images
Computer Modeling in Engineering & Sciences
images

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

Yan Hong1, Bai-Ni Guo2,* and Feng Qi3

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)=limn→∞n!nz∏k=0n(z+k),z∈C∖{0,−1,−2,…}.

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

Iν(z)=∑n=0∞1n!Γ(ν+n+1)(z2)2n+ν,z∈C, (1)

where ν∈C∖{−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(z−w+1),z,w,z−w∈C∖{−1,−2,…};0,z∈C∖{−1,−2,…},w,z−w∈{−1,−2,…}.

Concretely and explicitly, the power series expansion

[Iν(z)]2=∑k=0∞1[Γ(ν+k+1)]2(2k+2νk)(z2)2k+2ν

was listed [5], For ν∈C∖{−1,−2,…} and r,z∈C, the main result in [5], reads that

[Iν(z)]r=∑k=0∞1Γ(ν+k+1)[Γ(ν+1)]r−1Bk(ν,r)k!(z2)2k+rν, (2)

where

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

and

∑k=0∞bk+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+1r2−1ν+1r,B3(ν,r)=(ν+2)(ν+3)(ν+1)2r3−3(ν+3)(ν+1)2r2+4(ν+1)2r,B4(ν,r)=(ν+2)(ν+3)(ν+4)(ν+1)3r4−6(ν+3)(ν+4)(ν+1)3r3+(ν+4)(19ν+41)(ν+1)3(ν+2)r2−6(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)k∑j=1k(kj)j(r+1)−kΓ(ν+j+1)Γ(ν+k−j+1)Bk−j(ν,r)

in which the sequence bk(ν) for k∈N is not involved.

In [8], Theorem 2.3, alternative recursive relations

Bk(ν,r+1)=(r+1)Γ(ν+k+1)Γ(ν+1)k∑j=1k(kj)kBk−j(ν,r)Γ(ν+j+1)Γ(ν+k−j+1)

and

Bk(ν,r+1)=Bk(ν,r)+Γ(ν+k+1)Γ(ν+1)∑j=1k(kj)Bk−j(ν,r)Γ(ν+j+1)Γ(ν+k−j+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=1∞akk!zk)=∑n=0∞Bn(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 n∈N denote the zeros of Jν(z)zv, where Jν(z) is the first kind Bessel function which can be represented as

Jν(z)=(z2)ν∑n=0∞(−1)n(z/2)2nn!Γ(ν+n+1),z∈C,

where ν∈C∖{−1,−2,…} is called the order of Jν(z). The Bessel zeta function

ζν(q)=∑n=1∞1jν,nq (5)

for q > 1 was originally introduced and studied in [11–14]. 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)k−1(k−1)!rζν(2k))

and

Bk(ν,r)Γ(ν+k+1)=r∑j=0k−1(−1)kk!22j+2ζν(2j+2)(k−1j)Bk−j−1(ν,r)Γ(ν+k−j).

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 [15–17]. 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(k−1)!Γ(ν+k+1)(ν+1)Γ(ν+2)22k−2ζν+1(2k−2),k≥2 (8)

and confirms that, due to the second value in (7), the sequence bk(ν) for k∈N 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 k∈N. 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 k∈N.

Theorem 2.1. For k∈N and ν∈C∖{−1,−2,…}, the sequence bk+1(ν) can be determinantally represented as

bk+1(ν)=(2k)!!(2k−1)!!(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(i−2)!!2(i−1)/21Γ(ν+i−12+1))1≤i≤2k+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)i−j2(i−1j−1)(i−j−1)!!2(i−j)/21Γ(ν+|i−j|2+2))1≤i≤2k+1,1≤j≤2k (11)

is a (2k+1)×(2k) matrix.

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

∑k=0∞bk+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=0∞bk+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=1∞bk+1(ν)Γ(ν+k+2)1k!(t2)2k.

