# The Hyperspherical Functions of a Derivative

Authors: Nenad Cakic; Duško Letic; Branko Davidovic
Journal: Abstract and Applied Analysis
 (2010)

---

## Abstract

We present the results of theoretical research of the generalized hypersherical function (HS) by generalizing two known functions related to the sphere hypersurface and hypervolume and the recurrent relation between them. By introducing two-dimensional degrees of freedomk and n  (and the third, radius r), we develop the derivative functions for all three arguments and the possibilities of their use. The symbolical evolution, numerical experiment, and graphical presentation of functions are realized using the Mathcad Professional and Mathematica softwares.


---

## Body

## 1. Introduction

The  hyperspherical function (HS) is a hypothetical function related to multidimensional space and generalization of the sphere geometry. This function is primarily formed on the basis of the interpolating power of the gamma function. It belongs to the group of special functions, so its testing, besides gamma, is performed on the basis of the related functions, such asΓ-gamma, ψ-psi, β-beta, erf, and so forth. Its most significant value  is in its generalizing from discrete to continuous [1]. In addition, we move from the field of natural integers of the geometry sphere dimensions—degrees of freedom, to the set of real and nonintegral values, thus, we obtain the prerequisites for a more concise analysis of this function. In this paper the analysis is focused on the infinitesimal calculus application of the hyperspherical function that is given in its generalized form. For the development of hyperspherical and other functions of the multidimensional objects, see: Bishop and Whitlock [2], Collins [3], Conway [4], Dodd and Coll [5], Hinton [6], Hocking and Young [1], Manning [7], Maunder [8], Neville [9], Rohrmann and Santos [10], Rucker [11], Maeda et al. [12], Sloane [13], Sommerville [14], Weels [15], and others; see [16–22]. Nowadays, the research of the hypersperical functions is represented both in Euclid’s and Riemann’s geometry and topology (Riemann’s and Poincare’s sphere, multidimensional potentials, theory of fluids, nuclear (atomic) physics, hyperspherical black holes, etc.)


## 2. The Hyperspherical Functions of a Derivative

### 2.1. The Hyperspherical Funcional Matrix

The former results, as it is known, [5, 23], give the functions of the hyperspherical surface (n=2) and volume (n=3) therefore, we have(2.1)HS(k,2,r)=2πkrk-1Γ(k/2)=∂∂rHS(k,3,r),HS(k,3,r)=∫0rHS(k,2,r)dr=πkrkΓ(k/2+1).
In general, we have
(2.2)HS(k,n,r)=∫0rHS(k,n-1,r)dr=∬0r⋯∫0rS(k,r)drdr⋯drn-2=∬0r⋯∫0r2πk/2rk-1Γ(k/2)drdr⋯drn-2.
Thus, we give the definition of the hyperspherical function [24].Definition 2.1.
The hyperspherical function with two degrees of freedomk and n is defined as
(2.3)HS(k,n,r)=2πkrk+n-3Γ(k)Γ(k+n-2)Γ(k/2)(k,n∈R).
This is a function of three variables and two degrees of freedom k and n and the radius of the hypersphere r. In real spherical entities (Figure 1) such as, diameter, circumference or cycle surface; followed by sphere surface and volume, only the variable r exists.Figure 1
Moving through the vector of real surfaces left: by deducting one degree of freedomk of the surface sphere, we obtain circumference, and for two (degrees), we obtain the point 2. Moving through the vector of real solids (right column), that is, by deduction of one degree of freedom k from the solid sphere, we obtain a circle (disk), and for two (degrees), we get a line segment or diameter.By keeping the property that the derivatives on the radiusr generate new functions (the HS matrix columns), we perform “movements” to lower or higher degrees of freedom, from the starting nth, on the basis of the following recurrent relations:
(2.4)∂∂rHS(k,n,r)=HS(k,n-1,r),HS(k,n+1,r)=∫0rHS(k,n,r)dr.
This property is fundamental and hypothetically also holds also for objects out of this submatrix of six elements (Figures 1 and 2). The derivation example (right) shows that, we have obtained the zeroth (n=0) degree of freedom, if we perform the derivation of the nth degree. With the defined value of the derivative, the HS function is also valid in the complex domain of the hyperspherical matrix. Namely, the following expressions hold:
(2.5)∂n∂rnHS(k,n,r)=HS(k,0,r)or∂2n∂r2nHS(k,n,r)=HS(k,-n,r).
On the basis of the general hyperspherical function, we obtain the appropriate matrix Mk×n (k,n∈ℜ), where we also give the concrete values for the selected submatrix 9×9.Figure 2
The submatrixHS(k,n,r) of the function for k∈-3,-2,…,5 and n∈-2,-1,…,6 with six highlighted characteristic functions (undef. are undefined values, predominately singular ±∞ of this function value).For example ifn=2 and k∈N, we obtain the following relation (for k=0,5¯):
(2.6)∂4∂r4HS(k,2,r)=HS(k,-2,r),respectively,∂4∂r4[022πr4πr22π2r38π2r43]=[0000064π2].
The matrix [M]k×n has the property that every vector of the ith column is a derivative with respect to the radius of the next vector in the sequence according to Figure 2. This recursion property originates from the starting assumptions (2.1). Therefore, we have the following theorem.Theorem 2.2.
For the columns of the matrix[M]k×n, the following equality holds:
(2.7)[M]〈n-1〉=∂∂r[M]〈n〉.For example, for two adjacent columns of the matrix[M]〈n-1〉 and [M]〈n〉, one obtains the recurrent vectors(2.8)[M]〈n-1〉=∂∂r[rn-3Γ(n-2)2rn-2Γ(n-1)2πrn-1Γ(n)8πrnΓ(n+1)⋮2πkrk+n-3Γ(k)Γ(k+n-2)Γ(k/2)]=[rn-4Γ(n-3)2rn-3Γ(n-2)2πrn-2Γ(n-1)8πrn-1Γ(n)⋮2πkrk+n-4Γ(k)Γ(k+n-1)Γ(k/2)](n∈R).
One obtains interesting results on the basis of horizontal (n) or vertical (k) degrees of freedom. Therefore, one obtains the following:
(2.9)2πkrkkΓ(k/2)|k=3∨8πrnΓ(n+1)|n=3∨2πkrk+n-3Γ(k)Γ(k+n-2)Γ(k/2)|k=3∧n=3⟹43πr3.
Consequently, this property is fundamental, because it leads to the same result on the basis of two special formulas, or using only one general.


