Notes on Number Theory and Discrete Mathematics ISSN 30 3 Vol., 0, No., 6 66 Infinite arctangent sums involving Fibonacci and Lucas numbers Kunle Adegoke Department of Physics, Obafemi Awolowo University Ile-Ife, 000 Nigeria e-mail: adegoke00@gmail.com Abstract: We derive numerous infinite arctangent summation formulas involving Fibonacci and Lucas numbers. While most of the results obtained are new, a couple of celebrated results appear as particular cases of the more general formulas derived here. Keywords: Fibonacci numbers, Lucas numbers, Lehmer s formula, Arctangent sums, Infinite sums. AMS Classification: B39, Y60. Introduction It is our goal, in this work, to derive infinite arctangent summation formulas involving Fibonacci and Lucas numbers. The results obtained will be found to be of a more general nature than one finds in earlier literature. Previously known results containing arctangent identities and/or infinite summation involving Fibonacci numbers can be found in references [,, 3, 4, ] and references therein. In deriving the results in this paper, the main identities employed are the trigonometric addition formula y x xy + x tan y, which holds for either R and xy > 0 or R, xy < 0 and < xy, and the following identities which resolve products of Fibonacci and Lucas numbers 6
F u v F u+v = F u u v F v, L u v L u+v = L u + u v L v, L u F v = F v+u + u F v u, F u L v = F v+u u F v u, L u L v = L u+v + u L v u, F u v F u+v = L u u v L v. Also we shall make repeated use of the following identities connecting Fibonacci and Lucas numbers: F u = F u L u, L u u = F u, F u L u = 4 u+, L u + u = L u. Identities and 3 or their variations can be found in [6, 7, 8]. On notation: G i, i non-negative integers, denotes generalized Fibonacci numbers defined through the second order recurrence relation G i = G i + G i, where the boundary terms G 0 and G need to be specified. When G 0 = 0 and G =, we have the Fibonacci numbers, denoted F i, while when G 0 = and G =, we have the Lucas numbers, denoted L i. Throughout this paper, the principal value of the arctangent function is assumed. Also φ = + / denotes the golden ratio. a b c d e f 3a 3b 3c 3d Preliminary result Taking x = G mr+n m and y = G mr+n in the arctangent addition formula, Eq., gives Gmr+n G mr+n m G mr+n G mr+n m + G mr+n m G mr+n. 4 Summing each side of Eq. 4 from r = p Z to r = N Z + and noting that the summation of the terms on the right hand side telescopes, we obtain N Gmr+n G mr+n m G mr+n G mr+n m + Now taking limit as N, we have G mp+n m Theorem: For R, n, m, p Z, m 0 holds Gmr+n G mr+n m G mr+n G mr+n m + G mp+n m G mn+n.. 6 7
3 Main results 3. G F in Eq. 6, that is, G 0 = 0, G = Choosing m = 4j and n = k + j and using identities a and d we prove Corollary. For R, j, k, p Z and j 0 holds F j L 4jr+k F4jr+k F j +, 7 F 4jp+k j while taking m = 4j and n = k + j and using identities a and c we prove Corollary. For R and j, k, p Z holds L j F 4jr r+k F4jr r+k F j +. 8 F 4jp p+k j 3. G L in Eq. 6, that is, G 0 =, G = Choosing m = 4j and n = k + j and using identities b and f we prove Corollary 3. For R, j, k, p Z and j 0 holds Fj F 4jr+k L 8jr+4k L 4j +, 9 L 4jp+k j while taking m = 4j and n = k + j and using identities b and e we prove Corollary 4. For R and j, k, p Z holds L j L 4jr r+k L 8jr 4r+4k L 4j +. 0 L 4jp p+k j+ 4 Particular cases and special values Different combinations of the parameters, j, k and p in the above corollaries yield a variety of interesting particular cases. In this section we will consider some of the possible choices. 4. Results from Corollary 4.. = F j, p = and k = 0 in Eq. 7 The choice = F j, p = and k = 0 in Eq. 7 gives 8
F j L j L 4jr F 4jr F j + F j Thus, at j =, we obtain the special value Lj. 4.. = L j, p = and k = 0 in Eq. 7 The above choice gives L4r = π F4r 4. L j F j L 4jr F 4jr F j + L j Fj. 3 At j =, Eq. is reproduced, while at j = we have the special value 9L8r = π F8r 4. 4 Note that Eqs. and 4 are special cases of Eq. 8 below, at j = and j =, respectively. 4..3 = F j, k = j and p = 0 in Eq. 7 This choice gives which, at j =, gives the special value F j L 4jr j = π F4jr j, 4..4 = F j and p = in Eq. 7 This choice gives At k = 0 in Eq. 7 we have F j L 4jr+k F 4jr+k L4r = π F4r. 6 Fj. 7 F j+k F j L 4jr = π F4jr 4. 8 Note that Eqs. and 4 are special cases of Eq. 8 at j = and j =, respectively. 9