This implies that

b1(ν)=12(ν+1)limt→0tIν(t)Iν+1(t)−2

and

bk+1(ν)=2k−1(2k−1)!!Γ(ν+k+2)(ν+1)Γ(ν+2)limt→0[tIν(t)Iν+1(t)](2k) (12)

for k∈N. From (1), it follows that

tIν(t)Iν+1(t)=2∑k=0∞1k!Γ(ν+k+1)(t2)2k∑k=0∞1k!Γ(ν+k+2)(t2)2k→2(ν+1),t→0.

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

Let

ϕν(t)=∑k=0∞1k!Γ(ν+k+1)(t2)2k

and

φν(t)=∑k=0∞1k!Γ(ν+k+2)(t2)2k.

Then

[ϕν(t)](ℓ)=12ℓ∑k=ℓ2+1−(−1)ℓ2∞(2k)!(2k−ℓ)!k!Γ(ν+k+1)(t2)2k−ℓ→{(2m−1)!!2mΓ(ν+m+1),ℓ=2m0,ℓ=2m+1

and

[φν(t)](ℓ)=12ℓ∑k=ℓ2+1−(−1)ℓ2∞(2k)!(2k−ℓ)!k!Γ(ν+k+2)(t2)2k−ℓ→{(2m−1)!!2mΓ(ν+m+2),ℓ=2m0,ℓ=2m+1

as t→0, 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)0⋯00p′(t)q′(t)q(t)⋯00p′′(t)q′′(t)(21)q′(t)⋯00⋮⋮⋮⋱⋮⋮p(k−2)(t)q(k−2)(t)(k−21)q(k−3)(t)⋯q(t)0p(k−1)(t)q(k−1)(t)(k−11)q(k−2)(t)⋯(k−1k−2)q′(t)q(t)p(k)(t)q(k)(t)(k1)q(k−1)(t)⋯(kk−2)q′′(t)(kk−1)q′(t) |(13)

for k≥0. 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

limt→0[ tIν(t)Iν+1(t) ](2k)=2limt→0[ ϕν(t)φν(t) ](2k)=2limt→0(−1)2kφν2k+1(t)| ϕν(t)φν(t)0⋯00ϕ′ν(t)φ′ν(t)φν(t)⋯00ϕ″ν(t)φ″ν(t)(21)φ′ν(t)⋯00⋮⋮⋮⋱⋮⋮ϕν(2k−2)(t)φν(2k−2)(t)(2k−21)φν(2k−3)(t)⋯φν(t)0ϕν(2k−1)(t)φν(2k−1)(t)(2k−11)φν(2k−2)(t)⋯(2k−12k−2)φ′ν(t)φν(t)ϕν(2k)(t)φν(2k)(t)(2k1)φν(2k−1)(t)⋯(2k2k−2)φ″ν(t)(2k2k−1)φ′ν(t) |=2φν2k+1(0)| ϕν(0)φν(0)0⋯00ϕ′ν(0)φ′ν(0)φν(0)⋯00ϕ″ν(0)φ″ν(0)(21)φ′ν(0)⋯00⋮⋮⋮⋱⋮⋮ϕν(2k−2)(0)φν(2k−2)(0)(2k−21)φν(2k−3)(0)⋯φν(0)0ϕν(2k−1)(0)φν(2k−1)(0)(2k−11)φν(2k−2)(0)⋯(2k−12k−2)φ′ν(0)φν(0)ϕν(2k)(0)φν(2k)(0)(2k1)φν(2k−1)(0)⋯(2k2k−2)φ″ν(0)(2k2k−1)φ′ν(0) |=2[ Γ(ν+2) ]2k+1| 1Γ(ν+1)1Γ(ν+2)0⋯00001Γ(ν+2)⋯0012Γ(ν+2)12Γ(ν+3)0⋯00⋮⋮⋮⋱⋮⋮(2k−3)!!2k−1Γ(ν+k)(2k−3)!!2k−1Γ(ν+k+1)0⋯1Γ(ν+2)000(2k−11)(2k−3)!!2k−1Γ(ν+k+1)⋯01Γ(ν+2)(2k−1)!!2kΓ(ν+k+1)(2k−1)!!2kΓ(ν+k+2)0⋯(2k2k−2)12Γ(ν+3)0 |.

