Families of fibonacci and lucas sums via the moments of a random variable

In this paper we show how the expectation of a particular random variable gives rise to an infinite series whose coefficients are certain functions of the Fibonacci numbers. A general result follows from this which, when specialized to varying degrees, leads both to well-known and lesser-known identities. Also, by considering a different random variable, we go on to obtain corresponding results for the Lucas numbers. Finally, we look at series arising from higher moments.