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 Computer Modeling in Engineering & Sciences

DOI: 10.32604/cmes.2021.016431

ARTICLE

Determinantal Expressions and Recursive Relations for the Bessel Zeta Function and for a Sequence Originating from a Series Expansion of the Power of Modified Bessel Function of the First Kind

1College of Mathematics and Physics, Inner Mongolia University for Nationalities, Tongliao, 028043, China
2School of Mathematics and Informatics, Henan Polytechnic University, Jiaozuo, 454003, China
3School of Mathematical Sciences, Tiangong University, Tianjin, 300387, China
*Corresponding Author: Bai-Ni Guo. Email: bai.ni.guo@gmail.com Dedicated to Professor Dr. Mourad E. H. Ismail at University of Central Florida in USA
Received: 05 March 2021; Accepted: 07 June 2021

Abstract: In the paper, by virtue of a general formula for any derivative of the ratio of two differentiable functions, with the aid of a recursive property of the Hessenberg determinants, the authors establish determinantal expressions and recursive relations for the Bessel zeta function and for a sequence originating from a series expansion of the power of modified Bessel function of the first kind.

Keywords: Determinantal representation; recursive relation; series expansion; first kind modified Bessel function; Bessel zeta function; Pochhammer symbol; gamma function; Hessenberg determinant

1  Introduction and Motivations

Recall from [1], and [2,3] that the classical Euler gamma function Γ(z) is defined by

Γ(z)=limnn!nzk=0n(z+k),zC{0,1,2,}.

Recall from [4], that the modified Bessel function of the first kind Iν(z) is represented by

Iν(z)=n=01n!Γ(ν+n+1)(z2)2n+ν,zC, (1)

where νC{1,2,} is said to be the order of Iν(z). Recall from [1], that the generalized combinatorial number (or say, generalized binomial coefficient) is denoted and defined by

(zw)={Γ(z+1)Γ(w+1)Γ(zw+1),z,w,zwC{1,2,};0,zC{1,2,},w,zw{1,2,}.

Concretely and explicitly, the power series expansion

[Iν(z)]2=k=01[Γ(ν+k+1)]2(2k+2νk)(z2)2k+2ν

was listed [5], For νC{1,2,} and r,zC, the main result in [5], reads that

[Iν(z)]r=k=01Γ(ν+k+1)[Γ(ν+1)]r1Bk(ν,r)k!(z2)2k+rν, (2)

where

Bk(ν,r)=r(ν+k)ν+1Bk1(ν,r)+Γ(ν+2)kj=2kbj(ν)Γ(ν+j+2)(ν+kj)Bkj(ν,r) (3)

and

k=0bk+1(ν)Γ(ν+k+2)xkk!=1Γ(ν+2)[xν+1Iν(2x)Iν+1(2x)2] (4)

with the convention that the sum is zero if the starting index exceeds the finishing index. By the way, in the paper [6], there are new conclusions and applications on series expansions of powers of several fundamental elementary functions. In [5], the first five expressions for Bk(ν, r) were listed as follows:

B0(ν,r)=1,B1(ν,r)=r,B2(ν,r)=ν+2ν+1r21ν+1r,B3(ν,r)=(ν+2)(ν+3)(ν+1)2r33(ν+3)(ν+1)2r2+4(ν+1)2r,B4(ν,r)=(ν+2)(ν+3)(ν+4)(ν+1)3r46(ν+3)(ν+4)(ν+1)3r3+(ν+4)(19ν+41)(ν+1)3(ν+2)r26(5ν+11)(ν+1)3(ν+2)r.

In [7], the recursive relation (3) was simplified as

Bk(ν,r)=Γ(ν+k+1)Γ(ν+1)kj=1k(kj)j(r+1)kΓ(ν+j+1)Γ(ν+kj+1)Bkj(ν,r)

in which the sequence bk(ν) for kN is not involved.

In [8], Theorem 2.3, alternative recursive relations

Bk(ν,r+1)=(r+1)Γ(ν+k+1)Γ(ν+1)kj=1k(kj)kBkj(ν,r)Γ(ν+j+1)Γ(ν+kj+1)

and

Bk(ν,r+1)=Bk(ν,r)+Γ(ν+k+1)Γ(ν+1)j=1k(kj)Bkj(ν,r)Γ(ν+j+1)Γ(ν+kj+1)

were derived via a probabilistic interpretation of the series expansion of powers of a general series.

