Generating Function
A generating function 
is a formal power series
 |
(1)
|
whose coefficients give the sequence ![<span style=]()
{a_0,a_1,...}" src="https://mathworld.wolfram.com/images/equations/GeneratingFunction/Inline2.gif" style="height:15px; width:66px" />.
The Wolfram Language command GeneratingFunction[expr, n, x] gives the generating function in the variable 
for the sequence whose 
th term is expr. Given a sequence of terms, FindGeneratingFunction[![<span style=]()
{" src="https://mathworld.wolfram.com/images/equations/GeneratingFunction/Inline5.gif" style="height:15px; width:5px" />a1, a2, ...![<span style=]()
}" src="https://mathworld.wolfram.com/images/equations/GeneratingFunction/Inline6.gif" style="height:15px; width:5px" />, x] attempts to find a simple generating function in 
whose 
th coefficient is 
.
Given a generating function, the analytic expression for the 
th term in the corresponding series can be computing using SeriesCoefficient[expr, ![<span style=]()
{" src="https://mathworld.wolfram.com/images/equations/GeneratingFunction/Inline11.gif" style="height:15px; width:5px" />x, x0, n![<span style=]()
}" src="https://mathworld.wolfram.com/images/equations/GeneratingFunction/Inline12.gif" style="height:15px; width:5px" />]. The generating function 
is sometimes said to "enumerate" 
(Hardy 1999, p. 85).
Generating functions giving the first few powers of the nonnegative integers are given in the following table.
There are many beautiful generating functions for special functions in number theory. A few particularly nice examples are
for the partition function P, where 
is a q-Pochhammer symbol, and
for the Fibonacci numbers 
.
Generating functions are very useful in combinatorial enumeration problems. For example, the subset sum problem, which asks the number of ways 
to select 
out of 
given integers such that their sum equals 
, can be solved using generating functions.
The generating function of 
of a sequence of numbers 
is given by the Z-transform of 
in the variable 
(Germundsson 2000).
REFERENCES:
Bender, E. A. and Goldman, J. R. "Enumerative Uses of Generating Functions." Indiana U. Math. J. 20, 753-765, 1970/1971.
Bergeron, F.; Labelle, G.; and Leroux, P. "Théorie des espèces er Combinatoire des Structures Arborescentes." Publications du LACIM. Québec, Montréal, Canada: Univ. Québec Montréal, 1994.
Cameron, P. J. "Some Sequences of Integers." Disc. Math. 75, 89-102, 1989.
Doubilet, P.; Rota, G.-C.; and Stanley, R. P. "The Idea of Generating Function." Ch. 3 in Finite Operator Calculus (Ed. G.-C. Rota). New York: Academic Press, pp. 83-134, 1975.
Germundsson, R. "Mathematica Version 4." Mathematica J. 7, 497-524, 2000.
Graham, R. L.; Knuth, D. E.; and Patashnik, O. Concrete Mathematics: A Foundation for Computer Science, 2nd ed. Reading, MA: Addison-Wesley, 1994.
Harary, F. and Palmer, E. M. Graphical Enumeration. New York: Academic Press, 1973.
Hardy, G. H. Ramanujan: Twelve Lectures on Subjects Suggested by His Life and Work, 3rd ed. New York: Chelsea, p. 85, 1999.
Lamdo, S. K. Lectures on Generating Functions. Providence, RI: Amer. Math. Soc., 2003.
Leroux, P. and Miloudi, B. "Généralisations de la formule d'Otter." Ann. Sci. Math. Québec 16, 53-80, 1992.
Riordan, J. Combinatorial Identities. New York: Wiley, 1979.
Riordan, J. An Introduction to Combinatorial Analysis. New York: Wiley, 1980.
Rosen, K. H. Discrete Mathematics and Its Applications, 4th ed. New York: McGraw-Hill, 1998.
Sloane, N. J. A. and Plouffe, S. "Recurrences and Generating Functions." §2.4 in The Encyclopedia of Integer Sequences. San Diego, CA: Academic Press, pp. 9-10, 1995.
Stanley, R. P. Enumerative Combinatorics, Vol. 1. Cambridge, England: Cambridge University Press, p. 63, 1996.
Viennot, G. "Une Théorie Combinatoire des Polynômes Orthogonaux Généraux." Publications du LACIM. Québec, Montréal, Canada: Univ. Québec Montréal, 1983.
Wilf, H. S. Generatingfunctionology, 2nd ed. New York: Academic Press, 1994.
