The Dimensional Fluxes of the Hypercylindrical Function

Abstract

We present the results of the theoretical researches of some characteristics of the generalized hyperspherical function with two degrees as independent dimensions. Primarily are given the answers on the quantification of the dimensional potentials (fluxes) of this function in the domain of the integer natural numbers. Beside them, we have got the solutions for some continual fluxes of the contour hypercylindrical (HY) functions on the basis. The symbolical evaluation and numerical verification of the series values and integrals are realized through the program packages Mathcad Professional and Mathematica.

1. Introduction

Hypercylinder function is the hypothetical function connected to multidimensional space. The most significant value of this function is in the fact that it originated [1] on the properties of the cylindrical entities: point, diameter, square, surface, and volume of cylinder. Another property is generalizing of these functions from discretion to continuum. It belongs to the group of special functions, so its testing is performed on the basis of known functions of these types: gamma (Γ), psi (𝜓₀), error function (erf), and the like.

Definition 1.1. The generalized hypercylindrical function is defined by equality [2] ()=2HY𝑘,𝑛,𝑟𝜋^𝑘−1𝑟^{𝑘+𝑛−3}Γ(𝑘+1)Γ()Γ))()𝑘+𝑛−2((𝑘+1/2𝑘,𝑛∈ℜ,𝑟∈𝑁, (1.1) where Γ(𝑧) is the gamma function.

Thanks to the interpolation properties of the gamma function, we can analytically pass from the field of the natural integer values on the set of real and noninteger values with which there is concurrence of events both for its graphic interpretation and more concise mathematical analysis. It is developed on the basis of the two freedom degrees 𝑘 and 𝑛, as special (vectors) dimensions, besides 𝑟 radius, as implicitly included freedom degree for every hypercylinder (Figures 1 and 2). The dominant theorem that is set is the one that refers to the recurrent property of this function (when the height is ℎ=2𝑟). It implicitly includes that the left vectors (𝑛=2,1,0,−1,−2,…) of the 𝑀[HY]_𝑘𝑥𝑛 matrix columns in Figure 1 we get on the basis of the reverent vector (𝑛=3) deduction, and the right vectors (𝑛=4,5,6,7,8,…) on the basis of integrals on 𝑟 radius [2]𝜕()()()=𝜕𝑟HY𝑘,𝑛,𝑟=HY𝑘,𝑛−1,𝑟,HY𝑘,𝑛+1,𝑟^𝑟₀HY(𝑘,𝑛,𝑟)𝑑𝑟. (1.2)

Figure 1 

The functional submatrix 𝑀[HY(𝑘,𝑛,𝑟)] for the freedom degrees 𝑘∈−4,−3,…,4 and 𝑛∈−1,0,…,7 and with six characteristics hypercylindrical functions (undef. are nondefined, most often singular values of this function).

Figure 2 

The surface graphic of the hypercylindrical function with the constant radius 𝑟=1 and the selected fields of freedom degrees (−4<𝑛≤5) and (−2<𝑘≤5) (using the software Mathematica).

To the development of the theory of the multidimensional objects, especially have contributed: Conway and Sloane [3], Gwak et al. [4], Hinton [5], Hocking and Young [6], Manning [7], Maunder [8], Neville [9], Von R. Rucker [10], Sommerville [11], Sun and Bowman [12], and the others, and to its testing, Ramanujan and Hardy [16, 17]. Today the researches of the hypercylindrical function are represented both in Euclid's and Riemann's geometry (molecular dynamics, neural networks, hypercylindrical black holes and the like).