In [9], the complete Bell polynomials, denoted by Bn(a1,a2,,an), are defined by

exp(k=1akk!zk)=n=0Bn(a1,a2,,an)znn!.

By the way, in the article [10], some new results on the Bell polynomials of the second kind are surveyed and reviewed. Let jν,n for nN denote the zeros of Jν(z)zv, where Jν(z) is the first kind Bessel function which can be represented as

Jν(z)=(z2)νn=0(1)n(z/2)2nn!Γ(ν+n+1),zC,

where νC{1,2,} is called the order of Jν(z). The Bessel zeta function

ζν(q)=n=11jν,nq (5)

for q > 1 was originally introduced and studied in [1114]. In [8], there are the following special values:

ζν(2)=14(ν+1),ζν(4)=116(ν+1)3,ζν(6)=116(ν+1)4(2ν+3),ζν(8)=10ν+11256(ν+1)6(2ν2+7ν+6). (6)

Theorems 3.1 and 3.2 in [8] read that

Bk(ν,r)=22kΓ(ν+k+1)Γ(ν+1)Bk(0!rζν(2),1!rζν(4),,(1)k1(k1)!rζν(2k))

and

Bk(ν,r)Γ(ν+k+1)=rj=0k1(1)kk!22j+2ζν(2j+2)(k1j)Bkj1(ν,r)Γ(ν+kj).

Corollary 4.2 in [8] confirms that Bk(ν, r) is a polynomial in r of degree k.

One of the reasons why ones investigated the series expansion (2) and the sequence Bk(ν, r) is that the products of the first kind Bessel functions and of the first kind modified Bessel functions appear frequently in problems of statistical mechanics and plasma physics considered in [1517]. This reason has been mentioned in [5,7].

In the papers [8,18] and in Entry A131490 of The On-Line Encyclopedia of Integer Sequences, the sequence bk+1(ν) generated in (4) has been studied. In [8], there are two concrete values

b1(ν)=1andb2(ν)=1ν+1. (7)

Theorem 5.1 in [8] reads that

(1)k+1bk+1(0)=k!(k+1)!22kζ1(2k).

Corollary 5.2 in [8] asserts that the number bk+1(0) is an integer. Theorem 5.4 in [8] reads that

bk(ν)=(1)k(k1)!Γ(ν+k+1)(ν+1)Γ(ν+2)22k2ζν+1(2k2),k2 (8)

and confirms that, due to the second value in (7), the sequence bk(ν) for kN is not an integer sequence.

In this paper, we will establish determinantal expressions and recursive relations of the sequences bk+1(ν), bk+1(0), and ζν(2k) for kN. It is clear that, if all elements in determinants are closed forms, determinantal expressions are also closed forms.

2  Determinantal Representations via Ratios of Gamma Functions

We are now in a position to establish determinantal expressions of the sequences bk+1(ν), bk+1(0), and ζν(2k) for kN.

Theorem 2.1. For kN and νC{1,2,}, the sequence bk+1(ν) can be determinantally represented as

bk+1(ν)=(2k)!!(2k1)!!(ν+k+1k)[Γ(ν+2)]2k+1ν+1|P2k+1,1(ν)Q2k+1,2k(ν)|(2k+1)×(2k+1), (9)

where

P2k+1,1(ν)=(1(1)i2(i2)!!2(i1)/21Γ(ν+i12+1))1i2k+1 (10)

with ( −1)!! = 1 is a (2k + 1) × 1 matrix and, with the convention (nm)= 0 for n < m,

Q2k+1,2k(ν)=(1+(1)ij2(i1j1)(ij1)!!2(ij)/21Γ(ν+|ij|2+2))1i2k+1,1j2k (11)

is a (2k+1)×(2k) matrix.

Proof. Replacing 2x by t in the power series expansion (4) yields

k=0bk+1(ν)Γ(ν+k+2)1k!(t2)2k=1Γ(ν+2)[t2(ν+1)Iν(t)Iν+1(t)2],tIν(t)Iν+1(t)=4(ν+1)+2(ν+1)Γ(ν+2)k=0bk+1(ν)Γ(ν+k+2)1k!(t2)2k,tIν(t)Iν+1(t)=2(ν+1)[2+b1(ν)]+2(ν+1)Γ(ν+2)k=1bk+1(ν)Γ(ν+k+2)1k!(t2)2k.

