p-ADIC CONGRUENCES FOR GENERALIZED FIBONACCI SEQUENCES

is the ordinary formal power series generating function for the sequence {yn+i}„>0 (cf. [12]. Furthermore, it is easy to see [1] that when the discriminant A = X +4ju ofP(t) is nonnegative and X & 0, the ratios yn+l I yn converge (in the usual archimedean metric on U) to a reciprocal root a of P(t). In this article we show that ratios of these y n also exhibit rapid convergence properties relating to P(t) in the/?-adic metrics on Q. Precisely, we prove that for all primes/? and all positive integers m the ratios y r ly r_, converge/?-adically in Z; this is shown via congruences that extend those predicted by the theory of formal group laws (cf. [2], [7], [10]) or the theory of /?-adic hypergeometric functions (cf [13]). When/? does not divide ymA, these ratios converge to the quadratic character of A modulo /?; otherwise, the limit is p or zero. Moreover, when p>3 and/? divides A, one obtains a supercongruence (cf. [2], [5], and eqs. (1.6), (3.8) below). These results are then used to give formal-group-law interpretations of some generalized Lucas sequences {Xn} = {ylnlyn\, and of the sequence {7^} = {F5n / (5Fn)} (where {Fn} is the familiar Fibonacci sequence associated to X = ju = 1) which has been studied in [3]. The results are as follows.