# Orthogonal and diagonal dimension fluxes of hyperspherical function

- Dusko Letic
^{1}, - Nenad Cakic
^{2}, - Branko Davidovic
^{3}Email author and - Ivana Berkovic
^{4}

**2012**:22

https://doi.org/10.1186/1687-1847-2012-22

© Letic et al; licensee Springer. 2012

**Received: **17 May 2011

**Accepted: **1 March 2012

**Published: **1 March 2012

## 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*

*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 hypersphere. The dominant theorem is the one that relates to the recurrent property of this function [6].

*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].

## 2. Dimensional potentials--the fluxes of *HS* function

### 2.1. Vertical dimensional flux of hypersphere function

Definition 2.1 **.** The discrete dimensional potential or the hypersphere function *flux* is the sum of all separate functions in the (sub)matrix of this function that we expand for integer or real degrees of freedom (1.3).

*HS*function. The first phase is to define the value of infinite series of functions classified in columns (vectors) of the submatrix

*M*[

*HS*]

_{ kxn }(

*k, n*∈

*N*). This is also the definition of

*vertical dimensional fluxes*of

*HS*function. The first value to be calculated relates to the fourth columns (

*n*= 3) of this submatrix (1.3). From this, we obtain

*erf*(

*z*) is an error function. When

*k*is even (0,2,4,...), respectively, odd (1,3,5,...), that series can be classified as dichotomous, so we obtain two complementary series

*k*= 2

*b*) and odd members (

*k*= 2

*b*+1, where

*b*∈

*N*). In this sense we get

*n*< 3). We connect the sphere hypervolume (

*n*= 3) with its hypersurface (

*n*= 2). In this sense a new vector flux follows

*n*= 1), a series is obtained in view of the previous, therefore

*n*= 0, the series value is found on the basic of deducing, so it follows that

*n*= 3, series are found by inverse operations, i.e., by recurrent relation in view of integrating along radius

*r*. Consequently,

*erf*(

*z*) = -

*ierf*(

*iz*).

*b*" members (or as a series with the incomplete gamma function). In that sense, the value of the series with even numbers is

### 2.2. Integral solvability on the base of the incomplete gamma function

*n*= 3 is the easiest one to solve, and it represents the base for calculating fluxes of higher degrees of freedom (

*n*> 0), through integration of previously obtained results. In that sense, this procedure is possible by using the series where the incomplete gamma function. The second integral (2.2) is reduced to known terms, and one among them is [28]

*n*= 5 is realized by integrating the expressions

*n*= 6 is

*b*) + 3 < -1^b ≠ -1^

*r*= ∞. If the conditions are not met, this integral is indefinite. Some values of these discrete and continuous fluxes (for

*r*= 1) are given in Table 1.

Values of discrete and continuous fluxes.

Degree of freedom (n) | $\sum _{k=0}^{\infty}HS\left(k,n,1\right)\approx $ | $\underset{0}{\overset{\infty}{\int}}HS\left(k,n,1\right)dk\approx $ |
---|---|---|

0 | 16962.1740457 | 16962.3520362 |

1 | 2117.56926532 | 2117.48007283 |

2 | 291.022289825 | 291.104223905 |

3 | 45.9993260894 | 45.5712471365 |

4 | 8.71952109668 | 8.20993584833 |

5 | 1.87596579993 | 1.60128605246 |

6 | 0.40326040109 | 0.30739217922 |

7 | 0.07910676340 | 0.05435115208 |

8 | 0.01367865325 | 0.00860415949 |

9 | 0.00207449183 | 0.00121196056 |

10 | 0.00027764247 | 0.00015240808 |

11 | 0.00003309744 | 0.00001722662 |

12 | 0.00000354778 | 0.00000176333 |

$\vdots $ | $\vdots $ | $\vdots $ |

∞ | $\underset{n\to \infty}{\text{lim}}\sum _{k=0}^{\infty}HS\left(k,n,r\right)=0$ | $\underset{n\to \infty}{\text{lim}}\underset{0}{\overset{\infty}{\int}}HS\left(k,n,r\right)dk=0$ |

$\underset{n}{\Sigma}$ | 19427.858848843922 | - |

*n*. The dimensional fluxes can be studied as well for the complex part. So, for example, with the recurrence we get the series values for the negative degree of freedom

*n*= -2, as [30].

