Linear Recurrence Relations for Sums of Products of Two Terms

Abstract

For a sum of the form $\sum_k F({\boldsymbol n},k)G({\boldsymbol n},k)$, we set up two systems of equations involving shifts of $F({\boldsymbol n},k)$ and $G({\boldsymbol n},k)$. Then we solve the systems by utilizing the recursion of $F({\boldsymbol n},k)$ and the method of undetermined coefficients. From the solutions, we derive linear recurrence relations for the sum. With this method, we prove many identities involving Bernoulli numbers and Stirling numbers.