2. Dimensional Potentials: The Fluxes of the HY Function

2.1. Vertical Dimensional Flux of the Hypercylindrical Function

The discrete dimensional potential or flux of the hypercylindrical function presents the total of all single functions in the (sub)matrix of this function that develops for the integer natural freedom degrees. Formally, flux can be quantification by twofold series that covers this area of HY function. The first step is to define the value of the infinite succession of functions ordered in columns (vectors) of the submatrix 𝑀[HY]_𝑘𝑥𝑛(𝑘,𝑛∈𝑁). This is at the same time as well the definition of the vertical dimensional fluxes of HY function. The first value that is being calculated, refers to the fourth column (𝑛=3) of the submatrix 𝑀[HY]_𝑘𝑥𝑛 in Figure 1. In this case, it follows that the flux is equal ^∞_𝑘=0()=2HY𝑘,3,𝑟𝜋+2𝑟+4𝑟²+2𝜋𝑟³+83𝜋𝑟⁴+𝜋²𝑟⁴2+⋯+𝜀𝜋^𝑘−1𝑟^𝑘Γ((𝑘+1)/2). (2.1) The analytical value of this series is HY(^∞_𝑘=0𝑘,3,𝑟)=^∞_𝑘=02𝜋^𝑘−1𝑟^𝑘Γ))1((𝑘+1/2=2𝜋+𝑟𝑒^𝜋𝑟2erfc−𝑟√𝜋. (2.2) In the paper are used three known error functions as follows: erf(𝑧)-basic, erfc(𝑧)-cumulative and erfi(𝑧)-imaginary. When 𝑘 values are even (0,2,4,…), in other words odd ones (1,3,5,…), the series can be divided as dichotomous, so we can now obtain two complementary seriesHY(^∞_𝑘=0𝑘,3,1)=^∞_{𝑘=0,2,4,…}HY(𝑘,3,𝑟)+^∞_{𝑘=1,3,5,…}()1HY𝑘,3,𝑟=2𝜋+𝑟𝑒^𝜋𝑟2erf𝑟√𝜋+2𝑟𝑒^𝜋𝑟2. (2.3) Well, the result (2.3) can be presented in the from of series with even (𝑘=2𝑏) and odd members (𝑘=2𝑏+1) that complement (one another). In that sense follows^∞_𝑏=02𝜋^𝑏−1(2𝑟)^2𝑏𝑏!()!12𝑏=2𝜋+𝑟𝑒^𝜋𝑟2erf𝑟√𝜋,^∞_𝑏=02𝜋^𝑏𝑟^2𝑏+1𝑏!=2𝑟𝑒^𝜋𝑟2. (2.4) On the basis of the solution (2.3), as the starting and reference (one), and applying the reference relations (1.2), we can obtain the series values for lower freedom degrees (𝑛<3). So, we establish the connection of cylinder hypervolume (𝑛=3) with its hypersurface (𝑛=2). In that sense follows a new vector flux for 𝑛=2:𝜕𝜕𝑟^∞_𝑘=0HY(𝑘,3,𝑟)=^∞_𝑘=0HY(𝑘,2,𝑟), (2.5) so that HY(^∞_𝑘=0𝑘,2,𝑟)=22𝑟+𝑒^𝜋𝑟2erfc−𝑟√𝜋1+2𝜋𝑟². (2.6) Further, for the hypercylinder (𝑛=1), we obtain a series on the basis of the derivative of the previous series, so that𝜕𝜕𝑟^∞_𝑘=0HY(𝑘,2,𝑟)=^∞_𝑘=0HY(𝑘,1,𝑟)=421+𝜋𝑟²+𝜋𝑟𝑒^𝜋𝑟2erfc−𝑟√𝜋3+2𝜋𝑟². (2.7) For the zero dimension 𝑛=0, the series value is as well located on the derivative basis, so it follows that HY(^∞_𝑘=0𝑘,0,𝑟)=^∞_𝑘=02𝑟^𝑘−3𝜋^𝑘−1Γ(𝑘+1)Γ()Γ))𝑟𝑘−2((𝑘+1/2=4𝜋10+4𝜋𝑟²+𝑒^𝜋𝑟2erfc−𝑟√𝜋3+4𝜋𝑟²3+𝜋𝑟². (2.8) For greater freedom degree that 𝑛=3, series are found through an inverse operation, that is, by recurrent relation on the basis of integrating on the radius 𝑟. Well, HY(^∞_𝑘=0𝑘,𝑛+1,𝑟)=^𝑟₀^∞_𝑘=0HY(𝑘,𝑛,𝑟)𝑑𝑟, (2.9) and for the fourth dimension is valid the next integral form HY(^∞_𝑘=0𝑘,4,𝑟)=^𝑟₀^∞_𝑘=0HY(𝑘,3,𝑟)𝑑𝑟=^𝑟₀21𝜋+𝑟𝑒^𝜋𝑟2erf𝑟√𝜋=1𝑑𝑟𝜋𝑒^𝜋𝑟2erfc−𝑟√𝜋−1. (2.10) For greater freedom degree than 𝑛=3, series are found through a recurrent relation (1.2), by integrating on radiusHY(^∞_𝑘=0𝑘,4,𝑟)=^𝑟₀^∞_{𝑘=1,3,5,…}HY(𝑘,3,𝑟)𝑑𝑟+^𝑟₀^∞_{𝑘=0,2,4,…}HY(𝑘,3,𝑟)𝑑𝑟. (2.11) So, we come to the expressionHY(^∞_𝑘=0𝑘,4,𝑟)=^𝑟₀2𝑟𝑒^𝜋𝑟2𝑑𝑟+^𝑟₀^∞_𝑏=02^2𝑏+1𝜋^𝑏−1𝑟^2𝑏𝑏!