This implies that

b1(ν)=12(ν+1)limt0tIν(t)Iν+1(t)2

and

bk+1(ν)=2k1(2k1)!!Γ(ν+k+2)(ν+1)Γ(ν+2)limt0[tIν(t)Iν+1(t)](2k) (12)

for kN. From (1), it follows that

tIν(t)Iν+1(t)=2k=01k!Γ(ν+k+1)(t2)2kk=01k!Γ(ν+k+2)(t2)2k2(ν+1),t0.

Hence, we obtain b1(ν)=1, which confirms the first value in (7).

Let

ϕν(t)=k=01k!Γ(ν+k+1)(t2)2k

and

φν(t)=k=01k!Γ(ν+k+2)(t2)2k.

Then

[ϕν(t)]()=12k=2+1(1)2(2k)!(2k)!k!Γ(ν+k+1)(t2)2k{(2m1)!!2mΓ(ν+m+1),=2m0,=2m+1

and

[φν(t)]()=12k=2+1(1)2(2k)!(2k)!k!Γ(ν+k+2)(t2)2k{(2m1)!!2mΓ(ν+m+2),=2m0,=2m+1

as t0, where m{0}N and ( −1)!! = 1.

In [19], there exists a general formula

dkdtk[ p(t)q(t) ]=(1)kqk+1(t)| p(t)q(t)000p(t)q(t)q(t)00p(t)q(t)(21)q(t)00p(k2)(t)q(k2)(t)(k21)q(k3)(t)q(t)0p(k1)(t)q(k1)(t)(k11)q(k2)(t)(k1k2)q(t)q(t)p(k)(t)q(k)(t)(k1)q(k1)(t)(kk2)q(t)(kk1)q(t) |(13)

for k0. By the way, this formula has been extensively applied in recent years, see [20,21] and closely related references therein. Applying p(t) and q(t) in (13) to ϕν(t) and φν(t) results in

limt0[ tIν(t)Iν+1(t) ](2k)=2limt0[ ϕν(t)φν(t) ](2k)=2limt0(1)2kφν2k+1(t)| ϕν(t)φν(t)000ϕν(t)φν(t)φν(t)00ϕν(t)φν(t)(21)φν(t)00ϕν(2k2)(t)φν(2k2)(t)(2k21)φν(2k3)(t)φν(t)0ϕν(2k1)(t)φν(2k1)(t)(2k11)φν(2k2)(t)(2k12k2)φν(t)φν(t)ϕν(2k)(t)φν(2k)(t)(2k1)φν(2k1)(t)(2k2k2)φν(t)(2k2k1)φν(t) |=2φν2k+1(0)| ϕν(0)φν(0)000ϕν(0)φν(0)φν(0)00ϕν(0)φν(0)(21)φν(0)00ϕν(2k2)(0)φν(2k2)(0)(2k21)φν(2k3)(0)φν(0)0ϕν(2k1)(0)φν(2k1)(0)(2k11)φν(2k2)(0)(2k12k2)φν(0)φν(0)ϕν(2k)(0)φν(2k)(0)(2k1)φν(2k1)(0)(2k2k2)φν(0)(2k2k1)φν(0) |=2[ Γ(ν+2) ]2k+1| 1Γ(ν+1)1Γ(ν+2)000001Γ(ν+2)0012Γ(ν+2)12Γ(ν+3)000(2k3)!!2k1Γ(ν+k)(2k3)!!2k1Γ(ν+k+1)01Γ(ν+2)000(2k11)(2k3)!!2k1Γ(ν+k+1)01Γ(ν+2)(2k1)!!2kΓ(ν+k+1)(2k1)!!2kΓ(ν+k+2)0(2k2k2)12Γ(ν+3)0 |.

Accordingly, we acquire

bk+1(ν)=(2k)!!(2k1)!!(ν+k+1ν+1)[ Γ(ν+2) ]2k+1ν+1×| (1)!!201Γ(ν+1)(1)!!201Γ(ν+2)00000(1)!!201Γ(ν+2)001!!211Γ(ν+2)1!!211Γ(ν+3)000(2k3)!!2k11Γ(ν+k)(2k3)!!2k11Γ(ν+k+1)0(1)!!201Γ(ν+2)000(2k11)(2k3)!!2k11Γ(ν+k+1)0(1)!!201Γ(ν+2)(2k1)!!2k1Γ(ν+k+1)(2k1)!!2k1Γ(ν+k+2)0(2k2k2)1!!211Γ(ν+3)0 |

which can be rearranged as the form in (9). The proof of Theorem 2.1 is complete.