### 2.2. The Analysis of the Recurrent Potential Function of the Typezυ

The hyperspherical function, besides the gamma functions also contains the potential componentrk+n-3. The generalized equation of the hth derivative of the graded expression zυ  is [25]
(2.10)∂hzυ∂zh=Γ(υ+1)Γ(υ-h+1)zυ-h(-υ∉N).
Namely, it is known that the exponent with the basis of the HS function radius is υ=k+n-3. Now the mth derivative of the hyperspherical function with respect to the radius is defined by the relation
(2.11)∂m∂rmHS(k,n,r)=2πkΓ(k)Γ(k+n-2)Γ(k/2)⋅∂mrk+n-3∂rm.
Therefore, using (2.10), the partial derivative in that case is
(2.12)∂mrk+n-3∂rm=rk-m+n-3Γ(k+n-2)Γ(k-m+n-2).
After some transforming, we obtain the form
(2.13)∂m∂rmHS(k,n,m,r)=2rk-m+n-3πkΓ(k)Γ(k-m+n-2)Γ(k/2).
Using the known property Γ(z+1)=zΓ(z), we have the following theorem.Theorem 2.3.
Themth derivative of the hyperspherical function with respect to the radius r is
(2.14)∂m∂rmHS(k,n,m,r)=rk-m+n-3πkΓ(k+1)Γ(k-m+n-2)Γ(k/2+1).The above equation (2.14) is recurrent by nature, and with it, we obtain every derivative (for +m) and integral (for -m), depending on what position (n) we want to perform these operations. That way we can define and, in appropriate cases, use a unique operator that unites the operations of differentiating and integrating with respect to the radius of the HS function.Definition 2.4.
The unique operator that unites the operations of differentiating and integrating with respect to the radius of the hyperspherical function is given by:(2.15)∂±m∂r±mHS≡D±mHS,
where ∂/∂r≡D.
Symbolically, the integrating (more exactly, the operator∂-m/∂r-m) can be written in the form(2.16)∂-m∂r-mHS(k,n,m,r)=∬0r⋯∫0rHS(k,n,r)drdr⋯dr︸m=2rk+m+n-3πkΓ(k)Γ(k+m+n-2)Γ(k/2).
Because of the known characteristics of the gamma function, the value of the differential or integral degree m, does not have to be an integer, as, for example, with classical differentiating (integrating).


### 2.3. The Derivatives of the Dichotomic HS Functions

We now consider the case in which the hyperspherical function, with the added variablem, can be separated on even and odd members. We form the derivatives of the function with odd members by substituting the Legendre’s equivalent in the function HS(k,n,m,r). Consequently, for these members, we consider the following transformation:
(2.17)2πkrk-m+n-3Γ(k)Γ(k-m+n-2)Γ(k/2)→Γ(k/2)=21-kπ(k-1)!/((k-1)/2)!2kπ(k-1)/2rk-m+n-3(k-m+n-3)!(k-12)!.
From this, we obtain the function of the mth derivative for odd members (index 1).Theorem 2.5.
Themth derivative for odd members of the hyperspherical function with respect to r is
(2.18)∂m∂rmHS1(k,n,m,r)=2kπ(k-1)/2rk-m+n-3(k-m+n-3)!(k-12)!.The second, complementary function, with even members (index 2), is obtained on the basis of the second transformation(2.19)2πkrk-m+n-3Γ(k+1)Γ(k-m+n-2)Γ(k/2+1)→Γ(k/2+1)=(k/2)Γ(k/2)2πkrk-m+n-3(k-1)!(k-m+n-3)!(k/2-1)!.
Thus, we form the second, dichotomous function.Theorem 2.6.
Themth derivative for even members of the hyperspherical function with respect to r is
(2.20)∂m∂rmHS2(k,n,m,r)=2πkrk-m+n-3(k-1)!(k-m+n-3)!(k/2-1)!.Taking into consideration that the dichotomous functions as well as their adequate functional series are complementary, by differentiating the hyperspherical function the same characteristic is retained, but for all that we must perform a substitution of variables, for odd members,k=2b+1, and for even ones k=2b. So, the values of the complementary members, are, respectively.Corollary 2.7.
One has(2.21)∂m∂rmHS1(k,n,m,r)=∂m∂rmHS(2b+1,n,m,r)=22b+1πbr2b-m+n-2b!(2b-m+n-2)!,∂m∂rmHS2(k,n,m,r)=∂m∂rmHS(2b,n,m,r)=πbr2b-m+n-3(2b)!(2b-m+n-3)!b!.Examples 1.
(i)mis an integer: The derivative degree m can be integer, noninteger or consequently, from the field of real numbers. So, for example, we give its following representative values, which give the same result. In this case, we obtain the second derivative of the HS function related to the degree of freedom, n=5, as
(2.22)∂2∂r2HS1(k,5,2,r)=πb(2r)2b+1b!(2b+1)!,∂2∂r2HS2(k,5,2,r)=πbr2bb!,
and, in another case, by integrating the HS function, when n=2 (surfhypersphere)
(2.23)∂-1∂r-1HS1(k,2,-1,r)=πb(2r)2b+1b!(2b+1)!,∂-1∂r-1HS2(k,2,-1,r)=πbr2bb!.
(ii)m  is noninteger (fractional): With the use of noninteger (fractional) degrees of the derivative, for example, m=±1/2, starting with, for example, fixed degrees of freedom n=7/2 and n=5/2, we obtain the same results as with the procedure of integer differentiating/integrating (2.22) and (2.23). We obtain, respectively(2.24)∂1/2∂r1/2HS1(k,72,12,r)=πb(2r)2b+1b!(2b+1)!,∂1/2∂r1/2HS2(k,72,12,r)=πbr2bb!,
and with noninteger integrating (m<0)
(2.25)∂-1/2∂r-1/2HS1(k,52,-12,r)=πb(2r)2b+1b!(2b+1)!,∂-1/2∂r-1/2HS2(k,52,-12,r)=πbr2bb!.
The values (pairs) of the expressions (2.22), (2.23), (2.24), and (2.25) match the dichotomous values of the hyperspherical function for n=3 therefore, we have
(2.26)HS1(2b+1,3,r)=πb(2r)2b+1b!(2b+1)!,HS2(2b,3,r)=πbr2bb!.
By integrating the above complementary functions, we obtain the hyperspherical functions series, that describes the solid geometrical entities (n=3∧k∈N) in the form
(2.27)∑k=0∞HS(k,3,r)=∑k=0∞πkrkΓ(k/2+1)=∑b=0∞πb(2r)2b+1b!(2b+1)!+∑b=0∞πbr2bb!=eπr2erf⁡(rπ)+eπr2=eπr2[1+erf⁡(rπ)]=eπr2erfc⁡(rπ),
which is Mittag-Leffler’s [26], that is, Freden’s solution (1993) for the solid hyperspherical function series [27]. In the previous expression complementary error functions originated from the expression erf(z)+erfc(z)=1. By generalizing (2.4), we obtain the following differential equation:
(2.28)∂n∂rn∑k=0∞HS(k,n,r)-∑k=0∞HS(k,0,r)=0.
which describes the relation between the columns of the hyperspherical matrix for n∈N.


