p-ADIC INTERPOLATION OF THE FIBONACCI SEQUENCE VIA HYPERGEOMETRIC FUNCTIONS

Many authors have considered the problem of extending the Fibonacci sequence to arbitrary real or complex subscripts (cf. [1], [6], and references therein). Since the positive integers form a discrete subset of R the existence of multitudes of continuous functions f : R→ R such that f(n) = Fn for positive integers n is immediate and the question then becomes one of determining the various properties of such functions. In this paper we consider the extent to which the Fibonacci and Lucas sequences can be extended to arbitrary p-adic subscripts in a continuous way. In the process we determine several apparently new expressions, both p-adic and real, for the Fibonacci sequence in terms of hypergeometric functions and combinatorial sums. For example, Dilcher ([3], eq. (3.3)) has proved that for positive integers n,