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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.

Linear Recurrence Relations for Sums of Products of Two Terms