Related papers: Power Series Composition and Change of Basis
As was initially shown by Brent, exponentials of truncated power series can be computed using a constant number of polynomial multiplications. This note gives a relatively simple algorithm with a low constant factor.
This paper is a study of power series, where the coefficients are binomial expressions (iterated finite differences). Our results can be used for series summation, for series transformation, or for asymptotic expansions involving Stirling…
We transformed the generalized exponential power series to another functional form suitable for further analysis. By applying the Cauchy-Euler differential operator in the form of an exponential operator, the series became a sum of…
It is shown that the exponential of a complex power series up to order n can be implemented via (23/12+o(1))M(n) binary arithmetic operations over complex field, where M(n) stands for the (smoothed) complexity of multiplication of…
We derive and discuss a technique for manipulating power series which is complementary to standard procedures. We begin with the translation operator, but we express the operator as an infinite product instead of expanding it as a series…
The problem is to evaluate a polynomial in several variables and its gradient at a power series truncated to some finite degree with multiple double precision arithmetic. To compensate for the cost overhead of multiple double precision and…
This text provides very easy and short proofs of some basic properties of complex power series (addition, subtraction, multiplication, division, rearrangement, composition, differentiation, uniqueness, Taylor's series, Principle of…
Puiseux series are power series in which the exponents can be fractional and/or negative rational numbers. Several computer algebra systems have one or more built-in or loadable functions for computing truncated Puiseux series -- perhaps…
We provide algorithms computing power series solutions of a large class of differential or $q$-differential equations or systems. Their number of arithmetic operations grows linearly with the precision, up to logarithmic terms.
We present an algebraic algorithm that computes the composition of two power series in softly linear time complexity. The previous best algorithms are $\mathop{\mathrm O}(n^{1+o(1)})$ by Kedlaya and Umans (FOCS 2008) and an $\mathop{\mathrm…
A set of exactly computable orthonormal basis functions that are useful in computations involving constituent quarks is presented. These basis functions are distinguished by the property that they fall off algebraically in momentum space…
A rapid transformation is derived between spherical harmonic expansions and their analogues in a bivariate Fourier series. The change of basis is described in two steps: firstly, expansions in normalized associated Legendre functions of all…
We show that the compositions of positive integers may be interpreted in terms of powers of some power series, over arbitrary commutative ring. As consequences, several closed formulas for the compositions as well as for the generalized…
In this research, the Bernoulli polynomials are introduced. The properties of these polynomials are employed to construct the operational matrices of integration together with the derivative and product. These properties are then utilized…
We survey a family of polynomials that are very useful in all kinds of power series manipulations, and appearing more frequently in the literature. Applications to formal power series, generating functions and asymptotic expansions are…
We give an algorithm for reversion of formal power series, based on an efficient way to implement the Lagrange inversion formula. Our algorithm requires $O(n^{1/2}(M(n) + MM(n^{1/2})))$ operations where $M(n)$ and $MM(n)$ are the costs of…
Using notions of composita and composition of generating functions we obtain explicit formulas for Chebyshev polynomials, Legendre polynomials, Gegenbauer polynomials, Associated Laguerre polynomials, Stirling polynomials, Abel polynomials,…
We present several sequences involving harmonic numbers and the central binomial coefficients. The calculational technique is consists of a special summation method that allows, based on proper two-valued integer functions, to calculate…
Power transforms, such as the Box-Cox transform and Tukey's ladder of powers, are a fundamental tool in mathematics and statistics. These transforms are primarily used for normalizing and standardizing datasets, effectively by raising…
Certain triples of power series, considered by I. Macdonald, give a natural framework for many combinatorial and number theoretic sequences, such as the Stirling, Bernoulli and harmonic numbers and partitions of different kinds. The power…