Summation Formulas, Generating Functions, and Polynomial Division

Abstract

Summary We describe a general method that finds closed forms for partial sums of power series whose coefficients arise from linear recurrence relations. These closed forms allow one to derive a vast collection of identities involving the Fibonacci numbers and other related sequences. Although motivated by a polynomial long division problem, the method fits naturally into a standard generating function framework. We also describe an explicit way to calculate the generating function of the Hadamard product of two generating functions, a construction on power series which resembles the dot product. This allows one to use the method for many examples where the recurrence relation for the coefficients is not initially known.