### 2.4. The Gradient of the Hyperspherical Function

Taking into consideration its differentiability and multidimensionality,  we can apply the gradient on the hyperspherical function. As this function has three variablesk, n, and r, we obtain the solution of gradient functions ∇k,n,r in the following form:(2.29)∇k,n,r{HS(k,n,r)}=[∂∂kHS(k,n,r)∂∂nHS(k,n,r)∂∂rHS(k,n,r)]=HS(k,n,r)[ln⁡2rπ+12ψ0(k+12)-ψ0(k+n-2)ln⁡r-ψ0(k+n-2)1r(k+n-3)],
where  ψ0(z)=(d/dz)lnΓ(z) is digamma function.This function has a special use in determining the extreme values of the contour HS functions.


### 2.5. The Contour Graphics of the Derivatives of the Hyperspherical Functions

The derivative functionshs1=∂HS/∂k=∇k{HS} and sh1=∂HS/∂n=∇n{HS}, also belong to the family of the hyperspherical functions. According to the degrees of freedom k and n, we can split them into two groups of functions.Definition 2.8.
The function of the first derivative with respect to the degree of freedomk is
(2.30)hs1(k,n,r)=∂∂kHS(k,n,r)=HS(k,n,r)[ln⁡2rπ+12ψ0(k+12)-ψ0(k+n-2)].
The function hs1(k,n,r) has a special use in determining the maximum of the hyperspherical functions, for example, for a unit radius and the domain k∈ℜ (Figure 3). Here, on the basis of the known criteria, for each member of the derivative functions family we determine the maximum of the HS function by equaling its derivative with zero. Other than the maximum, there is the “optimal” value k0. Consequently
(2.31)hs1(k,n,r)=0,∂2∂k2HS(k,n,r)>0⟹max⁡HS(k,n,r)∧k0.Figure 3
The parallel presentation of the orginal hyperspherical functions (forn=2 and n=3) of the unit radius, and corresponding functions of their first derivatives for -25≤k≤20.Example 2.9.
For  the volume hyperspherical function with the radiusr=1, the derivative function is after transforming [24]
(2.32)hs1(k,3,1)=HS(k,3,1)2[ln⁡π-ψ0(k2+1)]=πkkΓ(k/2)(ln⁡π+γ-Hk/2),
where Hk/2 is a harmonious number. This number is obtained on the basis of the sum [28]
(2.33)Hk=∑j=1k1jorHk/2=∑j=1k/21j=1+12+⋯+k2.
The relation between the harmonious number and psi (digamma function) is the following:
(2.34)Hk=γ+ψ0(k+1),respectively,Hk/2=γ+ψ0(k2+1).
The graphical representation of the original HS(k,3,1) and its derivative are given in Figure 4.Figure 4
The contour function of the hyperspherical solidHS(3,n,1) with the maximum value maxHS and optimal degree of freedom k0 (for -2≤k≤18).


