restricted to ifiite series oly. I all applicatios g(, 1, will correspod to the Fiboacci sequece F, a subsequece of it or a sequece closely related to F. 3 Applicatios 3.1 First type: h(x a/x. Corollary 3.1 For a 0 let h(x a x g( F m+k, m 1, k 0. The by Theorem.1 Especially, for each m k ( ta 1 a(fm+m+k F m+k ( a ta 1. (10 a + F m+k F m+m+k F m+k ta 1 ( Fm+k (F m+m+k F m+k F m+k + F m+kf m+m+k π 4. (11
366 Robert Frotczak The case m 1 k 0 gives ( ta 1 af ta 1 (a. (1 a + F +1 F + From (1 isertig a 1, a 3/3, a 3, a 3 a α, respectively, we obtai the ifiite series evaluatios ( ta 1 F 1 + F +1 F + π 4, (13 ( ta 1 3F π 1 + 3F +1 F + 6, (14 ( ta 1 3F π 3 + F +1 F + 3, (15 ( ta 1 ( 3F 7 4 π 3 + F +1 F + 1, (16 ( ta 1 αf ta 1 (α. (17 α + 1 + F +1 F + Also, observe that we have the symmetric relatio ( ta 1 3F ( ta 1 3F. (18 3 + F +1 F + 1 + 3F +1 F + The case m k 0 i Eq. (10 gives Usig Cassii s idetity the relatio ( ta 1 a(f+ F ta 1 (a. (19 a + F F + F +1 F 1 ( 1 + F, 1 F F + F 1 F, 1, (see [3] the above expressio may be restated as ( a(f ta 1 +1 + F ta 1 (a. (0 a 1 + F+1
O ifiite series ivolvig Fiboacci umbers 367 For a 1 we get a series for π/4 ivolvig squared Fiboacci umbers: I a similar maer the case m k 1 gives where we have used the relatio (see [3] The case m k gives ( F ta 1 +1 + F π F+1 4. (1 ( a(f ta 1 + F ta 1 ( a, ( a + 1 + F+ F +1 F +1 + F, 0. ( ta 1 a(fm(+ F m(+1 ( a ta 1. (3 a + F m(+1 F m(+ F m From Corollary 3.1 the followig summatio idetity is obtaied straightforwardly by differetiatio: Corollary 3. For a 0, m 1, k 0 it holds that (F m+k F m+m+k a (F m+m+k F m+k (a + F m+k F m+m+k + a (F m+m+k F m+k F m+k. (4 + a F m+k Special cases are the ext expressios. The case m 1 k 0 gives F F +1 F + a F (a + F +1 F + + a F 1 1 + a. (5 Noticig that Eq. (5 (as well as (4 are also well defied for a 0, we immediately get Brousseau s series (5 as a special case of (5. Further explicit evaluatios of (5 are the idetities 4F F +1 F + F (1 + 4F +1 F + + 4F 1 5, (6 F F +1 F + F (1 + F +1 F + + F F F +1 F + α F (α + F +1 F + + α F 1, (7 1 α +, (8
368 Robert Frotczak as well as F F +1 F + α 4 F (α 4 + F +1 F + + α 4 F The case m k 0 gives the summatio formula from which 1 3(α + 1. (9 (F +1 1 a (F +1 + F (a 1 + F +1 + a (F +1 + F 1 1 + a, (30 (F +1 (F +1 + F F 4 +1 + (F +1 + F 1, (31 (F +1 3(F +1 + F (1 + F +1 + (F +1 + F 1 3, (3 are two special cases. Fially, the pair m k 1 produces the idetity (F + + 1 a (F + F (a + 1 + F + + a (F + F 4 + a. (33 3. Secod type: h(x ax + b. Corollary 3.3 For a > 0 let h(x ax + b. We will choose b such that h(g( 0 for 0. Let g( F m+k, m 1, k 0. The by Theorem.1 ( ta 1 a(f m+m+k F m+k π a F m+k F m+m+k + ab(f m+m+k + F m+k + b + 1 ta 1 (af m+k +b. (34 The case m 1 k 0 gives a idetity cotaiig four cosecutive Fiboacci umbers ( ta 1 af π a F +1 F + + abf +3 + b + 1 ta 1 (a + b, (35 from which the followig evaluatios are easily obtaied: The case a 1 b 0 gives Eq. (13. The case a b 0 gives ( ta 1 F π 1 + 4F +1 F + ta 1 (. (36
