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.
Additional information
Notes on contributors
Ethan Berkove
Ethan Berkove (MR Author ID: 608283) received his Ph.D. from the University of Wisconsin, Madison, and has taught at Lafayette College since 1999. He enjoys working collaboratively on engaging math problems wherever he finds them. His interests include hiking, biking, and reading (when he can find the time). He would like to thank his colleagues, Jonathan Bloom and Gary Gordon, as well as anonymous referees, for comments and conversations which greatly improved the exposition and content of this paper. He lives in Easton, Pennsylvania with his wife and two sons.
Michael A. Brilleslyper
Mike Brilleslyper (MR Author ID: 994411) received his Ph.D. from the University of Arizona. After a 21-year career on the faculty at the U. S. Air Force Academy, he is now the department chair of applied mathematics at Florida Polytechnic University. He enjoys teaching a variety of courses and collaborating with colleagues on interesting problems, many of which start out as undergraduate research projects. Mike and his wife MaryAnn (an amazing musician and teacher) enjoy exploring their new state and following the adventures of their two daughters: Emma and Meg.