### 2.6.The Extreme Values of the HS Function from the Degree of Freedom  k Point of View

On the basis of the above expressions (2.31) and (2.32), we define the greatest volume for the “optimal” dimension k0. We find that k0≈5.2569, and the corresponding volume is maxHS(k,3,1)=HS(k0,3,1)≈5.2777. For other hyperspherical functions, we can also determine the maximum values on the numerical bases (Table 1). However, for the degrees of freedom n≥5, the maximum is a function of its factorial.Table 1DimensionnDimensionk0maxHSErrorε010.700421168630701871.481745773695~019.377568346255670237.5057633480618−1.055·10−1327.25694640486057233.161194484961957.363·10−1535.2569464048605725.2777680211134113.516·10−1542.5065379841902551.054723774914521~0501/20⋮⋮⋮⋮n01/(n-3)!0So, we have(2.35)lim⁡k→0HS(k,5,1)=12,lim⁡k→0HS(k,6,1)=16,lim⁡k→0HS(k,7,1)=124,
and in general
(2.36)lim⁡k→0HS(k,n,1)=1Γ(n-2)=1(n-3)!,lim⁡k→0+HS(k,n,r)=rn-3(n-3)!.
On the following surface 3D graphic HS(k,n,1) (Figure 5) we notice the extreme, that is, the maximum value of this function with the 0th freedom degree n=0, which is a global maximum in the domain k,n∈N as well.Figure 5
The graphic of the hyperspherical function with the constant radiusr=1 and for the degrees of freedom 0<n≤4 and 0<k≤25.


### 2.7. The Extreme Values of the HS Function from the Degree of Freedomn Point of View

Definition 2.10.
The function of the first derivative with respect to the degree of freedomn, is
(2.37)sh1(k,n,r)=∂∂nHS(k,n,r)=HS(k,n,r)[ln⁡r-ψ0(k+n-2)].
The function sh1(k,n,r) is in fact the general solution of the HS function of the degree of freedom n.On the basis of the known differentiating procedure, besides the local maximum of the contour HS functions, we obtain the corresponding optimal valuesn0 as well. Consequently
(2.38)sh1(k,n,r)=0,∂2∂n2HS(k,n,r)>0⟹max⁡HS(k,n,r)=HS(k,n0,r)∧n0.
These functions can have a significant role with further research of hyperspherical functions. By calculating the extreme and optimal values of the HS function, we obtain Table 2.Table 2DimensionkDimensionn*maxHSErrorε03.4616321449683621.1291738854501331.08·10−1612.4616321449683622.2583477709002581.086·10−1521.4616321449683627.094808766311162~030.46163214496836228.379235065244632.665·10−154012π205032π20⋮⋮⋮⋮1006π5≈1836.18109011012821π5≈1865.262840120116π6≈1762.546850⋮⋮⋮⋮∞0limk→∞HS(k,0,1)=00We can conclude that the following optimal degrees of freedomn0 and k0(2.39)n0=0,6π5<k0<12821π5
generate a global value maximum of the hyperspherical function, taking only the domain of the natural numbers (k,n∈N).As in the previous analysis, we determine the value ofk0, the optimal vector is
(2.40)[k0n0]=[10.70042116863070]⟹max⁡HS(k,n,1)=HS(k0,n0,1)≈1871.481745773695.
The graphical representation of the original HS(3,n,1) and its derivate are given in Figure 6.Figure 6
The contour function of the hypersolidHS(3,n,1) with the local maximum maxHS≈28.379 and the optimal degree of freedom n0≈0.4616, for k∈N.


### 2.8. The Derivatives of the Hyperspherical Functions of the Higher Order with Respect to the Argumentk

The derivatives of the hyperspherical functions with respect to any argument retain the characteristics similar the original HS function. This is especially noticeable in the graphical representation. In addition, it would be interesting to test the regularities in the structure of this function of the higher derivation degree. In the following, we give, analytically and graphically, the derivative functions on argumentk (the more complex case) and n (the simpler case).The second derivative of the hyperspherical function with respect to argumenk is:
(2.41)hs2(k,n,r)=∂2∂k2HS(k,n,r)=HS(k,n,r){ψ1(k)-ψ1(k+n-2)+[ψ0(k+n-2)-12ψ0(k+12)-ln⁡2rπ]2-14ψ1(k2)}.
Here: ψ0(z)=(d/dz)lnΓ(z) digamma and ψ1(z)=(d2/dz2)lnΓ(z) trigamma function. The concrete value of this derivative for the selected parameters is the following:
(2.42)∂2∂k2HS(3,3,1)=2π9[(3γ-8)ln⁡π+γ(3γ2-8)+6(γ+ln⁡2π-83)ln⁡2+34(2ln2π-π2)+523].
The mixed derivative of the hyperspherical function is
(2.43)∂2∂k∂nHS(k,n,r)=HS(k,n,r){ψ0(k+n-2)[ψ0(k+n-2)-12ψ0(k+12)-ln⁡2r2π]+ln⁡r[ln⁡2rπ+12ψ0(k+12)]-ψ1(k+n-2)}.Example 2.11.
After introducing some values ofk and n it follows that
(2.44)∂2∂k∂nHS(3,3,1)=π9[(ln⁡4π+γ)(6γ+11)-2(π2+8γ)+1373].
The fifth derivative of the hyperspheric function on argument n and for its unique radius is
(2.45)∂5∂n5HS(k,n,r)=-HS(k,n,r){(ψ0(k+n-2)-ln⁡r)5-10(ψ0(k+n-2)-ln⁡r)3ψ1(k+n-2)+10(ψ0(k+n-2)-ln⁡r)2ψ2(k+n-2)+15(ψ0(k+n-2)-ln⁡r)2×ψ12(k+n-2)-5ψ3(k+n-2)(ψ0(k+n-2)-ln⁡r)-10ψ1(k+n-2)ψ2(υ+1)+ψ4(k+n-2)(ψ0(k+n-2)-ln⁡r)5}.
Namely, the fifth derivative values of this function for degrees of freedom k=0, n=0 are
(2.46)∂5∂k5HS(0,0,1)=sh5(0,0,1)=40(2γ-3)ζ(3)+10γ[π2(3-γ)+6γ(1-γ(1-γ6))]-π2(10-ζ(2)).
Generally, the hyperspherical function of the mth derivative on the argument n can be defined on the basis of the product for poligamma (psi) of the polynominal of the mth degree and the hyperspherical function. Its general form is the following:
(2.47)shm(k,n,r)=∂m∂nmHS(k,n,r)=HS(k,n,r)⋅fm{ψ(m-1)(k+n-2),r}.
So, for example, the functions of the fifth derivative are in Figure 7.Figure 7
The graphics of the fifth derivative of the hyperspherical functions with respect to the argumentn.