(2𝑏)!𝑑𝑟. (2.12) Solving the integrals and using the relation on dichotomous series [2], we obtain the dimension flux of the fourth dimension ^∞_𝑘=0()=𝑒HY𝑘,4,𝑟^𝜋𝑟2−1𝜋+^∞_𝑏=02^2𝑏+1𝜋^𝑏−1𝑟^2𝑏+1𝑏!(2𝑏+1)!. (2.13) For the freedom degree of 𝑛=5, series can be found through the integrating of the obtained solution (2.10) HY(^∞_𝑘=0𝑘,5,𝑟)=^𝑟₀1𝜋𝑒^𝜋𝑟2erfc−𝑟√𝜋−1𝑑𝑟. (2.14) The general solution is known, and it is 𝑒^𝑏𝑧2()√erfc𝑎𝑧𝑑𝑧=𝜋erf𝑧√𝑏2√𝑏−1𝑏√𝜋^ℵ_𝑘=0𝑎^2𝑘+1Γ𝑘+1,−𝑏𝑧²𝑏^𝑘𝑘!(2𝑘+1)+𝐶. (2.15) After settling the expression, we get the concrete solution HY(^∞_𝑘=0𝑘,5,𝑟)=^∞_𝑘=02𝑟^𝑘+2𝜋^𝑘−1Γ(𝑘+1)Γ()Γ))=1𝑘+3((𝑘+1/2𝜋erfi𝑟√𝜋21−𝑟+𝜋^∞_𝑘=0(−1)^2𝑘+1Γ2𝑘+11−𝑘+1,−𝜋𝑟².𝑘! (2.16) For the same freedom degree, the series can be found with the integrating of the dichotomous expressionHY(^∞_𝑘=0𝑘,5,𝑟)=^𝑟₀𝑒^𝜋𝑟2−1𝜋𝑑𝑟+^𝑟₀^∞_𝑏=02^2𝑏+1𝜋^𝑏−1𝑟^2𝑏+1𝑏!(2𝑏+1)!𝑑𝑟. (2.17) So that^∞_𝑘=0()=1HY𝑘,5,𝑟𝜋erfi𝑟√𝜋2−𝑟+^∞_𝑏=02^2𝑏+1𝜋^𝑏−1𝑟^2(𝑏+1)𝑏!(2𝑏+2)!. (2.18) The same result is obtained as well on the basis of the complementary dichotomous series HY(^∞_𝑘=0𝑘,5,𝑟)=^∞_𝑏=02𝜋^𝑏𝑟^2𝑏+3()+2𝑏+2)(2𝑏+3𝑏!^∞_𝑏=02^2𝑏+1𝜋^𝑏−1𝑟^2(𝑏+1)𝑏!(2𝑏+2)!. (2.19) Here, we use the imaginary error function that is equal to [14](𝑧)()=2erfi=−𝑖⋅erf𝑖⋅𝑧√𝜋^∞_𝑘=0𝑧^2𝑘+1𝑘!(2𝑘+1). (2.20) For the last analysed vector flux, through similar procedures follow: HY(^∞_𝑘=0𝑘,6,𝑟)=^𝑟₀^∞_𝑏=02𝜋^𝑏𝑟^2𝑏+3(2𝑏+2)(2𝑏+3)𝑏!𝑑𝑟+^𝑟₀^∞_𝑏=02^2𝑏+1𝜋^𝑏−1𝑟^2(𝑏+1)𝑏!(2𝑏+2)!𝑑𝑟. (2.21) It can be presented that the first subintegral member of the dichotomous series (2.19) is equal to ^∞_𝑏=02𝜋^𝑏𝑟^2𝑏+3()=12𝑏+2)(2𝑏+3𝑏!𝜋erfi𝑟√𝜋2−𝑟. (2.22) So, its integral is ^𝑟₀1𝜋erfi𝑟√𝜋21−𝑟𝑑𝑟=2𝜋1−𝑒^𝜋𝑟2𝜋+𝑟erfi𝑟√𝜋−𝑟². (2.23) The dimensional flux for the sixth freedom degree is now ^∞_𝑘=02𝑟^𝑘+3𝜋^𝑘−1Γ(𝑘+1)Γ()Γ))=1𝑘+4((𝑘+1/22𝜋1−𝑒^𝜋𝑟2𝜋+𝑟erfi𝑟√𝜋−𝑟²+^∞_𝑏=02^2𝑏+1𝜋^𝑏−1𝑟^2𝑏+3𝑏!(2𝑏+3)!. (2.24) For each freedom degree the recurrent relation would be formulated as HY(^∞_𝑘=0𝑘,𝑛,𝑟)=^𝑟₀^∞_𝑘=0,2,…HY(𝑘,𝑛−1,𝑟)+^∞_𝑘=1,3,…HY(𝑘,𝑛−1,𝑟)𝑑𝑟. (2.25) The similar formulation would be related as well to the recurrent relation of the type ^∞_𝑘=0()=𝜕HY𝑘,𝑛,𝑟𝜕𝑟^∞_𝑘=0,2,…HY(𝑘,𝑛+1,𝑟)+^∞_𝑘=1,3,…HY(𝑘,𝑛+1,𝑟). (2.26) It can be supposed that the values of the vector fluxes are less and less with the increasing of the freedom degree 𝑛, so that the limit values are equal lim_𝑛→∞^∞_𝑘=0HY(𝑘,𝑛,𝑟)=0,respectively,lim_𝑛→∞^∞₀HY(𝑘,𝑛,𝑟)𝑑𝑘=0. (2.27) The systematized numerical values of the discrete and continual fluxes, (for 𝑟=1), are given in Table 1.