### 2.3. Fluxes on the base of hypersphere matrix series

*M*[

*HS*]

_{ kxn }submatrix series. For example, by expanding the series for

*k*= 3, the flux would contain the following members

*r*= 1), are given in Table 2.

Values of discrete and continual fluxes

Degree of freedom (k) | $\sum _{n=0}^{\infty}HS\left(k,n,r\right)=$ | $\sum _{n=0}^{\infty}HS\left(k,n,1\right)\approx $ | $\underset{0}{\overset{+\infty}{\int}}HS\left(k,n,1\right)dn\approx $ |
---|---|---|---|

0 |
| 2.71828182846 | 2.89982256317 |

1 | 2 | 5.43656365692 | 5.24809906025 |

2 | 2 | 17.0794684453 | 17.6417407306 |

3 | 8 | 68.3178737814 | 56.964225268 |

4 | 12 | 203.505142758 | 139.918441638 |

5 | 64 | 453.706079704 | 271.32230045 |

6 | 60 | 812.172812098 | 437.960809928 |

7 | $\sum _{n=0}^{\infty}\frac{768{\pi}^{3}{r}^{n+4}}{\Gamma \left(n+5\right)}$ | 1229.10258235 | 611.722905550 |

8 | $\sum _{n=0}^{\infty}\frac{1680{\pi}^{4}{r}^{n+5}}{\Gamma \left(n+6\right)}$ | 1628.04409715 | 759.633692941 |

9 | $\sum _{n=0}^{\infty}\frac{12288{\pi}^{4}{r}^{n+6}}{\Gamma \left(n+7\right)}$ | 1933.28876014 | 855.051695653 |

10 | $\sum _{n=0}^{\infty}\frac{30240{\pi}^{5}{r}^{n+7}}{\Gamma \left(n+8\right)}$ | 2093.93742907 | 884.975895298 |

11 | $\sum _{n=0}^{\infty}\frac{245760{\pi}^{5}{r}^{n+8}}{\Gamma \left(n+9\right)}$ | 2095.29352414 | 851.441651487 |

12 | $\sum _{n=0}^{\infty}\frac{665280{\pi}^{6}{r}^{n+9}}{\Gamma \left(n+10\right)}$ | 1956.27052708 | 768.011397877 |

13 | $\sum _{n=0}^{\infty}\frac{5898240{\pi}^{6}{r}^{n+10}}{\Gamma \left(n+11\right)}$ | 1717.51550066 | 653.938458847 |

$\vdots $ | $\vdots $ | $\vdots $ | $\vdots $ |

50 | $\sum _{n=0}^{\infty}HS\left(50,n,r\right)$ | 0.00000002078 | 0.00000000526 |

$\vdots $ | $\vdots $ | $\vdots $ | $\vdots $ |

∞ | $\underset{k\to \infty}{\text{lim}}\sum _{n=0}^{\infty}HS\left(k,n,r\right)=0$ | 0 | $\underset{k\to \infty}{\text{lim}}\underset{0}{\overset{\infty}{\int}}HS\left(k,n,r\right)dn=0$ |

$\underset{k}{\Sigma}$ | 19427.858848843922 | - |

### 2.4. Some continuous fluxes

*n*. From the standpoint of functional analysis, the most interesting series of the matrix

*M*[

*HS*]

_{ k, n }is the one that relates to the degrees of freedom

*k*= 2 and

*k*= 3. The first series includes the known functions for the circumference (2

*πr*) and the surface of circle (

*πr*

^{2}). The members of the second series are the surface functions (4

*πr*

^{2}) and sphere volume ($\frac{4}{3}\pi {r}^{3}$). The same series are interesting as well for continuous fluxes. The continuous natural flux for the hypersphere surface is analyzed on the base of integrals, instead of series. This integral is specific, because its subintegral function is the reciprocal

*gamma*function. Its value, as it is known, is equal to the value of Fransen-Robinson constant [31]

*n*, a more general dimensional volume hypersphere flux follows on the base of Ramanujan-Hardy's integral [26].