## 2.1. The Hyperspherical Funcional Matrix

The former results, as it is known, [5, 23], give the functions of the hyperspherical surface (n=2) and volume (n=3) therefore, we have(2.1)HS(k,2,r)=2πkrk-1Γ(k/2)=∂∂rHS(k,3,r),HS(k,3,r)=∫0rHS(k,2,r)dr=πkrkΓ(k/2+1).
In general, we have
(2.2)HS(k,n,r)=∫0rHS(k,n-1,r)dr=∬0r⋯∫0rS(k,r)drdr⋯drn-2=∬0r⋯∫0r2πk/2rk-1Γ(k/2)drdr⋯drn-2.
Thus, we give the definition of the hyperspherical function [24].Definition 2.1.
The hyperspherical function with two degrees of freedomk and n is defined as
(2.3)HS(k,n,r)=2πkrk+n-3Γ(k)Γ(k+n-2)Γ(k/2)(k,n∈R).
This is a function of three variables and two degrees of freedom k and n and the radius of the hypersphere r. In real spherical entities (Figure 1) such as, diameter, circumference or cycle surface; followed by sphere surface and volume, only the variable r exists.Figure 1
Moving through the vector of real surfaces left: by deducting one degree of freedomk of the surface sphere, we obtain circumference, and for two (degrees), we obtain the point 2. Moving through the vector of real solids (right column), that is, by deduction of one degree of freedom k from the solid sphere, we obtain a circle (disk), and for two (degrees), we get a line segment or diameter.By keeping the property that the derivatives on the radiusr generate new functions (the HS matrix columns), we perform “movements” to lower or higher degrees of freedom, from the starting nth, on the basis of the following recurrent relations:
(2.4)∂∂rHS(k,n,r)=HS(k,n-1,r),HS(k,n+1,r)=∫0rHS(k,n,r)dr.
This property is fundamental and hypothetically also holds also for objects out of this submatrix of six elements (Figures 1 and 2). The derivation example (right) shows that, we have obtained the zeroth (n=0) degree of freedom, if we perform the derivation of the nth degree. With the defined value of the derivative, the HS function is also valid in the complex domain of the hyperspherical matrix. Namely, the following expressions hold:
(2.5)∂n∂rnHS(k,n,r)=HS(k,0,r)or∂2n∂r2nHS(k,n,r)=HS(k,-n,r).
On the basis of the general hyperspherical function, we obtain the appropriate matrix Mk×n (k,n∈ℜ), where we also give the concrete values for the selected submatrix 9×9.Figure 2
The submatrixHS(k,n,r) of the function for k∈-3,-2,…,5 and n∈-2,-1,…,6 with six highlighted characteristic functions (undef. are undefined values, predominately singular ±∞ of this function value).For example ifn=2 and k∈N, we obtain the following relation (for k=0,5¯):
(2.6)∂4∂r4HS(k,2,r)=HS(k,-2,r),respectively,∂4∂r4[022πr4πr22π2r38π2r43]=[0000064π2].
The matrix [M]k×n has the property that every vector of the ith column is a derivative with respect to the radius of the next vector in the sequence according to Figure 2. This recursion property originates from the starting assumptions (2.1). Therefore, we have the following theorem.Theorem 2.2.
For the columns of the matrix[M]k×n, the following equality holds:
(2.7)[M]〈n-1〉=∂∂r[M]〈n〉.For example, for two adjacent columns of the matrix[M]〈n-1〉 and [M]〈n〉, one obtains the recurrent vectors(2.8)[M]〈n-1〉=∂∂r[rn-3Γ(n-2)2rn-2Γ(n-1)2πrn-1Γ(n)8πrnΓ(n+1)⋮2πkrk+n-3Γ(k)Γ(k+n-2)Γ(k/2)]=[rn-4Γ(n-3)2rn-3Γ(n-2)2πrn-2Γ(n-1)8πrn-1Γ(n)⋮2πkrk+n-4Γ(k)Γ(k+n-1)Γ(k/2)](n∈R).
One obtains interesting results on the basis of horizontal (n) or vertical (k) degrees of freedom. Therefore, one obtains the following:
(2.9)2πkrkkΓ(k/2)|k=3∨8πrnΓ(n+1)|n=3∨2πkrk+n-3Γ(k)Γ(k+n-2)Γ(k/2)|k=3∧n=3⟹43πr3.
Consequently, this property is fundamental, because it leads to the same result on the basis of two special formulas, or using only one general.