Table 1 

The dimensional fluxes can be studied as well for the complex part. So, for example with recurrence we get the series value for negative freedom degrees, and so for 𝑛=−1, follows [2]HY(^∞_𝑘=0𝑘,−1,𝑟)=^∞_𝑘=02𝑟^𝑘−4𝜋^𝑘−1Γ(𝑘+1)Γ()Γ))2𝑘−3((𝑘+1/2=8𝜋1+2𝜋𝑟²4+𝜋𝑟²+𝜋𝑟𝑒^𝜋𝑟2erfc−𝑟√𝜋15+4𝜋𝑟²5+𝜋𝑟². (2.28) For more lower freedom degree, that is 𝑛=−2, the flux is more complex and is defined by the next analytical value ^∞_𝑘=0()=𝜕HY𝑘,−2,𝑟𝜕𝑟^∞_𝑘=0HY(𝑘,−1,𝑟)=^∞_𝑘=02𝑟^𝑘−5𝜋^𝑘−1Γ(𝑘+1)Γ(𝑘−4)Γ((𝑘+1)/2)=8𝜋²2𝑟3+2𝜋𝑟²11+2𝜋𝑟²+𝑒^𝜋𝑟2erfc−𝑟√𝜋×2𝜋𝑟²2𝜋𝑟²15+2𝜋𝑟²+45+15. (2.29) So, for example, the flux for radius 𝑟=1/2 is ∑^∞_𝑘=0HY(𝑘,−2,1/2)≈43770.014449969689. The other values, also can be established on the basis of the recurrent relations (2.25), in other words (2.26).

2.2. The Fluxes on the Basis of the Series of the Hypercylindrical Functional Matrix

The discrete dimensional fluxes can be calculated as well on the “horizontal line”, that is, adding functions values on the submatrix series 𝑀[HY]_𝑘,𝑛 (Figure 1). For example, through the series development for 𝑘=3 the flux would contain the next members: HY(^∞_𝑛=03,𝑛,𝑟)=12𝜋+12𝜋𝑟+6𝜋𝑟²+2𝜋𝑟³+12𝜋𝑟⁴+⋯+𝜀12𝜋𝑟^𝑛Γ(𝑛+1). (2.30) Some values of discrete and continual fluxes (for 𝑟=1) are given in Table 2.

