“generating function”的概念、定义、翻译、参考文献-科学参考

generating function

Mathematics

The power series G(x), where

$G (x) = g_{0} + g_{1} x + g_{2} x^{2} + g_{3} x^{3} + \dots,$

is the generating function for the infinite sequence g₀, g₁, g₂, g₃,…. (Notice that it is convenient here to start the sequence with a term with subscript 0.) Such power series can be manipulated algebraically, and it can be shown, for example, that

$\begin{matrix} \frac{1}{1 - x} = 1 + x + x^{2} + x^{3} + \dots, \\ \frac{1}{{(1 - x)}^{2}} = 1 + 2 x + 3 x^{2} + 4 x^{3} + \dots . \end{matrix}$

Hence, 1/(1−x) and 1/(1−x)² are the generating functions for the sequences 1, 1, 1, 1,…and 1, 2, 3, 4,…, respectively.

The Fibonacci sequence F₀, F₁, F₂,…is given by F₀ = 1, F₁ = 1, and F_n + 2 = F_n + 1+ F_n. It can be shown that the generating function for this sequence is 1/(1−x−x²).

The use of generating functions enables sequences to be handled concisely and algebraically. A difference equation for a sequence can lead to an equation for the corresponding generating function, and the use of partial fractions, for example, may then lead to a formula for the n‐th term of the sequence.

Probability and moment generating functions are very powerful tools in statistics.

Statistics

A function of an arbitrary variable (usually t) which, when expanded as a power series (in t), yields coefficients of interest to statisticians and others. Examples are the moment-generating function and the probability-generating function.

单词	generating function
释义	generating function Mathematics The power series G(x), where $G (x) = g_{0} + g_{1} x + g_{2} x^{2} + g_{3} x^{3} + \dots,$ is the generating function for the infinite sequence g₀, g₁, g₂, g₃,…. (Notice that it is convenient here to start the sequence with a term with subscript 0.) Such power series can be manipulated algebraically, and it can be shown, for example, that $\begin{matrix} \frac{1}{1 - x} = 1 + x + x^{2} + x^{3} + \dots, \\ \frac{1}{{(1 - x)}^{2}} = 1 + 2 x + 3 x^{2} + 4 x^{3} + \dots . \end{matrix}$ Hence, 1/(1−x) and 1/(1−x)² are the generating functions for the sequences 1, 1, 1, 1,…and 1, 2, 3, 4,…, respectively. The Fibonacci sequence F₀, F₁, F₂,…is given by F₀ = 1, F₁ = 1, and F_n + 2 = F_n + 1+ F_n. It can be shown that the generating function for this sequence is 1/(1−x−x²). The use of generating functions enables sequences to be handled concisely and algebraically. A difference equation for a sequence can lead to an equation for the corresponding generating function, and the use of partial fractions, for example, may then lead to a formula for the n‐th term of the sequence. Probability and moment generating functions are very powerful tools in statistics. Statistics A function of an arbitrary variable (usually t) which, when expanded as a power series (in t), yields coefficients of interest to statisticians and others. Examples are the moment-generating function and the probability-generating function.
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