Generating Function

A Power Series

$$f(x)=\sum_{n=0}^\infty a_nx^n$$

whose Coefficients give the Sequence $\{a_0$, $a_1$, ...$\}$. The Mathematica ${}^{\scriptstyle\circledRsymbol}$ (Wolfram Research, Champaign, IL) function DiscreteMath`RSolve`PowerSum gives the generating function of a given expression, and ExponentialPowerSum gives the exponential generating function.

Generating functions for the first few powers are

$$\matrix{ 1:\hfill &{x\over 1-x}\hfill &=x+x^2+x^3+\ldots\hf... ...^2+10x+1)\over(1-x)^5}\hfill &=x+16x^2+81x^3+\ldots.\hfill\cr}$$

See also Moment-Generating Function, Recurrence Relation

References

Wilf, H. S. Generatingfunctionology, 2nd ed. New York: Academic Press, 1990.