## 2.2. The Analysis of the Recurrent Potential Function of the Typezυ

The hyperspherical function, besides the gamma functions also contains the potential componentrk+n-3. The generalized equation of the hth derivative of the graded expression zυ  is [25]
(2.10)∂hzυ∂zh=Γ(υ+1)Γ(υ-h+1)zυ-h(-υ∉N).
Namely, it is known that the exponent with the basis of the HS function radius is υ=k+n-3. Now the mth derivative of the hyperspherical function with respect to the radius is defined by the relation
(2.11)∂m∂rmHS(k,n,r)=2πkΓ(k)Γ(k+n-2)Γ(k/2)⋅∂mrk+n-3∂rm.
Therefore, using (2.10), the partial derivative in that case is
(2.12)∂mrk+n-3∂rm=rk-m+n-3Γ(k+n-2)Γ(k-m+n-2).
After some transforming, we obtain the form
(2.13)∂m∂rmHS(k,n,m,r)=2rk-m+n-3πkΓ(k)Γ(k-m+n-2)Γ(k/2).
Using the known property Γ(z+1)=zΓ(z), we have the following theorem.Theorem 2.3.
Themth derivative of the hyperspherical function with respect to the radius r is
(2.14)∂m∂rmHS(k,n,m,r)=rk-m+n-3πkΓ(k+1)Γ(k-m+n-2)Γ(k/2+1).The above equation (2.14) is recurrent by nature, and with it, we obtain every derivative (for +m) and integral (for -m), depending on what position (n) we want to perform these operations. That way we can define and, in appropriate cases, use a unique operator that unites the operations of differentiating and integrating with respect to the radius of the HS function.Definition 2.4.
The unique operator that unites the operations of differentiating and integrating with respect to the radius of the hyperspherical function is given by:(2.15)∂±m∂r±mHS≡D±mHS,
where ∂/∂r≡D.
Symbolically, the integrating (more exactly, the operator∂-m/∂r-m) can be written in the form(2.16)∂-m∂r-mHS(k,n,m,r)=∬0r⋯∫0rHS(k,n,r)drdr⋯dr︸m=2rk+m+n-3πkΓ(k)Γ(k+m+n-2)Γ(k/2).
Because of the known characteristics of the gamma function, the value of the differential or integral degree m, does not have to be an integer, as, for example, with classical differentiating (integrating).


## 2.3. The Derivatives of the Dichotomic HS Functions

We now consider the case in which the hyperspherical function, with the added variablem, can be separated on even and odd members. We form the derivatives of the function with odd members by substituting the Legendre’s equivalent in the function HS(k,n,m,r). Consequently, for these members, we consider the following transformation:
(2.17)2πkrk-m+n-3Γ(k)Γ(k-m+n-2)Γ(k/2)→Γ(k/2)=21-kπ(k-1)!/((k-1)/2)!2kπ(k-1)/2rk-m+n-3(k-m+n-3)!(k-12)!.
From this, we obtain the function of the mth derivative for odd members (index 1).Theorem 2.5.
Themth derivative for odd members of the hyperspherical function with respect to r is
(2.18)∂m∂rmHS1(k,n,m,r)=2kπ(k-1)/2rk-m+n-3(k-m+n-3)!(k-12)!.The second, complementary function, with even members (index 2), is obtained on the basis of the second transformation(2.19)2πkrk-m+n-3Γ(k+1)Γ(k-m+n-2)Γ(k/2+1)→Γ(k/2+1)=(k/2)Γ(k/2)2πkrk-m+n-3(k-1)!(k-m+n-3)!(k/2-1)!.
Thus, we form the second, dichotomous function.Theorem 2.6.
Themth derivative for even members of the hyperspherical function with respect to r is
(2.20)∂m∂rmHS2(k,n,m,r)=2πkrk-m+n-3(k-1)!(k-m+n-3)!(k/2-1)!.Taking into consideration that the dichotomous functions as well as their adequate functional series are complementary, by differentiating the hyperspherical function the same characteristic is retained, but for all that we must perform a substitution of variables, for odd members,k=2b+1, and for even ones k=2b. So, the values of the complementary members, are, respectively.Corollary 2.7.
One has(2.21)∂m∂rmHS1(k,n,m,r)=∂m∂rmHS(2b+1,n,m,r)=22b+1πbr2b-m+n-2b!(2b-m+n-2)!,∂m∂rmHS2(k,n,m,r)=∂m∂rmHS(2b,n,m,r)=πbr2b-m+n-3(2b)!(2b-m+n-3)!b!.Examples 1.
(i)mis an integer: The derivative degree m can be integer, noninteger or consequently, from the field of real numbers. So, for example, we give its following representative values, which give the same result. In this case, we obtain the second derivative of the HS function related to the degree of freedom, n=5, as
(2.22)∂2∂r2HS1(k,5,2,r)=πb(2r)2b+1b!(2b+1)!,∂2∂r2HS2(k,5,2,r)=πbr2bb!,
and, in another case, by integrating the HS function, when n=2 (surfhypersphere)
(2.23)∂-1∂r-1HS1(k,2,-1,r)=πb(2r)2b+1b!(2b+1)!,∂-1∂r-1HS2(k,2,-1,r)=πbr2bb!.
(ii)m  is noninteger (fractional): With the use of noninteger (fractional) degrees of the derivative, for example, m=±1/2, starting with, for example, fixed degrees of freedom n=7/2 and n=5/2, we obtain the same results as with the procedure of integer differentiating/integrating (2.22) and (2.23). We obtain, respectively(2.24)∂1/2∂r1/2HS1(k,72,12,r)=πb(2r)2b+1b!(2b+1)!,∂1/2∂r1/2HS2(k,72,12,r)=πbr2bb!,
and with noninteger integrating (m<0)
(2.25)∂-1/2∂r-1/2HS1(k,52,-12,r)=πb(2r)2b+1b!(2b+1)!,∂-1/2∂r-1/2HS2(k,52,-12,r)=πbr2bb!.
The values (pairs) of the expressions (2.22), (2.23), (2.24), and (2.25) match the dichotomous values of the hyperspherical function for n=3 therefore, we have
(2.26)HS1(2b+1,3,r)=πb(2r)2b+1b!(2b+1)!,HS2(2b,3,r)=πbr2bb!.
By integrating the above complementary functions, we obtain the hyperspherical functions series, that describes the solid geometrical entities (n=3∧k∈N) in the form
(2.27)∑k=0∞HS(k,3,r)=∑k=0∞πkrkΓ(k/2+1)=∑b=0∞πb(2r)2b+1b!(2b+1)!+∑b=0∞πbr2bb!=eπr2erf⁡(rπ)+eπr2=eπr2[1+erf⁡(rπ)]=eπr2erfc⁡(rπ),
which is Mittag-Leffler’s [26], that is, Freden’s solution (1993) for the solid hyperspherical function series [27]. In the previous expression complementary error functions originated from the expression erf(z)+erfc(z)=1. By generalizing (2.4), we obtain the following differential equation:
(2.28)∂n∂rn∑k=0∞HS(k,n,r)-∑k=0∞HS(k,0,r)=0.
which describes the relation between the columns of the hyperspherical matrix for n∈N.


