Some remarkable infinite product identities involving Fibonacci and Lucas numbers∗ Kunle Adegoke † Department of Physics and Engineering Physics, Obafemi Awolowo University, Ile-Ife, Nigeria Abstract By applying the classic telescoping summation formula and its vari- ants to identities involving inverse hyperbolic tangent functions hav- ing inverse powers of the golden ratio as arguments and employing subtle properties of the Fibonacci and Lucas numbers, we derive interesting general infinite product identities involving these num- bers. 1 Introduction The following striking (in Frontczak’s description [1]) infinite product iden- tity involving Fibonacci numbers, ∞ F +1 2k+2 =3 , F2k+2 1 Yk=1 − originally obtained by Melham and Shannon [2], is an n =1= q case of a more general identity (to be derived in this present paper): arXiv:1702.08321v2 [math.NT] 29 Apr 2017 ∞ F + F q F L (2n−1)(2k+2q) 2q(2n−1) = (2n−1)(2k−1) (2n−1)2k . F(2n−1)(2k+2q) F2q(2n−1) L(2n−1)(2k−1)F(2n−1)2k Yk=1 − Yk=1 ∗AMS Classification Numbers : 11B39, 11Y60 †
[email protected],
[email protected] 1 + The above identity is valid for q N0, n Z , where we use N0 to denote the set of natural numbers including∈ zero.∈ Here the Fibonacci numbers, Fn, and Lucas numbers, Ln, are defined, for n N , as usual, through the recurrence relations F = F + F , with ∈ 0 n n−1 n−2 F0 =0, F1 =1 and Ln = Ln−1 + Ln−2, with L0 =2, L1 =1. Our goal in this paper is to derive several general infinite product identi- ties, including the one given above.