The Hadamard product (denoted by∗) of two power series A(x) =a0+a1x+a2x2+···and B(x) =b0+b1x+b2x2+··· is the power series A(x)∗B(x) =a0b0+a1b1x+a2b2x2+···. Although it is well known that the Hadamard product of two rational functions is also rational, a closed form expression of the Hadamard product of rational functions has not been found. Since any rational power series can be expanded by partial fractions as a polynomial plus a sum of power series of the form 1/(1−ax)m+1, to find the Hadamard product of any two rational power series it is sufficient to find the Hadamard product 1/(1−ax)m+1∗ 1/(1−bx)n+1 = (1 +ax)m∗(1+bx)n/(1−abx)m+n+1.The Hadamard product of negative powers of quadratic polynomials have also been derived.

Author Bio

The author of the paper is Ishan Kar, who is a junior at Prospect High School, Saratoga. He became interested in Combinatorics as a freshman and chanced upon this project trying to solve a problem with a typo in Fibonacci Quarterly. With his advisor, Professor Ira Gessel, he developed this three-step simple method to solve the Hadamard product of two rational generating functions and presented his work at the Joint Mathematics Meeting of the AMS/MAA in 2021. After finishing high school, he plans to study mathematics and computer science in a four year college. His other interests include tennis, running his non-profit organization and traveling.