Table 2 

2.3. Some Continual Fluxes of the Hypercylindrical Function

The trend of the distribution of the vector fluxes is increasing, and then asymptotically falling, with the linear growth of the freedom degree 𝑛. From the standpoint of the functional analysis the most interesting series of the matrix 𝑀[HY]_𝑘,𝑛 are the ones referring to the freedom degrees 𝑘=2 and 𝑘=3. The first series (𝑘=2) covers the known functions for the square size (8𝑟) and surface (4𝑟²). The members of the following series are, among the others, the cylinder functions of the surface (6𝜋𝑟²) and volume (2𝜋𝑟³) (Figure 1). The same series are interesting as well for calculatiing continual fluxes. So the continual natural flux for the hypercylinder surface is analysed in view 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's constant [15] 𝐹=^∞₀1Γ(𝑥)𝑑𝑥=𝑒+^∞₀𝑒^−𝑛𝜋²+ln²𝑛𝑑𝑛≈2.8077702420285. (2.31) The integral values of the wanted flux is now ∫^∞₀HY(2,𝑛,1)𝑑𝑛=∑^∞_𝑛=0HY(2,𝑛,1)+∫^∞₀(8𝑒^−𝑛/(𝜋²+ln²𝑛))𝑑𝑛, what concretely for the unit radius is ^∞₀8Γ(𝑛)𝑑𝑛=8𝑒+^∞₀𝑒^−𝑛𝜋²+ln²𝑛𝑑𝑛=8𝐹≈22.462161936228. (2.32) In respect to the continual dimension 𝑛, the more generalized cylinder volume flux follows in view of Ramanujan-Hardy's integral [16, 17]:^∞₀𝑦^𝑥Γ(𝑥+1)𝑑𝑥=𝑒^𝑦−^∞₀𝑒^−𝑥𝑦𝑥𝜋²+ln²𝑥𝑑𝑥. (2.33) Ramanujan defined this integral, and it was analytically intensified by Hardy. In that sense, the previous expression can be applied in calculating the hypercylindrical function flux, when 𝑘=3, asHY(^∞₀3,𝑛,1)𝑑𝑛=^∞₀12𝜋𝑟^𝑛Γ()𝑒𝑛+1𝑑𝑛=12𝜋^𝑟−^∞₀𝑒^−𝑛𝑟𝑛𝜋²+ln²𝑛𝑑𝑛for𝑥=𝑛,𝑦=𝑟. (2.34) The integral can be calculated as well as the difference between the series and the integral with the value (for 𝑟=1)^∞₀12𝜋𝑟^𝑛Γ(𝑛+1)𝑑𝑛=^∞_𝑛=012𝜋𝑟^𝑛−Γ(𝑛+1)^∞₀12𝜋𝑒^−𝑛𝑟𝑛(𝜋²+ln²𝑛)𝑑𝑛_𝑟=1≈278.6085228836. (2.35)

2.4. The Progressions of the Vector Fluxes

The whole dimension flux in the freedom degree domain with the natural numbers, is obtained in the result of double amount with which are considered the integer values of the hypercylindrical function HY(𝑘,𝑛,𝑟), for all 𝑘,𝑛,𝑟≥0. This double series must be convergent, and this characteristic is in the function of hypercylinder radius. The flux can be watched as well for every column 𝑀[HY]_𝑘,𝑛 of the matrix individually. So, there is for the 𝑛th column (in the mark ⟨𝑛⟩), the flux in the form of series Φ^⟨𝑛⟩_HY(𝑘,𝑛,𝑟)=^∞_𝑘=0HY(𝑘,𝑛,𝑟). (2.36)

2.5. The Orthogonal Dimensional Fluxes

These fluxes are the fluxes of the all columns or of all series of the matrix 𝑀[HY]_𝑘,𝑛. As the number of columns, that is of the series infinite, the total flux is as follows.

Definition 2.1. The dimensional flux (potentional) of the functional matrix with two freedom degrees 𝑘 and 𝑛 is defined as a double series

