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In combinatorics, the binomial transform is a sequence transformation (i.e., a transform of a sequence) that computes its forward differences. It is closely related to the Euler transform, which is the result of applying the binomial transform to the sequence associated with its ordinary generating function.
Sequence transformations include linear mappings such as discrete convolution with another sequence and resummation of a sequence and nonlinear mappings, more generally. They are commonly used for series acceleration , that is, for improving the rate of convergence of a slowly convergent sequence or series .
The formula for an integration by parts is () ′ = [() ()] ′ ().. Beside the boundary conditions, we notice that the first integral contains two multiplied functions, one which is integrated in the final integral (′ becomes ) and one which is differentiated (becomes ′).
Given an arithmetic function, one can generate a bi-infinite sequence of other arithmetic functions by repeatedly applying the first summation. For example, if one starts with Euler's totient function φ, and repeatedly applies the transformation process, one obtains: φ the totient function; φ ∗ 1 = I, where I(n) = n is the identity function
Aitken's delta-squared process is an acceleration of convergence method and a particular case of a nonlinear sequence transformation. A sequence X = ( x n ) {\textstyle X=(x_{n})} that converges to a limiting value ℓ {\textstyle \ell } is said to converge linearly , or more technically Q-linearly, if there is some number μ ∈ ( 0 , 1 ...
Two classical techniques for series acceleration are Euler's transformation of series [1] and Kummer's transformation of series. [2] A variety of much more rapidly convergent and special-case tools have been developed in the 20th century, including Richardson extrapolation, introduced by Lewis Fry Richardson in the early 20th century but also known and used by Katahiro Takebe in 1722; the ...
In mathematics, the discrete Fourier transform (DFT) converts a finite sequence of equally-spaced samples of a function into a same-length sequence of equally-spaced samples of the discrete-time Fourier transform (DTFT), which is a complex-valued function of frequency. The interval at which the DTFT is sampled is the reciprocal of the duration ...
The linear sequence transformation can be applied to the divergent sequences of partial sums of divergent series to give those series generalized sums. An infinite matrix ( a i , j ) i , j ∈ N {\displaystyle (a_{i,j})_{i,j\in \mathbb {N} }} with complex -valued entries defines a regular matrix summability method if and only if it satisfies ...