Accordingly, we acquire

bk+1(ν)=(2k)!!(2k−1)!!(ν+k+1ν+1)[ Γ(ν+2) ]2k+1ν+1×| (−1)!!201Γ(ν+1)(−1)!!201Γ(ν+2)0⋯0000(−1)!!201Γ(ν+2)⋯001!!211Γ(ν+2)1!!211Γ(ν+3)0⋯00⋮⋮⋮⋱⋮⋮(2k−3)!!2k−11Γ(ν+k)(2k−3)!!2k−11Γ(ν+k+1)0⋯(−1)!!201Γ(ν+2)000(2k−11)(2k−3)!!2k−11Γ(ν+k+1)⋯0(−1)!!201Γ(ν+2)(2k−1)!!2k1Γ(ν+k+1)(2k−1)!!2k1Γ(ν+k+2)0⋯(2k2k−2)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 k∈N, the sequence bk+1(0) can be determinantally represented as

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

where

P2k+1,1(0)=(1−(−1)i2(i−2)!!2(i−1)/21Γ(i+12))1≤i≤2k+1

and

Q2k+1,2k(0)=(1+(−1)i−j2(i−1j−1)(i−j−1)!!2(i−j)/21Γ(2+|i−j|2))1≤i≤2k+1,1≤j≤2k

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 k∈N and ν∈C∖{−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)!!(2k−1)!!(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 z∈C 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)=∏ℓ=0n−1(z+ℓ)={z(z+1)⋯(z+n−1),n≥1;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 k∈N and ν∈C∖{−1,−2,…}, the sequence bk+1(ν) can be determinantally represented as

bk+1(ν)=2k(ν+2)k(2k−1)!!×| 1(ν+1)01(ν+2)00⋯00001(ν+2)0⋯0012(ν+1)112(ν+2)10⋯00⋮⋮⋮⋱⋮⋮(2k−3)!!2k−1(ν+1)k−1(2k−3)!!2k−1(ν+2)k−10⋯1(ν+2)0000(2k−11)(2k−3)!!2k−1(ν+2)k−1⋯01(ν+2)0(2k−1)!!2k(ν+1)k(2k−1)!!2k(ν+2)k0⋯(2k2k−2)12(ν+2)10 |.(17)

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

limt→0[tIν(t)Iν+1(t)](2k)=2(ν+1)×|(−1)!!20Γ(ν+1)Γ(ν+1)(−1)!!20Γ(ν+2)Γ(ν+2)0⋯0000(−1)!!20Γ(ν+2)Γ(ν+2)⋯001!!21Γ(ν+1)Γ(ν+2)1!!21Γ(ν+2)Γ(ν+3)0⋯00⋮⋮⋮⋱⋮⋮(2k−3)!!2k−1Γ(ν+1)Γ(ν+k)(2k−3)!!2k−1Γ(ν+2)Γ(ν+k+1)0⋯(−1)!!20Γ(ν+2)Γ(ν+2)000(2k−11)(2k−3)!!2k−1Γ(ν+2)Γ(ν+k+1)⋯0(−1)!!20Γ(ν+2)Γ(ν+2)(2k−1)!!2kΓ(ν+1)Γ(ν+k+1)(2k−1)!!2kΓ(ν+2)Γ(ν+k+2)0⋯(2k2k−2)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 k∈N, the sequence bk+1(0) can be determinantally represented as

