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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''

The sum of the power series

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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

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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.

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

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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:

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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" />.

====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, [[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

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and the (formal) [[Dirichlet series|Dirichlet series]]

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