O ifiite series ivolvig Fiboacci umbers 369 The case a 1 b 1 gives ( ta 1 F π F +1 F + + F +3 + ta 1 (. (37 The case a α b 0 gives ( ta 1 αf π 1 + α F +1 F + ta 1 (α. (38 Comparig with Eq. (17 we arrive at ( ta 1 αf π 1 + α F +1 F + ( ta 1 αf. (39 α + F +1 F + The case a α b β gives ( ta 1 αf π α F +1 F + F +3 + β + 4. (40 As a fial example i this subclass, take a α b β to get ( ta 1 α F π α 4 F +1 F + + F +3 + 3β ta 1 (3. (41 The case m k 0 gives ( ta 1 a(f+1 + F π a F +1 + ab(4f + F + F+1 a + b + 1 ta 1 (a+b. (4 Choosig a 1 b 0 i the above equatio establishes Eq. (1. The case m k 1 gives ( ta 1 a(f+ F π a F + + ab(f + F+1 + F+ + a + b + 1 ta 1 (a+b, (43 from which ( F ta 1 + F π F+ + ta 1 (, (44 ( (F ta 1 + F π F+ + 5 4, (45 are two illustrative examples. Differetiatio of Eq. (34 with respect to a b will result i the followig summatios:
370 Robert Frotczak Corollary 3.4 Let a b be defied as above. We set x 1 F m+m+k F m+k x F m+k F m+m+k x 3 F m+m+k + F m+k. The a x 1 x (b + 1x 1 (a x + abx 3 + b + 1 + a x 1 a x 1 x 3 + abx 1 (a x + abx 3 + b + 1 + a x 1 F m+k 1 + (af m+k + b, (46 1 1 + (af m+k + b. (47 I case m 1 k 0 the sums reduce to a F F +1 F + (b + 1F (a F +1 F + + abf +3 + b + 1 + a F 1 1 + (a + b, (48 a F F +3 + abf (a F +1 F + + abf +3 + b + 1 + a F 1 1 + (a + b. (49 We cotiue by statig a few explicit examples of the above idetities: F F +1 F + F (F +1 F + + F +3 + + F 1 5, (50 F F +1 F + 3F 1 (F +1 F + + F +3 + 3 + F 9, (51 3F F +1 F + 4F 1 (3F +1 F + + 3F +3 + 4 + 3F 13, (5 F F +1 F + F 1 (F +1 F + + 1 + F 3, (53 α 4 F F +1 F + 3β F 1 (α 4 F +1 F + + F +3 + 3β + α 4 F 10, (54 F F +3 1 (F +1 F + + 1 + F. (55
O ifiite series ivolvig Fiboacci umbers 371 It is also worth to metio that the choice b a i (48 (49 gives a F F +1 F + (a + 1F 1, (56 (a F +1 F + a F +3 + a + 1 + a F a F F +3 a F (a F +1 F + a F +3 + a + 1 + a F Note that both series are idepedet of a. 1. (57 Explicit evaluatios i case m are limited to three examples: (F+ F(4F + + 3 (4F+ + 5 + 4(F+ F 17, (58 (F + F (F + + 1 (F + + 3 + (F + F 9, (59 (F+ F(α F+ β (α F+ F+ F+1 F + 4 + α (F+ F 3α. (60 Remark 3.5 It is clear that cosiderig h(x ax +bx+c more advaced series ivolvig ta 1 (a + b + c may be derived from the Theorem. We do ot go i this directio, however. Istead, we focus o fuctioal forms of h(x g( which will be able to produce alteratig series. This is doe ext. 3.3 Third type: h(x ax r. Corollary 3.6 For r a iteger, r 1, a 0 let h(x ax r. Let further g( F +k /F, k 1. The lim g( α k ( a(f r ta 1 F+1+k r F +1F r +k r FF r +1 r + a F+k r F ta 1 (aα kr ta 1 (afk+1, r (61 +1+k r or equivaletly ( a(f r ta 1 F+1+k r F +1F r +k r FF r +1 r + a F+k r F ta 1 (aα kr π sg(a, (6 +1+k r where sg(x deotes the sig fuctio of x. Especially, for each k 1 r 1 we have the remarkable idetities ( α kr ta 1 (FF r +1+k r F +1F r +k r α kr FF r +1 r + F+k r F π +1+k r 4, (63