Theorem 2.2. For kN, the sequence bk+1(0) can be determinantally represented as

bk+1(0)k+1=(2k)!!(2k1)!!|P2k+1,1(0)Q2k+1,2k(0)|(2k+1)×(2k+1), (14)

where

P2k+1,1(0)=(1(1)i2(i2)!!2(i1)/21Γ(i+12))1i2k+1

and

Q2k+1,2k(0)=(1+(1)ij2(i1j1)(ij1)!!2(ij)/21Γ(2+|ij|2))1i2k+1,1j2k

with ( −1)!! = 1 and the convention (nm)=0 for n ¡ m.

Proof. This can be deduced from taking ν= 0 in Theorem 2.1 and reformulating it for intuitive and visual beauty.

Theorem 2.3. For kN and νC{1,2,}, the values at q = 2k of the Bessel zeta function ζν+1(q) can be determinantally represented as

ζν+1(2k)=(1)k+1[Γ(ν+2)]2k+1(2k)!|P2k+1,1(ν)Q2k+1,2k(ν)|(2k+1)×(2k+1), (15)

where the matrices P2k+1, 1(ν) and Q2k+1, 2k(ν) are defined by (10) and (11), respectively.

Proof. Combining (8) with (9) in Theorem 2.1 results in

bk+1(ν)=(1)k+1k!Γ(ν+k+2)(ν+1)Γ(ν+2)22kζν+1(2k)=(2k)!!(2k1)!!(ν+k+1k)[Γ(ν+2)]2k+1ν+1|P2k+1,1(ν)Q2k+1,2k(ν)|(2k+1)×(2k+1).

Further simplifying gives (15). The proof of Theorem 2.3 is complete.

3  Determinantal Representations via the Pochhammer Symbols

For zC and n{0}N, the Pochhammer symbol (z)n, or say, the rising factorial (z)n, is defined in [6,22] and [1], by

(z)n=Γ(z+n)Γ(z)==0n1(z+)={z(z+1)(z+n1),n1;1,n=0. (16)

In terms of the Pochhammer symbol (z)n defined by (16), we can rewrite Theorems 2.1–2.3 for intuitive and visual beauty respectively as follows:

Theorem 3.1. For kN and νC{1,2,}, the sequence bk+1(ν) can be determinantally represented as

bk+1(ν)=2k(ν+2)k(2k1)!!×| 1(ν+1)01(ν+2)0000001(ν+2)00012(ν+1)112(ν+2)1000(2k3)!!2k1(ν+1)k1(2k3)!!2k1(ν+2)k101(ν+2)0000(2k11)(2k3)!!2k1(ν+2)k101(ν+2)0(2k1)!!2k(ν+1)k(2k1)!!2k(ν+2)k0(2k2k2)12(ν+2)10 |.(17)

Proof. In the proof of Theorem 2.1, we can write

limt0[tIν(t)Iν+1(t)](2k)=2(ν+1)×|(1)!!20Γ(ν+1)Γ(ν+1)(1)!!20Γ(ν+2)Γ(ν+2)00000(1)!!20Γ(ν+2)Γ(ν+2)001!!21Γ(ν+1)Γ(ν+2)1!!21Γ(ν+2)Γ(ν+3)000(2k3)!!2k1Γ(ν+1)Γ(ν+k)(2k3)!!2k1Γ(ν+2)Γ(ν+k+1)0(1)!!20Γ(ν+2)Γ(ν+2)000(2k11)(2k3)!!2k1Γ(ν+2)Γ(ν+k+1)0(1)!!20Γ(ν+2)Γ(ν+2)(2k1)!!2kΓ(ν+1)Γ(ν+k+1)(2k1)!!2kΓ(ν+2)Γ(ν+k+2)0(2k2k2)1!!21Γ(ν+2)Γ(ν+3)0|.

Substituting this equation into (12) and considering the definition in (16) lead to (17). The proof of Theorem 3.1 is complete.

Theorem 3.2. For kN, the sequence bk+1(0) can be determinantally represented as

bk+1(0)=12[2(k+1)]!!(2k1)!!×|(1)!!0!!(1)!!1×0!!00000(1)!!1×0!!001!!2!!1!!2×2!!000(2k3)!!(2k2)!!(2k3)!!k(2k2)!!0(1)!!1×0!!000(2k11)(2k3)!!k(2k2)!!0(1)!!1×0!!(2k1)!!(2k)!!(2k1)!!(k+1)(2k)!!0(2k2k2)1!!2×2!!0|, (18)

where ( −1)!! = 1.

