Hurwitzian Continued Fractions Containing a Repeated Constant and An Arithmetic Progression

We prove an explicit formula for infinitely many convergents of Hurwitzian continued fractions that repeat several copies of the same constant and elements of one arithmetic progression, in a quasi-periodic fashion. The proof involves combinatorics and formal Laurent series. Using very little analysis we can express their limits in terms of (modified) Bessel functions and Fibonacci polynomials. The limit formula is a generalization of Lehmer’s theorem that implies the continuous fraction expansions of e and tan(1), and it can also be derived from Lehmer’s work using Fibonacci polynomial identities. We completely characterize those implementations of our limit formula for which the parameter of each Bessel function is the half of an odd integer, allowing them to be replaced with elementary functions. Introduction It is a remarkable property of infinite continued fractions that they often define a sequence whose limit is easier to describe than the individual entries. Even for [1, 1, . . .], the simplest example, the limit is easily found by solving a quadratic equation, it takes a bit longer to find a formula for the convergents, in terms of Fibonacci numbers. The class of Hurwitzian continued fractions of the form [α, . . . , α } {{ } r , β0, α, . . . , α, β0 + β1 · n } {{ } d ]n=1 that we propose investigating seems to be no different in this regard. For the special case r = 0, D. H. Lehmer [9] proved a formula for the limit in terms of Bessel functions, that can be verified very easily, after having guessed the correct answer. Using another result from [9] and some Fibonacci polynomial identities, it is not hard to generalize Lehmer’s formula to Hurwitzian continued fractions of the above form (see Remark 2.10). The resulting generalization has several famous special cases, the most famous ones being Euler’s formula for e and the formula for tan(1). On the other hand, the only research regarding the convergents themselves seems to be the work [8] of D. N. Lehmer (D. H. Lehmer’s father!), who proved congruences for their numerators and denominators by induction, in somewhat greater generality, but without stating the values of the convergents in an explicit fashion. Our work contains such an explicit formula, stated in Section 2 and proved in Section 4. A variant of the Euler-Mindig formulas using shifted partial denominators yields a summation formula with many vanishing terms. This leads to a compact recurrence Date: November 11, 2012. 2000 Mathematics Subject Classification. Primary 05A10; Secondary 05A15, 11A55, 30B10, 33A40.