# Category Archives: generating functions

## A Request for a Proof of a Binomial Identity

A few weeks ago I received an email from Professor Steve Drekic at the University of Waterloo. He asked if I knew of a way to prove the following binomial identity: (He told me later that he wanted it to … Continue reading

## Six Proofs of a Binomial Identity

I’m a big fan of proving an identity in multiple ways, as I think each perspective gives additional insight into why the identity is true.  In this post we’ll work through six different proofs of the binomial identity . 1. … Continue reading

## Solving Second-Order Difference Equations with Constant Coefficients: Part II

In this post we’ll discuss the missing case from last month’s derivation of the solution for second-order difference equations with constant coefficients; namely, the case in which the characteristic equation has a repeated root rather than two distinct roots. Once … Continue reading

## Solving Second-Order Difference Equations with Constant Coefficients

Suppose you have a linear, homogeneous second-order difference equation with constant coefficients of the form .  The solution procedure is as follows: “Guess” a solution of the form .  Then substitute this guess into the difference equation to obtain .  … Continue reading

## Generating Function for the Reciprocals of the Central Binomial Coefficients

In this post we generalize the result from the last post to find the generating function for the reciprocals of the central binomial coefficients.  As we did with that one, we start with the beta integral expression for : . Now, … Continue reading