Generating function

2020 Mathematics Subject Classification: Primary: 05A15 [MSN][ZBL]

generatrix, of a sequence $ \{ a _ {n} ( x) \} $ of numbers or functions

The sum of the power series

$$ F ( x , w ) = \sum _ { n= 0} ^ { \infty} a _ {n} ( x) w^ {n} $$

with positive radius of convergence. If the generating function is known, then properties of the Taylor coefficients of analytic functions are used in the study of the sequence $ \{ a _ {n} ( x) \} $. The generating function

$$ F ( x , w ) = \sum _ { n= 0} ^ \infty P _ {n} ( x) w ^ {n} ,\ x \in (a , b ) , $$

exists, under certain conditions, for polynomials $ \{ P _ {n} ( x) \} $ that are orthogonal over some interval $ ( a , b ) $ with respect to a weight $ h ( x) $. For classical orthogonal polynomials the generating function can be explicitly represented in terms of the weight $ h ( x) $, and it is used in calculating values of these polynomials at individual points, as well as in deriving identity relations between these polynomials and their derivatives.

In probability theory, the generating function of a random variable $ \xi $ taking integer values $ \{ n \} _ {0} ^ \infty $ with probabilities $ \{ p _ \xi ( n) \} $ is defined by

$$ F ( \xi , z ) = \sum _ { n= 0} ^ \infty p _ \xi ( n) z ^ {n} ,\ | z | \leq 1 . $$

Using the generating function one can compute the probability distribution of $ \xi $, its mathematical expectation and its variance:

$$ p _ \xi ( n) = \frac{1}{n!} F ^ { ( n) } ( \xi , 0 ) ,\ \ {\mathsf E} \xi = F ^ { \prime } ( \xi , 1 ) , $$

$$ {\mathsf D} \xi = F ^ { \prime\prime } ( \xi , 1 ) + F ^ { \prime } ( \xi , 1 ) - [ F ^ { \prime } ( \xi , 1 ) ] ^ {2} . $$

The generating function of a random variable $ \xi $ can also be defined as the mathematical expectation of the random variable $ z ^ \xi $, i.e. $ F ( z , \xi ) = {\mathsf E} z ^ \xi $.

References

[1]	G. Szegö, "Orthogonal polynomials", Amer. Math. Soc. (1975)
[2]	P.K. Suetin, "Classical orthogonal polynomials", Moscow (1979) (In Russian)
[3]	W. Feller, "An introduction to probability theory and its applications", 1–2, Wiley (1957–1971)

Comments

Generating functions in the sense of formal power series are also often used. Other commonly used types of generating functions are, e.g., the exponential generating function

$$ F(x, w) = \sum_{n=0}^\infty \frac{a_{n}(x) w^{n}}{n!} $$

and the formal Dirichlet series

$$ F ( x , s ) = \ \sum _ { n= 1} ^ \infty \frac{a _ {n} ( x) }{n ^ {s} } . $$

Usually it is possible to justify manipulations with such functions regardless of convergence.