Proof. This can be deduced from letting ν= 0 in Theorem 3.1 and reformulating it for intuitive and visual beauty.

Theorem 3.3. For kN and νC{0,1,2,}, the sequence ζν(2k) can be determinantally represented as

ζν(2k)=(1)k+1ν(2k)!

bk+1(ν)=(1)k+1k!Γ(ν+k+2)(ν+1)Γ(ν+2)22kζν+1(2k)=2k(ν+2)k(2k1)!!×| 1(ν+1)01(ν+2)0000001(ν+2)00012(ν+1)112(ν+2)1000(2k3)!!2k1(ν+1)k1(2k3)!!2k1(ν+2)k101(ν+2)0000(2k11)(2k3)!!2k1(ν+2)k101(ν+2)0(2k1)!!2k(ν+1)k(2k1)!!2k(ν+2)k0(2k2k2)12(ν+2)10 |. (19)

Proof. Combining (8) with (17) in Theorem 3.1 results in

bk+1(ν)=(1)k+1k!Γ(ν+k+2)(ν+1)Γ(ν+2)22kζν+1(2k)=2k(ν+2)k(2k1)!!×| 1(ν+1)01(ν+2)0000001(ν+2)00012(ν+1)112(ν+2)1000(2k3)!!2k1(ν+1)k1(2k3)!!2k1(ν+2)k101(ν+2)0000(2k11)(2k3)!!2k1(ν+2)k101(ν+2)0(2k1)!!2k(ν+1)k(2k1)!!2k(ν+2)k0(2k2k2)12(ν+2)10 |.

Further simplifying gives (19). The proof of Theorem 3.3 is complete.

4  Recursive Relations

In this section, we establish recursive relations of the sequences bk+1(ν) and ( −1)k+12kζν(2k) for kN.

Theorem 4.1. For k2 and νC{1,2,}, the sequence bk+1(ν) has the recursive relation

bk+1(ν)=kν+1Γ(ν+k+2)Γ(ν+2)(2k1)!!=2k(2k22)(2k2+1)!!(23)!!Γ(ν+k+3)Γ(ν++1)b(ν) (20)

Consequently, the sequence ( −1)k+12kζν(2k) for k2 and νC{0,1,2,} satisfies the recursive relation

(1)k+12kζν(2k)=k(2k)!!Γ(ν+1)Γ(ν+k+1)Γ(ν+1)=2k(1)21ζν(22)(2k2+2)!!Γ(ν+k+2). (21)

Proof. Let D0 = 1 and

Dn=|e1,1e1,2000e2,1e2,2e2,300e3,1e3,2e3,300en2,1en2,2en2,3en2,n10en1,1en1,2en1,3en1,n1en1,nen,1en,2en,3en,n1en,n| (22)

for nN. In [23], it was proved that the Hessenberg determinant Dn for n0 satisfies D1 = e1, 1 and

Dn=r=1n(1)nren,r(j=rn1ej,j+1)Dr1 (23)

for n 1, where the product is 1 if the starting index exceeds the finishing index. Replacing n by 2k + 1 for k0, letting

ei,1=1(1)i2(i2)!!2(i1)/21Γ(ν+i12+1)

for 1 i 2k + 1, and taking

ei,j+1=1+(1)ij2(i1j1)(ij1)!!2(ij)/21Γ(ν+|ij|2+2)

for 1 i 2k + 1 and 1 j 2k in (23) yield

D2k+1=r=12k+1(1)2kr+1e2k+1,r(j=r2kej,j+1)Dr1=(1)2ke2k+1,1(j=12kej,j+1)D0+r=22k+1(1)2kr+1e2k+1,r(j=r2kej,j+1)Dr1=(2k1)!!2k1Γ(ν+k+1)j=12k1Γ(ν+2)r=22k+11+(1)r2(2kr2)(2kr+1)!!2(2kr+2)/21Γ(ν+|2kr+2|2+2)[j=r2k1Γ(ν+2)]Dr1=(2k1)!!2kΓ(ν+k+1)[Γ(ν+2)]2k=1k(2k22)(2k2+1)!!2k+1Γ(ν+k+3)[Γ(ν+2)]2k2+1D21=(2k1)!!2kΓ(ν+k+1)[Γ(ν+2)]2k(2k1)!!2kΓ(ν+k+2)[Γ(ν+2)]2k1D1=2k(2k22)(2k2+1)!!2k+1Γ(ν+k+3)[Γ(ν+2)]2k2+1D21=(2k1)!!2kΓ(ν+k+1)[Γ(ν+2)]2k(2k1)!!2kΓ(ν+k+2)[Γ(ν+2)]2k1Γ(ν+1)=2k(2k22)(2k2+1)!!2k+1Γ(ν+k+3)[Γ(ν+2)]2k2+1D21=(2k1)!!k2kΓ(ν+k+2)[Γ(ν+2)]2k=2k(2k22)(2k2+1)!!2k+1Γ(ν+k+3)[Γ(ν+2)]2k2+1D21

