Inverse trigonometric and hyperbolic identities,series and products for Lucas sequences

Abstract

In 1936 Lehmer observed that the inverse tangents of the reciprocals of the odd-indexed Fibonacci numbers add up exactly to a quarter of $π$. Many formulas of the same kind have been found since, each with its own proof: for the inverse tangent and for the inverse hyperbolic tangent, for sums and for differences, for the Fibonacci numbers and for the Lucas numbers. We show that these are not different results but one and the same computation, seen through three signs. A single two-term identity, valid for the generalized Fibonacci sequence and its companion Lucas sequence of any parameter, contains them all. From it follow, with short telescoping proofs, infinitely many series for $π$ whose speed of convergence can be chosen at will, series whose sums are multiples of the logarithm of the golden ratio and of its analogues, and infinite and finite products with closed values. The Fibonacci and Lucas cases are listed in full; the classical formulas of Lehmer, of Hoggatt and Ruggles, of Melham and Shannon and of Melham appear among them, and the Pell cases, among them those of Mahon and Horadam, the Jacobsthal cases, where the logarithm of the golden ratio is replaced by that of $2$, the Mersenne numbers and the Chebyshev polynomials follow by substitution.