37 Robert Frotczak ( 3α kr ta 1 (FF r +1+k r F +1F r +k r α kr FF r +1 r + 3F+k r F π +1+k r 6, (64 ( 3α kr ta 1 (FF r +1+k r F +1F r +k r 3α kr FF r +1 r + F+k r F π +1+k r 3, (65 ( ( 3α kr ta 1 (FF r +1+k r F +1F r +k r α kr FF r +1 r + (7 4 3F+k r F 5π +1+k r 1. (66 A direct cosequece of the Corollary is Remark 3.7 Let S(r, k, a deote the sum o the LHS of Eq. (6. The S(r, k, a possesses the followig symmetry property: If kr q d is a divisor of q icludig 1 q, the S(d, d, a S(d, d, a, where d q/d. For istace, we have that S(1,, a S(, 1, a or S(3, 1, a S(1, 3, a. Though the above results may be iterestig, the alteratig structure of the series is still hidde ot obvious. To make the structure more evidet we prove the followig Lemma: Lemma 3.8 For 0 k 1 F F +1+k F +1 F +k ( 1 +1 F k. (67 Furthermore, for each r we have the relatio r ( r FF r +1+k r F+1F r +k r ( 1 i(+1 F i i kf+1f r i r i +k. (68 i1 PROOF: To prove the first idetity we use iductio o k. For k 1 the statemet follows from Cassii s Theorem. Let the statemet be true for a fixed k > 1. The F F ++k F +1 F +1+k F (F +1+k + F +k F +1 (F +k + F +k 1 F F +1+k F +1 F +k + F F +k F +1 F +k 1 ( 1 +1 F k + ( 1 +1 F k 1 ( 1 +1 F k+1. The secod idetity follows from applyig the biomial formula. Remark 3.9 The first idetity of the Lemma is a o-symmetric geeralizatio of Cassii s Theorem. It may be see as a compaio of the symmetric geeralizatio due to Catala (see [3], p. 109: F k F +k F ( 1 +k+1 F k.
O ifiite series ivolvig Fiboacci umbers 373 Equipped with the fidigs of the Lemma, the case r 1 ca be stated as ( ta 1 a( 1 +1 F k ta 1 (aα k π sg(a. (69 F F +1 + a F +k F +1+k The subcase k 1 a 1/α gives ( ta 1 α( 1 π α F F +1 + F +1 F + 4. (70 Also, comparig the results for a 1 with that oe for a 1/α: ( ta 1 ( 1 +1 F F +1 + F +1 F + Further examples i this subcase are ( ta 1 α( 1 +1 ta 1 (α π α F F +1 + F +1 F + 4. (71 ta 1 ( 3( 1 +1 F F +1 + 3F +1 F + ta 1 ( 3α π 3, (7 ( 3( 1 ta 1 +1 3α ta 1 ( 3F F +1 + F +1 F + 3 π 6, (73 as well as ( ta 1 ( 3( 1 +1 F F +1 + (7 4 ta 1 (( 3α π 3F +1 F + 1, (74 The subcase k gives very similar results. Two explicit evaluatios are ( ta 1 α ( 1 π α 4 F F +1 + F + F +3 4. (75 ( 3α ta 1 ( 1 π 3α 4 F F +1 + F + F +3 3. (76 For a > 0, the subcase k 3 may be stated as ( ta 1 a( 1 π F F +1 + a F +3 F +4 ta 1 (aα 3. (77 I case r we obtai ( a(f ta 1 k + ( 1 +1 F k F +1 F +k FF +1 + a F+k F ta 1 (aα k π sg(a. (78 +1+k