Φ_HY(𝑘,𝑛,𝑟)=^∞_𝑛=0Φ^⟨𝑛⟩_HY(𝑘,𝑛,𝑟)=^∞∞_𝑛=0𝑘=0HY(𝑘,𝑛,𝑟). (2.37) When the certain number of members is calculated, the flux has the next form:Φ_HY(𝑘,𝑛,𝑟)=^∞∞_𝑛=0𝑘=0()𝑟HY𝑘,𝑛,𝑟=4𝜋10+4𝜋𝑟²+𝑒^𝜋𝑟2erfc−𝑟√𝜋3+4𝜋𝑟²3+𝜋𝑟²+421+𝜋𝑟²+𝜋𝑟𝑒^𝜋𝑟2erfc−𝑟√𝜋3+2𝜋𝑟²+22𝑟+𝑒^𝜋𝑟2erfc−𝑟√𝜋1+2𝜋𝑟²1+2𝜋+𝑟𝑒^𝜋𝑟2erfc−𝑟√𝜋+1𝜋𝑒^𝜋𝑟2erfc−𝑟√𝜋+1−1𝜋erfi𝑟√𝜋2+1−𝑟𝜋^2∞_𝑘=0(−1)^2𝑘+1Γ2𝑘+11−𝑘+1,−𝜋𝑟²+1𝑘!2𝜋1−𝑒^𝜋𝑟2𝜋+𝑟erfi𝑟√𝜋−𝑟²+⋯. (2.38) The flux on the matrix series in the domain of the natural number is defined as a double series, but with the changed sequence of summing. This dimensional flux is, thus, defined asΩ_HY(𝑘,𝑛,𝑟)=^∞∞_𝑘=0𝑛=0HY(𝑘,𝑛,𝑟). (2.39) In view of previously established members, the matrix flux has the following form:Ω_HY(𝑘,𝑛,𝑟)=^∞∞_𝑘=0𝑛=0()=HY𝑘,𝑛,𝑟2𝑒^𝑟𝜋+2𝑒^𝑟+8𝑒^𝑟+12𝜋𝑒^𝑟+64𝜋𝑒^𝑟−1+120𝜋²𝑒^𝑟−𝑟−1+768𝜋²𝑒^𝑟𝑟+𝑟−²2+1+^∞_𝑛=01680𝜋³𝑟^𝑛+4Γ()+𝑛+5^∞_𝑛=012288𝜋³𝑟^𝑛+5Γ(𝑛+6)+⋯. (2.40) The equivalence of the orthogonal dimensional fluxes means the equality of the double seriesΦ_HY(𝑘,𝑛,𝑟)=Ω_HY(𝑘,𝑛,𝑟). (2.41) So, for example, for 𝑟=1 the dimensional fluxes have the unambiguous numerical value that is: Φ_HY(𝑘,𝑛,1)=Ω_HY(𝑘,𝑛,1)=52812.977966365550. (2.42)

2.6. The Application of the Recurrent Operators with Defining the Diagonal Dimensional Fluxes of the 𝐻𝑌 Function

In the previous analyses the defining of the dimensional fluxes of the matrix 𝑀[HY]_𝑘,𝑛 is performed in view of the adding up of the HY function value for (on) the columns, in other words, the series of this matrix. The more detailed analysis would be very ample including the exponential function, then error functions erf(𝑧), erfc(𝑧), erfi(𝑧), the incomplete gamma function Γ(𝑎,𝑧) and the like. When we use the ideas of transition operators from the reference function into the destined HY function, in the functional hypercylindrical matrix 𝑀[HY]_𝑘,𝑛 we can also establish the dimensional fluxes values on the diagonals (2.44), whose total would present the whole flux for the matrix in the field of freedom degrees for natural numbers, that is, 𝑘,𝑛∈𝑁. Such matrix contains infinite number of elements. For reference functions we take HY functions on the positions of the first series of the matrix, and they are so-called zero HY functions: HY(0,0,𝑟),HY(0,1,𝑟),…,HY(0,𝑛,𝑟),… The destined functions are arranged on the “gradual” growth law (+Δ𝑘) and decrease (−Δ𝑛) of increments.

Definition 2.2. The flux series operator 𝜗(Δ𝑘,Δ𝑛,0) is defined by the relation 𝜗()=()Δ𝑘,Δ𝑛,0HY𝑘+Δ𝑘,𝑛+Δ𝑛,𝑟()=𝜋HY𝑘,𝑛,𝑟_Δ𝑘/2𝑟^{Δ𝑘+Δ𝑛}Γ(𝑘+𝑛−2)Γ(𝑘+Δ𝑘+1)Γ()Γ()Γ))Γ𝑘+1𝑘+𝑛+Δ𝑘+Δ𝑛−2((𝑘+Δ𝑘+1/2𝑘+12. (2.43)(2.44)