At k = j 0 in Eq. 7 we have yielding at j =, the special value F j L 4jr+j, 9 F4jr+j L j L4r+ F4r+ 3 Finally, taking limit of Eq. 7 as j, we obtain lim j F j L 4jr+k F 4jr+k 4.. = L 4j, p = 0 and k = j in Eq. 7. 0. φ k Another interesting particular case of Eq. 7 is obtained by setting = L 4j, p = 0 and k = j to obtain which at j = gives the special value Fj L4j L 4jr j L 4jr j = π, 3 L4r = π L 4r. 3 4..6 = L 4j, p = 0 and k = j in Eq. 7 In this case Corollary reduces to Fj L4j L 4jr L 8jr At j =, we have the special value L4j F j. 4 3 L4r 7 = L 8r. 4..7 = L j / and k = j in Eq. 7 Setting = L j / and k = j in Eq. 7 we have F4j Lj, 6 L 4jr+j F 4jp 60
which at p = gives and at p = 0 yields F4j L 4jr+j F j 7 F4j = π L 4jr j. 8 4..8 = L j /, p = 0 and k = j 0 in Eq. 7 The above choice yields F4j Lj. 9 L 4jr F j 4. Results from Corollary 4.. = F j and p = in Eq. 8 The above choice gives Fj F 4jr r+k At k = j in Eq. 30 we have the interesting formula Fj F 4jr r+j Fj. 30 F j+k L j. 3 Note that Eq. 3, at j =, includes Lehmer s result cited in [3, ] as a particular case. Setting j = in Eq. 30 we obtain F r+k F k. 3 Note again that Eq. 3 subsumes Lehmer s formula and the result of Melham p = in Eq.3. of [], at k = and at k = 0 respectively. Finally, taking limit j in Eq. 30, we obtain lim j Fj F 4jr r+k. 33 φ k 6
4.. = L j / and k = j in Eq. 8 The above choice gives L j F 4jr r+j L 4jr r+j Setting p = in Eq. 34, we find while choosing j = leads to L j F 4jr r+j L 4jr r+j which at p = 0 gives the special value L j. 34 F 4jp p, 3 F j Fr+, 36 L r+ F p Fr = π L r. 37 4..3 = L 4j and k = j in Eq. 8 The above substitutions give L4j L j F 4jr r+j L 4jr r+j At p = 0 in Eq. 38 we have, for positive integers j, giving, at j =, the special value L4j L j F 4jr r j+ L 4jr r j+ At p = in Eq. 38 we have, for positive integers j, L4j L j F 4jr r+6j 3 L 4jr r+6j 3 which gives, at j =, the special value L4j. 38 F 4jp p = π, 39 Fr = π L r. 40, 4 F4j F 8j 4 6
Fr+3. 4 L r+3 4.3 Results from Corollary 3 4.3. = L 4j, k = 0 and p = in Eq. 9 The above choice gives which, at j =, gives L4j F j F 4jr L 8jr L4j, 43 L j 7F 4r = tan 7. 44 L 4r 4.3. = L j and p = in Eq. 9 Setting = L j and p = in Eq. 9 gives F4j F 4jr+k Taking limit as j in Eq. 4 gives lim j F4j F 4jr+k Lj. 4 L j+k. 46 φ k 4.3.3 = F j, p = and k = 0 in Eq. 9 Setting = F j, p = and k = 0 in Eq. 9 we obtain which gives the special value at j =. F j F 4jr L 4jr Fj, 47 L j F 4r = tan, 48 L 4r 63
4.4 Results from Corollary 4 4.4. = L 4j and j = 0 = k in Eq. 0 With the above choice we obtain which gives rise, at p =, to the special value 3Lr 3, 49 L 4r L p 4.4. = L j and p = in Eq. 0 With the above choice we have k = 0 in Eq. gives L j which at j = gives the special value 3Lr L 4r L 4jr r+k F 4jr r+k L j = π 3. 0 Lj. L j+k L 4jr r = π F4jr r 4, L r = π Fr 4. 3 j = in Eq. leads to L r+k, 4 Fr+k L k+ which gives the special value L r+ Fr+ 4 at k =. Taking limit j in Eq., we obtain lim j L j 4.4.3 = L j and j = 0 = k in Eq. 0 This choice gives L 4jr r+k F 4jr r+k 64,. 6 φ k
L r, 7 Fr L p Note that Eqs. 3 and are special cases of 7 at p = and at p =. 4.4.4 = F j and j = 0 = k in Eq. 0 The above choice gives which at p = gives the special value L r, 8 L r L p. 9 Conclusion Using a fairly straightforward technique, we have derived numerous infinite arctangent summation formulas involving Fibonacci and Lucas numbers. In the present paper, only non-alternating series were considered. While most of the results obtained are new, a couple of celebrated results appear as particular cases of more general formulas derived in this paper. References [] Bragg, L. 00 Arctangent sums. The College Mathematics Journal, 34, 7. [] Hayashi, K. 003 Fibonacci numbers and the arctangent function. Mathematics Magazine, 763, 4. [3] Hoggatt Jr, V. E., & Ruggles, I. D. 964 A primer for the Fibonacci numbers: Part V. The Fibonacci Quarterly,, 46. [4] Mahon, J. M., Br., & Horadam, A. F. 98 Inverse trigonometrical summation formulas involving Pell polynomials. The Fibonacci Quarterly, 34, 39 34. [] Melham, R. S. & Shannon, A. G. 99 Inverse trigonometric and hyperbolic summation formulas involving generalized Fibonacci numbers. The Fibonacci Quarterly, 33, 3 40. [6] Basin, S. L. & Hoggatt Jr., V. E. 964 A primer for the Fibonacci numbers: Part I. The Fibonacci Quarterly,, 3 7. 6
[7] Dunlap, R. A. 003 The Golden Ratio and Fibonacci Numbers. World Scientific. [8] Howard, F. T. 003 The sum of the squares of two generalized Fibonacci numbers. The Fibonacci Quarterly, 4, 80 84. 66