*r*= 1),

### 2.5. Vector flux series

*HS*(

*k,n,r*), (

*k, n, r*≥ 0) are respected. This twofold series has to be convergent, and this property is in the function of hypersphere radius. As usual, calculating of total discrete flux is being performed with its unit value and the convergence is in that case provided, taking into consideration that the unit series on that condition are convergent. The flux can be considered also for each column

*M*[

*HS*]

_{ k, n }of the matrix, separately. So, we have for the

*n*th column (denoted by <

*n*>), the flux in the following form

### 2.6. Orthogonal dimensional flux

These fluxes are all columns or *M*[*HS*]_{
k, n
}. As the number of columns, respectively, series, is infinite, we introduce the following definition for the total flux.

**.**

*The dimensional flux of the functional matrix with two degrees of freedom k and n is defined as a double series*

*r*= 1 dimensional fluxes have unambiguous numerical value

### 2.7. The application of the recurring operators at defining diagonal dimension fluxes

*HS*matrix was performed on the basis of addition of the

*HS*function values on the columns, in regard to the series of the

*HS*matrix. The more detailed analysis would be very large scale, including the exponential function, error functions

*erf*(

*z*),

*erf*(

*z*), the incomplete gamma function Γ(

*a, z*), etc. When we use the idea of the transition operators from the reference function into the defining

*HS*function in the functional hyperspherical matrix, we can also establish the values of the dimensional fluxes on the diagonals (Figure 3), whose sum would present the overall flux for the matrix where the degree of freedom is in the domain of natural numbers, i.e.,

*k, n*∈

*N*. Such matrix contains an infinite number of elements. For the reference functions, we take

*HS*functions on the positions of the first series of the matrix, and they are the so-called zero

*HS*functions:

*HS*(0,0,

*r*),

*HS*(0,1,

*r*),...,

*HS*(0,

*n,r*),... The defining functions are placed according to the "gradual" law of growth (+Δ

*k*) and decline (-Δ

*n*).

*ϑ*(Δ

*k*, Δ

*n*, 0) is defined by the quotient [4]

*k*| = |-Δ

*n*| = 1, a new joint argument

*u*(Δ

*k =*Δ

*n = u*) is assigned to them. In addition to the starting value of the

*k*th degree of freedom is

*k*= 0, the operator theta becomes (2.4)

*n*th diagonal would be calculated in the form of a sum

*n*= 2 is calculated on the basis of the function limit value. Respecting that

*n*th diagonal, after reordering, can get a new form, and it is equivalent to the expression (2.7)

*n*= 4) we get

*n*= 5) it follows that

*r*= 1 and

*r*= 12 we obtain

*v*is a summing index by which the sequence of the matrix elements from left to right and from above to down along the diagonal is taken into consideration. The polynomial coefficients contain rational numbers and the graded constant

*π*. The first three coefficients are zero, so they are not included in the summation sequence. Its other values (

*v*= 0, 1,...,13) are given in Table 3

The polinomial coefficients

v | a |
---|---|

0 | 3 + 10 |

1 | 3 + 10 |

2 | $\frac{3}{2}+5\pi +38{\pi}^{2}$ |

3 | $\frac{1}{2}+\frac{5\pi}{3}+\frac{38{\pi}^{2}}{3}+20{\pi}^{3}$ |

4 | $\frac{1}{8}+\frac{5\pi}{12}+\frac{19{\pi}^{2}}{6}+37{\pi}^{3}$ |

5 | $\frac{1}{40}+\frac{\pi}{12}+\frac{19{\pi}^{2}}{30}+\frac{37{\pi}^{3}}{5}+14{\pi}^{4}$ |

