# Generating Function

The generating function for the sequence a_{1}, a_{2}, ...., a_{r}, .... of real numbers of a numeric functions (a_{0}, a_{1}, a_{2}, ...a_{r},...) is the infinite series.

A(z) = a_{0} + a_{1}z + a_{2}z^{2} + ...+ a_{r}z^{r} + ...

# Some Results

- Let a and b are any two discrete numeric functions and b = Î±a, then B(z) = Î± A(z) where B(z) is the generating function corresponding to numeric b and A(z) is that of a.

e.g., The generating function of the numeric function

is - For any three discrete numeric functions a, b and c, if c = a + b, then C(z) = A(z) + B(z) is the generating function representation.

e.g., The generating function of the numeric function a_{r}= 1^{r}+ 2^{r}(r â‰¥ 0)

is - Let a be a numeric function and A(z) its generating function. Let b be a numeric function such that
*b*= Î±_{r}^{r}*a*_{r}

Then, the generating function of b is

_{}

_{ }

B(z) = A(Î±z)

e.g., The generating function of the numeric function ar = 1, r â‰¥ 0 is - Let
*A*(*z*) be the generating function of a. Then, z^{1}A(z) is the generating function of S^{i}a for any positive integer i. - Let
*A*(*z*) be the generating function of a. Then, z^{â€“i}[A(z) â€“ a_{0}â€“ a_{1}z â€“ a_{2}z^{2}â€“ ...â€“a_{iâ€“1}z^{iâ€“1}] is the generating function of S^{â€“i}a.

e.g., The generating function of a_{r}= 3^{r+2}, z â‰¥ 0 is - For b = âˆ†a, the generating function is given by

and for b = âˆ‡ a B(z) = A(z) â€“ z A(z) - Let C = a*b, i.e., C is the convolution of two numeric functions and its generating function C (z) = A(z) Â· B(z) is the coefficient of z
^{r}in the product of