Series Representation of Power Function

Kolosov Petro (1)

1 Odessa National I.I.Mechnikov University

In this paper described numerical expansion of natural-valued power function $x^n$, in point $x=x_0$ where $(n, \ x_0)$ - natural numbers. Applying numerical methods, that is calculus of finite differences, namely, discrete case of Binomial expansion is reached. Received results were compared with solutions according to Newton’s Binomial theorem and MacMillan Double Binomial sum. Additionally, in Section 2 exponential function's $Exp(x), \ x \in \mathbb{N}$ representation is shown and relation between Pascal's triangle and hypercubes is shown in Section 3.

Mots clés: Exponentiation; Mathematical Series; Series Expansion; Derivative; Series representation; Cube (Algebra); Perfect cube; Hypercube; Binomial distribution; Discrete Mathematics; Applied Mathematics; Preprint; Monomial; Polynomial; Power function; Power series (mathematics); Power (mathematics); Exponential function; Divided difference; Diophantine equations; Pascal's triangle; Ordinary differential equation; Forward Finite Difference; Backward Finite Difference; Partial differential equation; Multinomial coefficient; High order finite difference; Partial derivative; Calculus of variations; High order derivative; Partial difference; Differential calculus; Binomial coefficient; Central Finite difference; Difference Equations; Differentiation; Finite differences; Binomial Series; Pascal’s triangle; Newton's interpolation formula; Numerical Differentiation; Pascal triangle; Multinomial theorem; Binomial theorem; Binomial expansion; Calculus; Differential equations; Number theory; Newton's Binomial Theorem; Binomial Sum; Numerical analysis; Mathematics; Mathematical analysis; Functional analysis; Finite difference; Numercal methods; Series

Domaines: Théorie des nombres [math.NT]; Analyse classique [math.CA]; Systèmes dynamiques [math.DS]; Combinatoire [math.CO]; Mathématiques générales [math.GM]

Kolosov Petro. Series Representation of Power Function. 2018. ⟨hal-01283042v7⟩