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.
Famous quotes containing the word functions:
“Empirical science is apt to cloud the sight, and, by the very knowledge of functions and processes, to bereave the student of the manly contemplation of the whole.”
—Ralph Waldo Emerson (18031882)