for k2. Further setting

D2k+1=(2k1)!!(2k)!!1(ν+k+1k)ν+1[Γ(ν+2)]2k+1bk+1(ν)

for kN produces

(2k1)!!(2k)!!1(ν+k+1k)ν+1[Γ(ν+2)]2k+1bk+1(ν)=(2k1)!!k2kΓ(ν+k+2)[Γ(ν+2)]2k=2k(2k22)(2k2+1)!!2k+1Γ(ν+k+3)[Γ(ν+2)]2k2+1(23)!!(22)!!1(ν+1)ν+1[Γ(ν+2)]21b(ν)

which can be simplified as (20).

Substituting (8) into (20) produces

(1)k+1k!Γ(ν+k+2)(ν+1)Γ(ν+2)22kζν+1(2k)=kν+1Γ(ν+k+2)(2k1)!!=2k(2k22)(2k2+1)!!(23)!!Γ(ν+k+3)(1)(1)!ν+1222ζν+1(22)

which can be rearranged as

(1)k+12kζν+1(2k)=k(2k)!!Γ(ν+2)Γ(ν+k+2)Γ(ν+2)=2k(1)21ζν+1(22)(2k2+2)!!Γ(ν+k+3).

The recursive relation (21) is thus proved. The proof of Theorem 4.1 is complete.

5  More Numerical Computation of the First Few Values

Via newly-established determinantal expressions (9), (14), (15), (17)(19), with the aid of the famous software Mathematica version 12.0, we numerically compute more special values of the sequences bk+1(ν), bk+1(0), and ζν(2k) for kN, which are supplements of those listed in (6) and (7), as follows:

ζν(4)=116(ν+1)2(ν+2),ζν(6)=132(ν+1)3(ν+2)(ν+3),ζν(8)=5ν+11256(ν+1)4(ν+2)2(ν+3)(ν+4),b3(ν)=2(ν+1)(ν+2),b4(ν)=12(ν+1)(ν+2)2,b5(ν)=24(5ν+16)(ν+1)(ν+2)3(ν+3).

We notice that the numerical computation of ζν(4), ζν(6), and ζν(8) here correct corresponding ones listed in (6).

Using the famous software Mathematica version 12.0, we plotted graphs of ζν(2k) for 1 k 4 on the interval ( −1, 9) in Fig. 1.

Figure 1: Graphs of ζν(2k) for 1 k 4 on the interval ( −1, 9)

6  Conclusions

In this paper, by virtue of a general formula (13) for derivatives of the ratio of two differentiable functions and with the aid of a recursive property (23) of the Hessenberg determinants (22), we establish six determinantal expressions (9), (14), (15), (17)(19), find two recursive relations (20) and (21) for the sequence bk+1(ν) defined by (4) and for the Bessel zeta function ζν(2k) defined by (5).

Acknowledgement: The authors thank 1. Jiaying Chen and Geng Li (Undergraduates Enrolled in 2018 at School of Mathematical Sciences, Tianjin Polytechnic University, China), for their valuable help downloading the papers [5,8,17] on 27 January 2021. 2. Christophe Vignat (Universite d’Orsay, France; Tulane University, USA; cvignat@tulane.edu) for his sending electronic version of the paper [8] on 28 January 2021. 3. Anonymous referees for their careful reading of, helpful suggestions to, and valuable comments on the original version of this paper.

Funding Statement: The first author, Mrs. Yan Hong, was partially supported by the Natural Science Foundation of Inner Mongolia (Grant No. 2019MS01007), by the Science Research Fund of Inner Mongolia University for Nationalities (Grant No. NMDBY15019), and by the Foundation of the Research Program of Science and Technology at Universities of Inner Mongolia Autonomous Region (Grant Nos. NJZY19157 and NJZY20119) in China.

Conflicts of Interest: The authors declare that they have no conflicts of interest to report regarding the present study.

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