6 | $\frac{1}{240}+\frac{\pi}{72}+\frac{19{\pi}^{2}}{180}+\frac{37{\pi}^{3}}{30}+\frac{97{\pi}^{4}}{5}$ |

7 | $\frac{1}{1680}+\frac{\pi}{504}+\frac{19{\pi}^{2}}{1260}+\frac{37{\pi}^{3}}{210}+\frac{97{\pi}^{4}}{35}+6{\pi}^{5}$ |

8 | $\frac{1}{13440}+\frac{\pi}{4032}+\frac{19{\pi}^{2}}{10080}+\frac{37{\pi}^{3}}{1680}+\frac{97{\pi}^{4}}{280}+\frac{575{\pi}^{5}}{84}$ |

9 | $\frac{1}{120960}+\frac{\pi}{36288}+\frac{19{\pi}^{2}}{90720}+\frac{37{\pi}^{3}}{15120}+\frac{97{\pi}^{4}}{2520}+\frac{575{\pi}^{5}}{756}+\frac{11{\pi}^{6}}{6}$ |

10 | $\frac{1}{1209600}+\frac{\pi}{362880}+\frac{19{\pi}^{2}}{907200}+\frac{37{\pi}^{3}}{151200}+\frac{97{\pi}^{4}}{25200}+\frac{115{\pi}^{5}}{1512}+\frac{2279{\pi}^{6}}{1260}$ |

11 | $\frac{1}{\text{13305600}}+\frac{\pi}{\text{3991680}}+\frac{19{\pi}^{2}}{\text{9979200}}+\frac{\text{37}{\pi}^{\text{3}}}{\text{1663200}}+\frac{\text{97}{\pi}^{\text{4}}}{\text{277200}}+\frac{\text{1}15{\pi}^{5}}{\text{16632}}+\frac{2279{\pi}^{6}}{13860}+\frac{13{\pi}^{7}}{30}$ |

12 | $\frac{1}{\text{159667200}}+\frac{\pi}{\text{47900160}}+\frac{19{\pi}^{2}}{\text{119750400}}+\frac{\text{37}{\pi}^{\text{3}}}{\text{19958400}}+\frac{\text{97}{\pi}^{\text{4}}}{\text{3326400}}+\frac{\text{1}15{\pi}^{5}}{\text{199584}}+\frac{2279{\pi}^{6}}{166320}+\frac{905{\pi}^{7}}{2376}$ |

*n*= 30 the double series of the diagonal flux gets the following structure:

*π*

^{ n }constant in the degrees of the series members, in contrast to vertical fluxes with the domination of function errors, where

*π*and

*e*are constants. The horizontal fluxes, as it was presented in (2.6), contain exponential functions. In the meantime, the total flux for the unique radius is convergent and can be calculated with considerably greater value

while, e.g., for *r* = 2 the flux value is substantially greater and its value is obtained as Π_{
HS
}≈ 1375905492.377.

## 3. Conclusion

*HS*(

*k,n,r*) is the function of three variables, its dependence is certainly also a variable

*k*, respectively,

*n*. In this article, we calculated several continual fluxes (2.4) and (2.5), for contour hyperspherical function, on the basis of Ramanujan-Hardy's integral. The dimensional flux calculating with diagonal algorithm is much simpler and faster to perform on a computer, because the total flux is now defined as convergent-graded series and it does not contain special functions as components. In any case its value is identical with the fluxes that are calculated on the base of series, i.e., the

*HS*matrix columns, so there is a numerically verified statement that

*k*= 3. In that case, we obtain the solution [20].

*k, n*∈

*N*) of the unit hyperspherical function

*HS*(

*k,n*,1) is equal to the value of twofold integral

and its solution can be looked for on numerical bases.

## Declarations

## Authors’ Affiliations

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