374 Robert Frotczak Agai, for each pair k a umerous special cases are immediately available. We restrict the presetatio of explicit examples to a few series: Take k 1 a 1 to get I a similar maer we get Also, for each a: ( 1 + ( 1 ta 1 F+1 π FF +1 + F+1F + ta 1 (α + 1. (79 ( α ta 1 ( 1 + ( 1 F+1 π α 4 FF +1 + F+1F + 4, (80 ( ta 1 3α ( 1 + ( 1 F+1 π α 4 FF +1 + 3F+1F + 6, (81 ( ta 1 3α ( 1 + ( 1 F+1 π 3α 4 FF +1 + F+1F + 3. (8 ( ta 1 a( 1 +1 ( a(1 + ( 1 ta 1 +1 F +1. F F +1 + a F + F +3 F F+1 + a F+1F + (83 Isertig k 3 a 1 a 3, respectively, results i the evaluatios ( 4( 1 + ( 1 ta 1 F +1 F +3 π FF +1 + F+3F +4 ta 1 ((α + 1 3, (84 ( 1( 1 + ( 1 ta 1 F +1 F +3 π FF +1 + 9F+3F +4 ta 1 (3(α + 1 3. (85 We cotiue with presetig series idetities that are derived from Corollary 3.6 by differetiatio with respect to a. Corollary 3.10 For r, k 1 set x 1 F r F r +1+k F r +1F r +k x FF r +1 r x 3 F+kF r +1+k. r The x 1 (x a x 3 (x + a x 3 + a x 1 α kr 1 + a α Fk+1 r. (86 kr 1 + a Fk+1 r
O ifiite series ivolvig Fiboacci umbers 375 Especially for a 0 we get the idetity F r F r +1+k F r +1F r +k F r F r +1 α kr F r k+1. (87 The case r 1 simplifies to ( 1 +1 F k (F F +1 a F +k F +1+k (F F +1 + a F +k F +1+k + a F k α k 1 + a α F k+1. (88 k 1 + a Fk+1 Settig a 0 we get for each k ( 1 +1 F F +1 αk F k+1 F k, (89 which i view of the formula α k αf k + F k 1 may be writte i the form For k 1 a 1 we get For k 1 a 1/ we get ( 1 +1 F F +1 α 1. (90 ( 1 +1 F+1 F+1(F + F + + 1 1 α + α. (91 ( 1 +1 (4F F +1 F +1 F + (4F F +1 + F +1 F + + 4 1 4α 5 5 α + 5. (9 For k 1 a 1/ α we get ( 1 +1 (αf F +1 F +1 F + (αf F +1 + F +1 F + + α 0. (93 The case r produces (F k + ( 1+1 F k F +1 F +k (F F +1 a F +k F +1+k (F F +1 + a F +k F +1+k + a (F k + ( 1+1 F k F +1 F +k α k 1 + a α Fk+1. (94 4k 1 + a Fk+1 4
376 Robert Frotczak For a 0 this simplifies to F k + ( 1 +1 F +1 F +k F F +1 (α 1(α k +F k+1 F k α+f k 1 (α 1. (95 Isertig k 1,, 3 we get 1 + ( 1 +1 F +1 F F +1 (α 1(α + 1 α, (96 1 + ( 1 +1 F +1 F + F F +1 1 + ( 1 +1 F +1 F +3 F F +1 (α 1(α + 3 3α, (97 (α 1(α + α 1. (98 As a fial example for this type of evaluatios we take the pair k 1 a 1 get after some algebra F 3 +1(F +1 + F (1 + ( 1 +1 F +1 (F F +1 + F +1F + + (1 + ( 1 +1 F +1 1 6. (99 I our fial applicatio of Theorem.1 we cosider series that cotai the expoetial fuctio therefore exhibit as very fast covergece. 3.4 Fourth type: h(x ae bx. Corollary 3.11 For a 0 b > 0 let h(x ae bx. Let further g( F +k, k 0. The Especially, for each b k ( a(e ta 1 bf +k+1 e bf +k ta 1 (ae bf k+1. (100 a + e bf +k+ ta 1 ( e b(f k+1 +F +k+1 e b(f k+1+f +k e bf k+1 + e bf +k+ π 4, (101 ( ta 1 3(e b(f k+1+f +k+1 e b(f k+1+f +k π 3e bf k+1 + e bf +k+ 3, (10 ta 1 ( 3(e b(f k+1+f +k+1 e b(f k+1+f +k e bf k+1 + 3e bf +k+ π 6, (103
