Hypergeometric Functions and Fibonacci Numbers

Hypergeometric functions are an important tool in many branches of pure and applied mathematics, and they encompass most special functions, including the Chebyshev polynomials. There are also well-known connections between Chebyshev polynomials and sequences of numbers and polynomials related to Fibonacci numbers. However, to my knowledge and with one small exception, direct connections between Fibonacci numbers and hypergeometric functions have not been established or exploited before. It is the purpose of this paper to give a brief exposition of hypergeometric functions, as far as is relevant to the Fibonacci and allied sequences. A variety of representations in terms of finite sums and infinite series involving binomial coefficients are obtained. While many of them are well known, some identities appear to be new. The method of hypergeometric functions works just as well for other sequences, especially the Lucas, Pell, and associated Pell numbers and polynomials, and also for more general secondorder linear recursion sequences. However, apart from the final section, we will restrict our attention to Fibonacci numbers as the most prominent example of a second-order recurrence. The idea and "philosophy" behind this paper is similar to that of R. Roy in [42] concerning binomial identities, though somewhat more limited in scope. It can be seen as an attempt to bring some partial order into the confusing abundance of formulas satisfied by Fibonacci numbers. For reasons of brevity and clarity, no attempt has been made to be complete, or to classify the many identities in the literature that are similar to, but still different from, those obtained in this paper. After each hypergeometric transformation, only the most immediate Fibonacci formula is given. Statements that a certain identity is apparently new should be taken with the necessary caution. Only The Fibonacci Quarterly has been checked to any degree of completeness, and even there it may be possible for some identities to have been overlooked. The author apologizes in advance for any missed or incomplete references. In spite of the relative absence of hypergeometric series from the pages of The Fibonacci Quarterly or related papers published elsewhere, it should be mentioned that they were occasionally used in somewhat different connections. The four papers that make most extensive use of hypergeometric functions are, to the best of my knowledge, by P. S. Bruckman [8], L. Carlitz [12], [13], and H. W. Gould [25]. To this we should add the article-length solution [44] by P. S. Bruckman to a problem in The Fibonacci Quarterly. The one direct connection to Fibonacci numbers that I could find is in the solution (by the proposer) of a problem by H.-J. Seiffert [43].