scholarly journals Series Representation of Power Function

2018 ◽  
Author(s):  
Petro Kolosov
Keyword(s):  
Numerical Methods ◽  
Power Function ◽  
Series Representation ◽  
Positive Integers

In this paper described numerical expansion of natural-valued power function xn, in point x = x0, where n; x0 - positive integers. Applying numerical methods, the calculus of nite dierences, particular pattern, that is sequence A287326 in OEIS, which shows the expansion of perfect cube n as row sum over k; 0 ≤ k ≤ n − 1 is generalized, obtained results are applied to show expansion of monomial n2m+1; m = 0; 1; 2; ..., N. Additionally, relation between Faulhaber's sum ∑nm and nite dierences of power are shown in section 4.

scholarly journals Series Representation of Power Function

2018 ◽  
Author(s):  
Petro Kolosov
Keyword(s):  
Numerical Methods ◽  
Power Function ◽  
Exponential Function ◽  
Finite Differences ◽  
Series Representation ◽  
Discrete Case ◽  
Natural Numbers ◽  
Binomial Theorem ◽  
Binomial Expansion

In this paper described numerical expansion of natural-valued power function xn, in point x = x0 where n, x0 - natural numbers. Apply- ing 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 Bi- nomial sum. Additionally, in section 4 exponential function’s ex representation is shown.

scholarly journals Series Representation of Power Function

2018 ◽  
Author(s):  
Petro Kolosov
Keyword(s):  
Numerical Methods ◽  
Power Function ◽  
Exponential Function ◽  
Finite Differences ◽  
Series Representation ◽  
Discrete Case ◽  
Natural Numbers ◽  
Binomial Theorem ◽  
Binomial Expansion

In this paper described numerical expansion of natural-valued power function xn, in point x = x0 where n, x0 - natural numbers. Apply- ing 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 Bi- nomial sum. Additionally, in section 4 exponential function’s ex representation is shown.

scholarly journals Series Representation of Power Function

2017 ◽  
Author(s):  
Kolosov Petro
Keyword(s):  
Numerical Methods ◽  
Power Function ◽  
Exponential Function ◽  
Finite Differences ◽  
Series Representation ◽  
Discrete Case ◽  
Natural Numbers ◽  
Binomial Theorem ◽  
Binomial Expansion

In this paper described numerical expansion of natural-valued power function xn, in point x = x0 where n, x0 - natural numbers. Apply- ing 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 Bi- nomial sum. Additionally, in section 4 exponential function’s ex representation is shown.

scholarly journals On generalized heat polynomials

1988 ◽  
Vol 11 (2) ◽  
pp. 393-400 ◽  
Author(s):  
C. Nasim
Keyword(s):  
Heat Equation ◽  
Power Function ◽  
Initial Temperature ◽  
Series Representation ◽  
Generalized Heat Equation ◽  
Heat Transform

We consider the generalized heat equation ofnthorder∂2u∂r2+n−1r∂u∂r−α2r2u=∂u∂t. If the initial temperature is an even power function, then the heat transform with the source solution as the kernel gives the heat polynomial. We discuss various properties of the heat polynomial and its Appell transform. Also, we give series representation of the heat transform when the initial temperature is a power function.

scholarly journals Series Representation of Power Function

2016 ◽  
Vol 07 (03) ◽  
pp. 327-333 ◽  
Author(s):  
Petro Kolosov
Keyword(s):  
Power Function ◽  
Series Representation

scholarly journals Series Representation of Power Function

2018 ◽  
Author(s):  
Petro Kolosov
Keyword(s):  
Power Function ◽  
Series Representation ◽  
Main Property ◽  
Similar Triangles ◽  
To Receive

In this paper we discuss a problem of generalization of binomial distributed triangle, that is sequence A287326 in OEIS. The main property of A287326 that it returns a perfect cube n as sum of n-th row terms over k; 0 ≤ k ≤ n−1 or 1 ≤ k ≤ n, by means of its symmetry. In this paper we have derived a similar triangles in order to receive powers m = 5; 7 as row items sum and generalized obtained results in order to receive every odd-powered monomial n2m+1; m ≥ 0 as sum of row terms of corresponding triangle. In other words, in this manuscript are found and discussed the polynomials Dm(n,k) and Um(n,k), such that, when being summed up over k in some range with respect to m and n returns the monomial n2m+1.

scholarly journals Series Representation of Power Function

2018 ◽  
Author(s):  
Petro Kolosov
Keyword(s):  
Power Function ◽  
Series Representation ◽  
Main Property ◽  
Similar Triangles ◽  
To Receive

In this paper we discuss a problem of generalization of binomial distributed triangle, that is sequence A287326 in OEIS. The main property of A287326 that it returns a perfect cube n as sum of n-th row terms over k; 0 ≤ k ≤ n−1 or 1 ≤ k ≤ n, by means of its symmetry. In this paper we have derived a similar triangles in order to receive powers m = 5; 7 as row items sum and generalized obtained results in order to receive every odd-powered monomial n2m+1; m ≥ 0 as sum of row terms of corresponding triangle.

Numerical Methods

2019 ◽  
Author(s):  
Rajesh Kumar Gupta
Keyword(s):  
Numerical Methods
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