O ifiite series ivolvig Fiboacci umbers 377 ( ( 3(e ta 1 b(f k+1 +F +k+1 e b(f k+1+f +k (7 4 π 3e bf k+1 + e bf +k+ 1. (104 For k 0 k 1 Eq. (100 simplifies to ( a(e ta 1 bf +1 e bf ta 1 (ae b, (105 a + e bf + from which the followig idetities are special cases: as well as ( e ta 1 F +1 e F ta 1 ( 1, (106 1 + e F + e ( e ta 1 F +1 e F ta 1 ( 1, (107 1 + e F + e ( α(e ta 1 F +1 e F ta 1 ( α, (108 α + e F + e ( e ta 1 (1+F +1 e (1+F π e + e F + 4, (109 ( ta 1 3(e (1+F+1 e (1+F π 3e + e F + 3, (110 ( ta 1 3(e (1+F+1 e (1+F π e + 3e F + 6, (111 ( ( 3(e ta 1 (1+F +1 e (1+F (7 4 π 3e + e F + 1. (11 For k Eq. (100 reduces to ( ae ta 1 bf + (e bf +1 1 ta 1 (ae b. (113 a + e bf +4 Differetiatio of Eq. (100 with respect to the parameter a gives
378 Robert Frotczak The Corollary 3.1 Set x 1 e bf +k+1 e bf +k, x e bf +k+. x 1 (x a (x + a + a x 1 e bf +k+ e bf k+1. (114 1 + a e bf k+1 Especially for a 0 we get the idetity e bf +k+1 e bf +k e bf k+1. (115 We fiish the study by presetig a few special cases of the Corollary i case k 0. For a 1 b 1 we get (e F +1 e F (e F + 1 (e F + + 1 + (e F +1 e F e e + 1. (116 For a e b 1 we get (e F +1 e F (e F + e (e F + + e + e (e F +1 e F 1 e. (117 For a 1/e b 1 the result is (e F +1 e F (e +F + 1 (e +F + + 1 + e (e F +1 e F e e 4 + 1. (118 4 Coclusio I this paper we have derived closed-form expressios for various types of ifiite series ivolvig Fiboacci umbers. As a direct cosequece ew series represetatios for π have bee established. The approach preseted is very geeral i the sese that alteratig o-alteratig series may be studied. Also, it is ot limited to a specific sequece. Thus similar results may be derived for Lucas umbers, Pell umbers or other sequeces. This is left for future research. Refereces [1] G. Boros V.H. Moll, Sums of arctagets some formulas of Ramauja, SCIENTIA Series A: Mathematical Scieces, 11 (005, 13-4.
O ifiite series ivolvig Fiboacci umbers 379 [] B.A. Brousseau, Fiboacci-Lucas Ifiite Series Research Topic, The Fiboacci Quarterly, 7 (1968, o., 11-17. [3] R.P. Grimaldi, Fiboacci Catala umbers: A Itroductio, Joh Wiley & Sos, New York, 01. http://dx.doi.org/10.100/9781118159743 [4] A.R. Guillot, Problem XXX, The Fiboacci Quarterly, 7 (1977, o., 57. [5] T. Koshy, Fiboacci Lucas Numbers with Applicatios, Joh Wiley & Sos, New York, 001. http://dx.doi.org/10.100/9781118033067 [6] R.S. Melham, Sums ivolvig Fiboacci Pell umbers, Portugaliae Mathematica, 56 (1999, o. 3, 309-317. [7] R.S. Melham A.G. Shao, Iverse trigoometric hyperbolic summatio formulas ivolvig geeralized Fiboacci umbers, The Fiboacci Quarterly, 33 (1995, o. 1, 3-40. [8] K. Adegoke, Ifiite arctaget sums ivolvig Fiboacci Lucas umbers, Notes o Number Theory Discrete Mathematics, 1 (015, o. 1, 56-66. Received: September 1, 015; Published: November 11, 015