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Generating function transformation information


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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Generating function transformation

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a transformation of a sequence's generating function provides a method of converting the generating function for one sequence into a generating function...

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Generating function

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in fact, the generating function is not actually regarded as a function, and the "variable" remains an indeterminate. Generating functions were first introduced...

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Canonical transformation

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ensuring that the generating function is a function of 2N + 1 independent variables. However, as a feature of canonical transformations, it is always possible...

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Legendre transformation

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involutive transformation on real-valued functions that are convex on a real variable. Specifically, if a real-valued multivariable function is convex...

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Infinitesimal transformation

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f being an analytic function. Concentrating on the operator part, it shows that D is an infinitesimal transformation, generating translations of the real...

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Monotonic function

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monotonic transformation (or monotone transformation) may also cause confusion because it refers to a transformation by a strictly increasing function. This...

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Dirichlet series

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of related derivative and series-based generating function transformations on the ordinary generating function of a sequence which effectively produces...

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Binomial transform

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binomial transform to the sequence associated with its ordinary generating function. The binomial transform, T, of a sequence, {an}, is the sequence...

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Stirling transform

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to the generating function identity f ( x ) = g ( log ⁡ ( 1 + x ) ) . {\displaystyle f(x)=g(\log(1+x)).} Binomial transform Generating function transformation...

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Transformation matrix

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_{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...

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Jacobi elliptic functions

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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...

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Probability density function

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a probability density function (PDF), density function, or density of an absolutely continuous random variable, is a function whose value at any given...

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Theta function

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x} This formula was discussed in the essay Square series generating function transformations by the mathematician Maxie Schmidt from Georgia in Atlanta...

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Operational transformation

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identifier of the site that has generated the operation. We can write the following inclusion transformation function: T(ins( p 1 , c 1 , s i d 1 {\displaystyle...

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Stirling numbers of the first kind

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"Zeta Series Generating Function Transformations Related to Generalized Stirling Numbers and Partial Sums of the Hurwitz Zeta Function". arXiv:1611.00957...

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Hypergeometric function

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hypergeometric function 2F1(a,b;c;z) is a special function represented by the hypergeometric series, that includes many other special functions as specific...

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Ramanujan theta function

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M. D. (2017). "Square series generating function transformations" (PDF). Journal of Inequalities and Special Functions. 8 (2). arXiv:1609.02803. Weisstein...

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Inverse transform sampling

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the inverse transformation method, or the Smirnov transform) is a basic method for pseudo-random number sampling, i.e., for generating sample numbers...

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Continuous uniform distribution

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height would be 1 15 . {\displaystyle {\tfrac {1}{15}}.} The moment-generating function of the continuous uniform distribution is: M X = E ( e t X ) = ∫...

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Laplace transform

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of generating functions (1814), and the integral form of the Laplace transform evolved naturally as a result. Laplace's use of generating functions was...

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Digital transformation

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chains. Represented by the TOP framework, digital transformation acts as a catalyst for generating and leveraging benefits. These benefits hold the potential...

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Function composition

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combinations of these functions forms a transformation group; and one says that the group is generated by these functions. A fundamental result in group theory...

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