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.
Ira M. Gessel
"A New Method To Compute The Hadamard Product Of Two Rational Functions,"
Rose-Hulman Undergraduate Mathematics Journal: Vol. 23:
1, Article 3.
Available at: https://scholar.rose-hulman.edu/rhumj/vol23/iss1/3