Cubo 3/2003 [antikvár]

From Fibonacci Numbers to Symmetric Functions Trueman MacHenry York University, 4700 Keele Street, Toronto, Ontario, Canada, M3J1P3 e-mail address: machenryQmathstat. yorku. ca (To the memory of my teacher. Max Dehn) Introduction In this essay we are interested in the not so well-known theory that lies behind a very farnous and interesting collection of results that have to do with the Fibonacci numbers. The interest in Fibonacci numbers has several facets, their interpretation as a growth sequence is of interest both in mathematics and in physics and biology. Their use in primality testing and their connection with the so-called Golden Ratio have been exploited by mxmber theorists. Recreational mathematics has been much interested in their striking numerical properties such as the fact that, if the numbers are suitably indexed, those indexed by primes are themselves prime, and pairs of numbers indexed by relatively prime numbers are themselves relatively prime; and, so 011 for results which are constantly being added to. The Fibonacci sequence has alsó been generalized to sequences of Fibonacci polynomials in two indetenninants which satisfy the same recursion formula [Rí, p.34 ]. There are many interesting relations among these polynomials which in turn produce the large number of well-advertised numerical relations [Ri], [HW], [Vo]. And, of course, there is a respected journal (ievoted to mathematics connected to them, The Fibonacci Quarterly. What is not so well-known is that this body of results is a part of a much more generál and far-reaching theory that impinges 011 the theory of multiplicative arithmetic functions, 011 the theory of equations and symmetric functions and even reaches intő representation theory of groups and algebras, and into other extensions of these subjects. That is, the lore of Fibonacci