Meanwhile, as increments for the absolute value are mutually equal and unit ones, that is, |Δ𝑘|=|−Δ𝑛|=1, to them is assigned a new joint argument 𝑢 (Δ𝑘=Δ𝑛=𝑢). In addition, the starting value of the 𝑘-og freedom degree states 𝑘=0, so the theta operator gets the form𝜃(𝑢,−𝑢,0)=2^𝑢√𝜋^𝑢Γ𝑢2+1. (2.45) The destined function is now calculated as()()()=2HY𝑢,𝑛−𝑢,𝑟=𝜃𝑢,−𝑢,0⋅HY0,𝑛,𝑟^𝑢+1√𝜋^𝑢−2𝑟^𝑛−3Γ()Γ𝑢𝑛−22+1. (2.46) The dimensional flux on the diagonal presents the total of ist particular nembers. So, for the first diagonal (with the mark ⟨0⟩) flux is equal Π^⟨0⟩(𝑘,𝑛,𝑟)=HY(0,0,𝑟)=0, (2.47) then for the second, Π^⟨1⟩(𝑘,𝑛,𝑟)=HY(0,1,𝑟)+HY(1,0,𝑟)=0. (2.48) For the third, Π^⟨2⟩(𝑘,𝑛,𝑟)=HY(0,2,𝑟)+HY(1,1,𝑟)+HY(2,0,𝑟)=0. (2.49) Or for the fourth, Π^⟨3⟩()()()()()2𝑘,𝑛,𝑟=HY0,3,𝑟+HY1,2,𝑟+HY2,1,𝑟+HY3,0,𝑟=10+𝜋+12𝜋. (2.50) The flux in the 𝑛th diagonal would be reckoned in the sum formΠ^⟨𝑛⟩(𝑘,𝑛,𝑟)=^𝑛_𝑢=0()=𝑟HY𝑢,𝑛−𝑢,𝑟^𝑛−3Γ(𝑛−2)^𝑛_𝑢=02^𝑢+1√𝜋^𝑢−2Γ𝑢2+1(𝑛≠2). (2.51) The flux for the value 𝑛=2 is calculated on the basis of the highest function value. Considering that the equivalence of the double factorial and trigonometric functions [18] is known 𝑢!!=2^{(1/4)[1+2𝑢−cos(𝜋𝑢)]}𝜋^{(1/4)[cos(𝜋𝑢)−1]}Γ𝑢2+1. (2.52) The expression (2.19) can after settling be written as well in the equivalent form Π^⟨𝑛⟩()=𝑟𝑘,𝑛,𝑟^𝑛−3(𝑛−3)!^𝑛_𝑢=0𝑢!!⁴(2𝜋)^2𝑢2𝜋^{3+cos(𝜋𝑢)}. (2.53) So, we get for the fifth diagonal (𝑛=4), using the expressions (2.51) or (2.53)Π^⟨4⟩(𝑘,𝑛,𝑟)=^𝑛_𝑢=0HY(𝑢,𝑛−𝑢,𝑟)=2𝑟𝜋5𝜋+38𝜋²+1. (2.54) For the sixth diagonal (𝑛=5) follows:Π^⟨5⟩(𝑘,𝑛,𝑟)=^𝑛=5_𝑢=0()=𝑟HY𝑢,𝑛−𝑢,𝑟²𝜋5𝜋+38𝜋²+60𝜋³+1. (2.55) As the number of diagonals is infinite, the total flux is formed as the series of all diagonal fluxes Π_HY(𝑘,𝑛,𝑟)=^∞_𝑛=0Π^⟨𝑛⟩()Π𝑘,𝑛,𝑟,orconcretely,_HY(𝑘,𝑛,𝑟)=^∞𝑛_𝑛=0𝑢=0HY(𝑢,𝑛−𝑢,𝑟)=^∞_𝑛=0𝑟^𝑛−3Γ(𝑛−2)^𝑛_𝑢=02^𝑢+1√𝜋^𝑢−1Γ𝑢+12. (2.56) For example, approximately, the flux for 𝑟=1 and 𝑛=11 it concretely isΠ_HY(𝑘,11,1)=109601+20160𝜋109601+4032143729910080𝜋+426437𝜋1680²+58497𝜋280³+2901𝜋28⁴+332𝜋⁵. (2.57) In the developed, form the total flux has the polynominal structure of membersΠ_HY()2𝑘,𝑛,𝑟=10+𝜋2+12𝜋+𝑟10+𝜋+76𝜋+𝑟²15+𝜋+38𝜋+60𝜋²+𝑟³53+1+3𝜋383𝜋+148𝜋²+𝑟⁴5+11212𝜋+37𝜋²+70𝜋³+𝑟⁵1+112+60𝜋1930𝜋+375𝜋²+5825𝜋³+112+𝑟⁶1+172+360𝜋19180𝜋+37𝜋30²+975𝜋³+42𝜋⁴+⋯. (2.58) The diagonal flux of the hypercylindrical function can be expressed by the series of general form Π_HY(𝑣,𝑟)=^∞_𝑣=0𝑎_𝑣𝑟^𝑣. (2.59) Here, 𝑟 is the summation index with which we take into consideration the order of elements from left to right and from top to bottom on (along) the diagonal (Figure 3). The polynomial coefficients contain the rational numbers and the graded constant 𝜋 (Table 3). The first three coefficients are equal to zero, so they are not put in the summation sequence. Its other values are (𝑣=0,1,…,12) given in Table 3.

