# Generating Function

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## generating function

[′jen·ə‚rād·iŋ ‚fəŋk·shən]*g*(

*x*,

*y*) corresponding to a family of orthogonal polynomials ƒ

_{0}(

*x*), ƒ

_{1}(

*x*),…, where a Taylor series expansion of

*g*(

*x*,

*y*) in powers of

*y*will have the polynomial ƒ

_{n }(

*x*) as the coefficient for the term

*y*

^{ n }.

*g*(

*y*), corresponding to a sequence

*a*

_{0},

*a*

_{1}, ⋯) where

*g*(

*y*) =

*a*

_{0}+

*a*

_{1}

*y*+

*a*

_{2}

*y*

^{2}+ …. Also known as ordinary generating function.

## Generating Function

A generating function of the sequence *f _{0}, f_{1},..., f_{n}* is the function

assuming that this power series converges for at least one nonzero value of *t*.

The sequence *f _{0}, f_{1},..., f_{n}*,… can be a sequence of numbers or of functions. In the latter case, the generating function depends not only on

*t*but also on the arguments of the functions

*f*For example, if

_{n}.*f*=

_{n}*aq*, where

^{n}*a*and

*q*are constants, the generating function is

If the fn are Fibonacci numbers—that is, if *f _{0} = 0, f_{1} = 1, f_{n+2} = f_{n+1} + f_{n}*—we have

If *f _{n} = T_{n}* (

*x*) are Chebyshev polynomials—that is, if

*T*

_{0}(

*x*) = 1

and *T _{n}* (

*x*) = cos (

*n*arc cos

*x*)—then

Knowledge of the generating function of a sequence often makes it easier to study the properties of the sequence. Generating functions are used in probability theory, in the theory of functions, and in the theory of invariants in algebra. Methods involving generating functions were first applied by P. Laplace to solve certain problems in probability theory.

### REFERENCES

Feller, W.*Vvedenie v teoriiu veroiatnostei i eeprilozheniia,*2nd ed., vols. 1–2. Moscow, 1967. (Translated from English.)

Natanson, I. P.

*Konstruktivnaia teoriia funktsii.*Moscow-Leningrad, 1949.