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''generatrix, of a sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043900/g0439001.png" /> of numbers or functions''
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''generatrix, of a sequence  $  \{ a _ {n} ( x) \} $
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of numbers or functions''
  
 
The sum of the power series
 
The sum of the power series
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043900/g0439002.png" /></td> </tr></table>
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$$
 +
F ( x , w )  = \sum _ { n= 0} ^ { \infty}  a _ {n} ( x) w^ {n}
 +
$$
  
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043900/g0439003.png" />. The generating function
+
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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043900/g0439004.png" /></td> </tr></table>
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$$
 +
F ( x , w )  = \sum _ { n= 0} ^  \infty  P _ {n} ( x) w  ^ {n} ,\  x \in (a , b ) ,
 +
$$
  
exists, under certain conditions, for polynomials <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043900/g0439005.png" /> that are orthogonal over some interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043900/g0439006.png" /> with respect to a weight <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043900/g0439007.png" />. For [[Classical orthogonal polynomials|classical orthogonal polynomials]] the generating function can be explicitly represented in terms of the weight <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043900/g0439008.png" />, 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.
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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|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|random variable]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043900/g0439009.png" /> taking integer values <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043900/g04390010.png" /> with probabilities <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043900/g04390011.png" /> is defined by
+
In probability theory, the generating function of a [[Random variable|random variable]] $  \xi $
 +
taking integer values $  \{ n \} _ {0}  ^  \infty  $
 +
with probabilities $  \{ p _  \xi  ( n) \} $
 +
is defined by
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043900/g04390012.png" /></td> </tr></table>
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$$
 +
F ( \xi , z )  = \sum _ { n= 0} ^  \infty  p _  \xi  ( n) z  ^ {n} ,\  | z |
 +
\leq  1 .
 +
$$
  
Using the generating function one can compute the probability distribution of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043900/g04390013.png" />, its mathematical expectation and its variance:
+
Using the generating function one can compute the probability distribution of $  \xi $,  
 +
its mathematical expectation and its variance:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043900/g04390014.png" /></td> </tr></table>
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$$
 +
p _  \xi  ( n)  =
 +
\frac{1}{n!}
 +
F ^ { ( n) } ( \xi , 0 ) ,\ \
 +
{\mathsf E} \xi  = F ^ { \prime } ( \xi , 1 ) ,
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043900/g04390015.png" /></td> </tr></table>
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$$
 +
{\mathsf D} \xi  = F ^ { \prime\prime } ( \xi , 1 ) + F ^ { \prime } ( \xi
 +
, 1 ) - [ F ^ { \prime } ( \xi , 1 ) ]  ^ {2} .
 +
$$
  
The generating function of a random variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043900/g04390016.png" /> can also be defined as the mathematical expectation of the random variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043900/g04390017.png" />, i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043900/g04390018.png" />.
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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====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> G. Szegö,   "Orthogonal polynomials" , Amer. Math. Soc. (1975)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> P.K. Suetin,   "Classical orthogonal polynomials" , Moscow (1979) (In Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> W. Feller,   "An introduction to probability theory and its applications" , '''1–2''' , Wiley (1957–1971)</TD></TR></table>
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<table><TR><TD valign="top">[1]</TD> <TD valign="top"> G. Szegö, "Orthogonal polynomials", Amer. Math. Soc. (1975)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> P.K. Suetin, "Classical orthogonal polynomials", Moscow (1979) (In Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> W. Feller, [[Feller, "An introduction to probability theory and its applications"|"An introduction to probability theory and its  applications"]], '''1–2''', Wiley (1957–1971)</TD></TR></table>
  
 +
====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
  
 +
$$
 +
F(x, w) = \sum_{n=0}^\infty \frac{a_{n}(x) w^{n}}{n!}
 +
$$
  
====Comments====
+
and the [[formal Dirichlet series]]
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
 
 
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043900/g04390019.png" /></td> </tr></table>
 
  
and the (formal) [[Dirichlet series|Dirichlet series]]
+
$$
 +
F ( x , s ) = \
 +
\sum _ { n=  1} ^  \infty 
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043900/g04390020.png" /></td> </tr></table>
+
\frac{a _ {n} ( x) }{n  ^ {s} }
 +
.
 +
$$
  
 
Usually it is possible to justify manipulations with such functions regardless of convergence.
 
Usually it is possible to justify manipulations with such functions regardless of convergence.

Latest revision as of 08:25, 16 March 2023


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.

How to Cite This Entry:
Generating function. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Generating_function&oldid=13920
This article was adapted from an original article by P.K. Suetin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article