## 2.4. The Gradient of the Hyperspherical Function

Taking into consideration its differentiability and multidimensionality,  we can apply the gradient on the hyperspherical function. As this function has three variablesk, n, and r, we obtain the solution of gradient functions ∇k,n,r in the following form:(2.29)∇k,n,r{HS(k,n,r)}=[∂∂kHS(k,n,r)∂∂nHS(k,n,r)∂∂rHS(k,n,r)]=HS(k,n,r)[ln⁡2rπ+12ψ0(k+12)-ψ0(k+n-2)ln⁡r-ψ0(k+n-2)1r(k+n-3)],
where  ψ0(z)=(d/dz)lnΓ(z) is digamma function.This function has a special use in determining the extreme values of the contour HS functions.


## 2.5. The Contour Graphics of the Derivatives of the Hyperspherical Functions

The derivative functionshs1=∂HS/∂k=∇k{HS} and sh1=∂HS/∂n=∇n{HS}, also belong to the family of the hyperspherical functions. According to the degrees of freedom k and n, we can split them into two groups of functions.Definition 2.8.
The function of the first derivative with respect to the degree of freedomk is
(2.30)hs1(k,n,r)=∂∂kHS(k,n,r)=HS(k,n,r)[ln⁡2rπ+12ψ0(k+12)-ψ0(k+n-2)].
The function hs1(k,n,r) has a special use in determining the maximum of the hyperspherical functions, for example, for a unit radius and the domain k∈ℜ (Figure 3). Here, on the basis of the known criteria, for each member of the derivative functions family we determine the maximum of the HS function by equaling its derivative with zero. Other than the maximum, there is the “optimal” value k0. Consequently
(2.31)hs1(k,n,r)=0,∂2∂k2HS(k,n,r)>0⟹max⁡HS(k,n,r)∧k0.Figure 3
The parallel presentation of the orginal hyperspherical functions (forn=2 and n=3) of the unit radius, and corresponding functions of their first derivatives for -25≤k≤20.Example 2.9.
For  the volume hyperspherical function with the radiusr=1, the derivative function is after transforming [24]
(2.32)hs1(k,3,1)=HS(k,3,1)2[ln⁡π-ψ0(k2+1)]=πkkΓ(k/2)(ln⁡π+γ-Hk/2),
where Hk/2 is a harmonious number. This number is obtained on the basis of the sum [28]
(2.33)Hk=∑j=1k1jorHk/2=∑j=1k/21j=1+12+⋯+k2.
The relation between the harmonious number and psi (digamma function) is the following:
(2.34)Hk=γ+ψ0(k+1),respectively,Hk/2=γ+ψ0(k2+1).
The graphical representation of the original HS(k,3,1) and its derivative are given in Figure 4.Figure 4
The contour function of the hyperspherical solidHS(3,n,1) with the maximum value maxHS and optimal degree of freedom k0 (for -2≤k≤18).


## 2.6.The Extreme Values of the HS Function from the Degree of Freedom  k Point of View

On the basis of the above expressions (2.31) and (2.32), we define the greatest volume for the “optimal” dimension k0. We find that k0≈5.2569, and the corresponding volume is maxHS(k,3,1)=HS(k0,3,1)≈5.2777. For other hyperspherical functions, we can also determine the maximum values on the numerical bases (Table 1). However, for the degrees of freedom n≥5, the maximum is a function of its factorial.Table 1DimensionnDimensionk0maxHSErrorε010.700421168630701871.481745773695~019.377568346255670237.5057633480618−1.055·10−1327.25694640486057233.161194484961957.363·10−1535.2569464048605725.2777680211134113.516·10−1542.5065379841902551.054723774914521~0501/20⋮⋮⋮⋮n01/(n-3)!0So, we have(2.35)lim⁡k→0HS(k,5,1)=12,lim⁡k→0HS(k,6,1)=16,lim⁡k→0HS(k,7,1)=124,
and in general
(2.36)lim⁡k→0HS(k,n,1)=1Γ(n-2)=1(n-3)!,lim⁡k→0+HS(k,n,r)=rn-3(n-3)!.
On the following surface 3D graphic HS(k,n,1) (Figure 5) we notice the extreme, that is, the maximum value of this function with the 0th freedom degree n=0, which is a global maximum in the domain k,n∈N as well.Figure 5
The graphic of the hyperspherical function with the constant radiusr=1 and for the degrees of freedom 0<n≤4 and 0<k≤25.


