Orthogonal and diagonal dimension fluxes of hyperspherical function
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RESEARCH
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Orthogonal and diagonal dimension fluxes of hyperspherical function Dusko Letic1, Nenad Cakic2, Branko Davidovic3* and Ivana Berkovic4 * Correspondence: iwtbg@beotel. net 3 Technical High School, Kragujevac, Serbia Full list of author information is available at the end of the article
Abstract In this paper, we present the theoretical research results of certain characteristics of the generalized hyperspherical function with two degrees of freedom as independent dimensions. Here, we primarily give the answers to the quantification of dimensional potentials (fluxes) of this function in the domain of natural numbers. In addition, we also give the solutions to continual fluxes of separate contour hyperspherical (HS) functions. The symbolical evaluation and numerical verification of the values of series and integrals are realized using MathCAD Professional and Mathematica. MSC 2010: 51M04; 33E99. Keywords: special function, hypersphere, dimension flux
1. Introduction The hypersphere function is a hypothetical function related to multi-dimensional space (see [1-3]). The most important aspect of this function is its connection to all functions that describe the properties of spherical entities: points, diameter, circumference, circle, surface, and volume of a sphere. The second property is the generalization of these functions from discrete to continuous. It belongs to the group of special functions, so its testing is being performed on the basis of known functions such as gamma (Γ), psi (ψ), and the like, so that its generalized, explicit form is the following [4]. Definition 1.1. The hyperspherical function [5]with two degrees of freedom k and n is defined as √ 2 π k r k+n−3 (k) HS(k, n, r) = k (k + n − 2) 2
(k, n ∈ Z, r ∈ N) ,
(1:1)
where Γ(z) is the gamma function. Using the fundamental properties of the gamma function, we advance from the domain of the natural values analytically to the set of real values for which we form the conditions for both its graphical interpretation and a more concise mathematical analysis. It is developed on the basis of two degrees of freedom k and n as vector dimensions, in addition to radius r, as an implied degree of freedom for every
© 2012 Letic et al; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Letic et al. Advances in Difference Equations 2012, 2012:22 http://www.advancesindifferenceequations.com/content/2012/1/22
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hypersphere. The dominant theorem is the one that relates to the recurrent property of this function [6]. It implies that the vectors on the left (n = 2, 1, 0, -1, -2,...) of the matrix M[HS]kxn (1.3) are obtained on the base of the reverent vector (n = 3) deduction, and the vectors on the right (n = 4, 5, 6, 7, 8,...) on the base of integrals by radius r [7]. ∂ HS(k, n, r) =
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