Examples of Generating Functions - Worked Example A: Basics - Bivariate Generating Functions

Bivariate Generating Functions

One can define generating functions in several variables, for series with several indices. These are often called super generating functions, and for 2 variables are often called bivariate generating functions.

For instance, since is the generating function for binomial coefficients for a fixed n, one may ask for a bivariate generating function that generates the binomial coefficients for all k and n. To do this, consider as itself a series (in n), and find the generating function in y that has these as coefficients. Since the generating function for is just, the generating function for the binomial coefficients is:

and the coefficient on is the binomial coefficient.

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