## 2.7. The Extreme Values of the HS Function from the Degree of Freedomn Point of View

Definition 2.10.
The function of the first derivative with respect to the degree of freedomn, is
(2.37)sh1(k,n,r)=∂∂nHS(k,n,r)=HS(k,n,r)[ln⁡r-ψ0(k+n-2)].
The function sh1(k,n,r) is in fact the general solution of the HS function of the degree of freedom n.On the basis of the known differentiating procedure, besides the local maximum of the contour HS functions, we obtain the corresponding optimal valuesn0 as well. Consequently
(2.38)sh1(k,n,r)=0,∂2∂n2HS(k,n,r)>0⟹max⁡HS(k,n,r)=HS(k,n0,r)∧n0.
These functions can have a significant role with further research of hyperspherical functions. By calculating the extreme and optimal values of the HS function, we obtain Table 2.Table 2DimensionkDimensionn*maxHSErrorε03.4616321449683621.1291738854501331.08·10−1612.4616321449683622.2583477709002581.086·10−1521.4616321449683627.094808766311162~030.46163214496836228.379235065244632.665·10−154012π205032π20⋮⋮⋮⋮1006π5≈1836.18109011012821π5≈1865.262840120116π6≈1762.546850⋮⋮⋮⋮∞0limk→∞HS(k,0,1)=00We can conclude that the following optimal degrees of freedomn0 and k0(2.39)n0=0,6π5<k0<12821π5
generate a global value maximum of the hyperspherical function, taking only the domain of the natural numbers (k,n∈N).As in the previous analysis, we determine the value ofk0, the optimal vector is
(2.40)[k0n0]=[10.70042116863070]⟹max⁡HS(k,n,1)=HS(k0,n0,1)≈1871.481745773695.
The graphical representation of the original HS(3,n,1) and its derivate are given in Figure 6.Figure 6
The contour function of the hypersolidHS(3,n,1) with the local maximum maxHS≈28.379 and the optimal degree of freedom n0≈0.4616, for k∈N.


## 2.8. The Derivatives of the Hyperspherical Functions of the Higher Order with Respect to the Argumentk

The derivatives of the hyperspherical functions with respect to any argument retain the characteristics similar the original HS function. This is especially noticeable in the graphical representation. In addition, it would be interesting to test the regularities in the structure of this function of the higher derivation degree. In the following, we give, analytically and graphically, the derivative functions on argumentk (the more complex case) and n (the simpler case).The second derivative of the hyperspherical function with respect to argumenk is:
(2.41)hs2(k,n,r)=∂2∂k2HS(k,n,r)=HS(k,n,r){ψ1(k)-ψ1(k+n-2)+[ψ0(k+n-2)-12ψ0(k+12)-ln⁡2rπ]2-14ψ1(k2)}.
Here: ψ0(z)=(d/dz)lnΓ(z) digamma and ψ1(z)=(d2/dz2)lnΓ(z) trigamma function. The concrete value of this derivative for the selected parameters is the following:
(2.42)∂2∂k2HS(3,3,1)=2π9[(3γ-8)ln⁡π+γ(3γ2-8)+6(γ+ln⁡2π-83)ln⁡2+34(2ln2π-π2)+523].
The mixed derivative of the hyperspherical function is
(2.43)∂2∂k∂nHS(k,n,r)=HS(k,n,r){ψ0(k+n-2)[ψ0(k+n-2)-12ψ0(k+12)-ln⁡2r2π]+ln⁡r[ln⁡2rπ+12ψ0(k+12)]-ψ1(k+n-2)}.Example 2.11.
After introducing some values ofk and n it follows that
(2.44)∂2∂k∂nHS(3,3,1)=π9[(ln⁡4π+γ)(6γ+11)-2(π2+8γ)+1373].
The fifth derivative of the hyperspheric function on argument n and for its unique radius is
(2.45)∂5∂n5HS(k,n,r)=-HS(k,n,r){(ψ0(k+n-2)-ln⁡r)5-10(ψ0(k+n-2)-ln⁡r)3ψ1(k+n-2)+10(ψ0(k+n-2)-ln⁡r)2ψ2(k+n-2)+15(ψ0(k+n-2)-ln⁡r)2×ψ12(k+n-2)-5ψ3(k+n-2)(ψ0(k+n-2)-ln⁡r)-10ψ1(k+n-2)ψ2(υ+1)+ψ4(k+n-2)(ψ0(k+n-2)-ln⁡r)5}.
Namely, the fifth derivative values of this function for degrees of freedom k=0, n=0 are
(2.46)∂5∂k5HS(0,0,1)=sh5(0,0,1)=40(2γ-3)ζ(3)+10γ[π2(3-γ)+6γ(1-γ(1-γ6))]-π2(10-ζ(2)).
Generally, the hyperspherical function of the mth derivative on the argument n can be defined on the basis of the product for poligamma (psi) of the polynominal of the mth degree and the hyperspherical function. Its general form is the following:
(2.47)shm(k,n,r)=∂m∂nmHS(k,n,r)=HS(k,n,r)⋅fm{ψ(m-1)(k+n-2),r}.
So, for example, the functions of the fifth derivative are in Figure 7.Figure 7
The graphics of the fifth derivative of the hyperspherical functions with respect to the argumentn.


## 3. Conclusion

The research of the hyperspherical function leaves the following problems unsolved: continual dimensions which include noninteger values, the zeroth dimension, the domain of the dimensions less than zero, consequently the domain of the more complex degrees of freedom, and so forth. In any case, the negative dimensions matrix demands special structuring and deeper mathematical analysis. The hyperspheres present one part of the multigeometrical objects. It is known that a sphere is the most symmetrical geometrical object. Some other objects have such characteristics but not entirely. Namely, a cube is symmetrical on orthogonal coordinates, if its centre is in the intersection of space diagonals, and the axes of the coordinate system are parallel to cube edges. Some other polyhedrons have this characteristic, too. A cube, because of orthogonality, retains the characteristic that its derivatives of the solid-entity functions create the function of surfentity. The objects that do not have the characteristic that recurrence is created on the basis of the derivative function, must introduce new degrees of freedom depending on the object complexity. The next object that has some characteristics similar to sphere is cylinder (this similarity is caused by the equality of the cylinder height and diameter). Its symmetry is dual and is not unambigous on all coordinate axes like the sphere is. These characteristics are the fundamental assumptions for defining the recurrent relations that would determine the relation between functions of some entities and both on rows and columns of a cube, that is, cylinder. The considered analysis of the multidimensional space and the formula of this geometry lead into the conclusion of its complexity and the relations with special functions and the other geometry fields. Special problems not analysed in this paper, refer to zeros and singular values of this function, and some ot them can be seen in the matrix (Figure2).



---

*Source: 364292-2010-12-19.xml*