Table 3 

Figure 3 

The representative submatrix (above) and the addition principle of the matrix members on the diagonal.

The series approximation with 13 coefficients in the decimal notation is in the form¹²_𝑣=0𝑎_𝑣𝑟^𝑣≈48,33+249,4𝑟+716,87𝑟²+1502,27𝑟³+2546,01𝑟⁴+3684,24𝑟⁵+4705,22𝑟⁶+5422,03𝑟⁷+5727,08𝑟⁸+5610,38𝑟⁹+5143,66𝑟¹⁰+4445,23𝑟¹¹+3642,42𝑟¹². (2.60) Approximately, the double series leads us to the solution that is very near to the correct one. Namely, for the unit radius and reducing on ∞∼𝑛=30 the double series of the diagonal flux develops the following structure:Π_HY(𝑢,𝑛,1)≈^𝑛=30𝑛_𝑛=0𝑢=0()=HY𝑢,𝑛−𝑢,𝑟739975398988375932899873137+27222173626045880401920000739975398988375932899873137𝜋136110868130229402009600000⁻¹+510736042137463255831073137𝜋+3581864950795510579200000238514305877004451811873137𝜋11342572344185783500800000²+395025082463267992919694289𝜋1890428724030963916800000³+103021449478899595399537𝜋986310638624850739200⁴+3457149785418899743999223𝜋94521436201548195840000⁵+2633045028485350315237𝜋270061246290137702400⁶+7044091324892514823027𝜋3375765578626721280000⁷+9964355179641216157𝜋26791790306561280000⁸+354387723451762783𝜋6251417738197632000⁹+1285336197080807𝜋170493211041753600¹⁰[phone]𝜋15786408429792000¹¹[phone]𝜋5464525994928000¹²[phone]𝜋390323285352000¹³[phone]𝜋130107761784000¹⁴≈52810,5103716. (2.61) The diagonal dimension flux is characteristic with its coefficients that contain the 𝜋^𝑛 constant in the graded series numbers, as distinguished from the vertical fluxes with the destination of the errors functions and 𝜋 and 𝑒 constants. The horizontal fluxes, as it is presented (2.40) contain the exponential functions. Meanwhile, the total flux for the unit radius is convergent and it can be calculated with considerably greater value. The value aberration (2.61) of the accurate value is only (or 0,00467%) ΔΠ=52812,97796636−52810,5103716≈2,467595. (2.62) The whole dimensional continual flux (𝑘,𝑛∈𝑁) of the unit hypercylindrical function HY(𝑘,𝑛,1) is equal to the value of the twofold integral^∞₀2𝜋^𝑘−1Γ(𝑘+1)Γ(𝑘+𝑛−2)Γ((𝑘+1)/2)𝑑𝑘𝑑𝑛, (2.63) and we find its solution on the analytical and numerical bases.

3. Conclusion

On view of the supposition of recurrent relations (1.2), namely, (2.25) and (2.26) that exist in the scope of the hypercylindrical function, we can calculate the discrete dimension flux of this function in the domain of integer freedom degrees. The quantitative flux value at the most depends of the formulated value of the hypercylinder radius. Meanwhile, as the function HY(𝑘,𝑛,𝑟) is the three variables function, its dependence is by all means restricted by the values of 𝑘 variables, namely 𝑛. In the paper are calculated several continual fluxes for the contour hypercylindrical functions, in view of Ramanujan-Hardy's integrals. With the continual flux in the domain 𝑘,𝑛∈0,∞ the problem is more complex, because we must, for its defining, to perform the twofold integration (2.63). Although it is not yet calculated, we suppose that its value is very close to the discrete flux, obtained in view of the double series.

Calculating the dimensional flux by the diagonal algorithm is much simpler and faster by computer, because the total flux is now defined as the convergent graded series and does not contain as components the special functions. In any case its value is identical with the fluxes that are calculated on the basis of the series, relatively to HY-matrix columns, so there is valid the numerically verified statement thatΦ_HY(𝑘,𝑛,𝑟)=Ω_HY(𝑘,𝑛,𝑟)=Π_HY(𝑘,𝑛,𝑟)_𝑟=1=52812.977966365550. (3.1) In any case, this solution is initial for solving the other dimensional fluxes, both for hypercylindrical and hyperspherical [3, 19], in other words hypercube function [20].