This article is about transformations of generating functions in mathematics. For generating functions (main article), see generating function. For generating functions in classical mechanics, see Generating function (physics). For generating function transformations in classical mechanics, see canonical transformation.
In mathematics, a transformation of a sequence's generating function provides a method of converting the generating function for one sequence into a generating function enumerating another. These transformations typically involve integral formulas applied to a sequence generating function (see integral transformations) or weighted sums over the higher-order derivatives of these functions (see derivative transformations).
Given a sequence, , the ordinary generating function (OGF) of the sequence, denoted , and the exponential generating function (EGF) of the sequence, denoted , are defined by the formal power series
In this article, we use the convention that the ordinary (exponential) generating function for a sequence is denoted by the uppercase function / for some fixed or formal when the context of this notation is clear. Additionally, we use the bracket notation for coefficient extraction from the Concrete Mathematics reference which is given by .
The main article gives examples of generating functions for many sequences. Other examples of generating function variants include Dirichlet generating functions (DGFs), Lambert series, and Newton series. In this article we focus on transformations of generating functions in mathematics and keep a running list of useful transformations and transformation formulas.
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a transformation of a sequence's generatingfunction provides a method of converting the generatingfunction for one sequence into a generating function...
in fact, the generatingfunction is not actually regarded as a function, and the "variable" remains an indeterminate. Generatingfunctions were first introduced...
ensuring that the generatingfunction is a function of 2N + 1 independent variables. However, as a feature of canonical transformations, it is always possible...
_{n})\end{bmatrix}}} For example, the function T ( x ) = 5 x {\displaystyle T(x)=5x} is a linear transformation. Applying the above process (suppose that...
of related derivative and series-based generatingfunctiontransformations on the ordinary generatingfunction of a sequence which effectively produces...
involutive transformation on real-valued functions that are convex on a real variable. Specifically, if a real-valued multivariable function is convex...
monotonic transformation (or monotone transformation) may also cause confusion because it refers to a transformation by a strictly increasing function. This...
f being an analytic function. Concentrating on the operator part, it shows that D is an infinitesimal transformation, generating translations of the real...
"Zeta Series GeneratingFunctionTransformations Related to Generalized Stirling Numbers and Partial Sums of the Hurwitz Zeta Function". arXiv:1611.00957...
x} This formula was discussed in the essay Square series generatingfunctiontransformations by the mathematician Maxie Schmidt from Georgia in Atlanta...
identifier of the site that has generated the operation. We can write the following inclusion transformationfunction: T(ins( p 1 , c 1 , s i d 1 {\displaystyle...
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identifying the transformation, γi is a multiplication factor common to these three functions, and the prime indicates the transformed function. The other...
to the generatingfunction identity f ( x ) = g ( log ( 1 + x ) ) . {\displaystyle f(x)=g(\log(1+x)).} Binomial transform Generatingfunction transformation...
M. D. (2017). "Square series generatingfunctiontransformations" (PDF). Journal of Inequalities and Special Functions. 8 (2). arXiv:1609.02803. Weisstein...
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∗ n t {\displaystyle n_{t+1}=f'(0)k*n_{t}} Using a moment-generating-functiontransformation M ( s ) = ∫ − ∞ ∞ e s x n ( x ) d x {\displaystyle M(s)=\int...
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