bk+1(0)=12[2(k+1)]!!(2k−1)!!×|(−1)!!0!!(−1)!!1×0!!0⋯0000(−1)!!1×0!!⋯001!!2!!1!!2×2!!0⋯00⋮⋮⋮⋱⋮⋮(2k−3)!!(2k−2)!!(2k−3)!!k(2k−2)!!0⋯(−1)!!1×0!!000(2k−11)(2k−3)!!k(2k−2)!!⋯0(−1)!!1×0!!(2k−1)!!(2k)!!(2k−1)!!(k+1)(2k)!!0⋯(2k2k−2)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 k∈N and ν∈C∖{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(2k−1)!!×| 1(ν+1)01(ν+2)00⋯00001(ν+2)0⋯0012(ν+1)112(ν+2)10⋯00⋮⋮⋮⋱⋮⋮(2k−3)!!2k−1(ν+1)k−1(2k−3)!!2k−1(ν+2)k−10⋯1(ν+2)0000(2k−11)(2k−3)!!2k−1(ν+2)k−1⋯01(ν+2)0(2k−1)!!2k(ν+1)k(2k−1)!!2k(ν+2)k0⋯(2k2k−2)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(2k−1)!!×| 1(ν+1)01(ν+2)00⋯00001(ν+2)0⋯0012(ν+1)112(ν+2)10⋯00⋮⋮⋮⋱⋮⋮(2k−3)!!2k−1(ν+1)k−1(2k−3)!!2k−1(ν+2)k−10⋯1(ν+2)0000(2k−11)(2k−3)!!2k−1(ν+2)k−1⋯01(ν+2)0(2k−1)!!2k(ν+1)k(2k−1)!!2k(ν+2)k0⋯(2k2k−2)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 k∈N.

Theorem 4.1. For k≥2 and ν∈C∖{−1,−2,…}, the sequence bk+1(ν) has the recursive relation

bk+1(ν)=kν+1−Γ(ν+k+2)Γ(ν+2)(2k−1)!!∑ℓ=2k(2k2ℓ−2)(2k−2ℓ+1)!!(2ℓ−3)!!Γ(ν+k−ℓ+3)Γ(ν+ℓ+1)bℓ(ν) (20)

Consequently, the sequence ( −1)k+12kζν(2k) for k≥2 and ν∈C∖{0,−1,−2,…} satisfies the recursive relation

(−1)k+12kζν(2k)=k(2k)!!Γ(ν+1)Γ(ν+k+1)−Γ(ν+1)∑ℓ=2k(−1)ℓ2ℓ−1ζν(2ℓ−2)(2k−2ℓ+2)!!Γ(ν+k−ℓ+2). (21)

Proof. Let D0 = 1 and

Dn=|e1,1e1,20…00e2,1e2,2e2,3…00e3,1e3,2e3,3…00⋮⋮⋮⋮⋮⋮en−2,1en−2,2en−2,3…en−2,n−10en−1,1en−1,2en−1,3…en−1,n−1en−1,nen,1en,2en,3…en,n−1en,n| (22)

for n∈N. In [23], it was proved that the Hessenberg determinant Dn for n≥0 satisfies D1 = e1, 1 and

Dn=∑r=1n(−1)n−ren,r(∏j=rn−1ej,j+1)Dr−1 (23)

for n ≥ 1, where the product is 1 if the starting index exceeds the finishing index. Replacing n by 2k + 1 for k≥0, letting

ei,1=1−(−1)i2(i−2)!!2(i−1)/21Γ(ν+i−12+1)

for 1 ≤ i ≤ 2k + 1, and taking

ei,j+1=1+(−1)i−j2(i−1j−1)(i−j−1)!!2(i−j)/21Γ(ν+|i−j|2+2)

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

D2k+1=∑r=12k+1(−1)2k−r+1e2k+1,r(∏j=r2kej,j+1)Dr−1=(−1)2ke2k+1,1(∏j=12kej,j+1)D0+∑r=22k+1(−1)2k−r+1e2k+1,r(∏j=r2kej,j+1)Dr−1=(2k−1)!!2k1Γ(ν+k+1)∏j=12k1Γ(ν+2)−∑r=22k+11+(−1)r2(2kr−2)(2k−r+1)!!2(2k−r+2)/21Γ(ν+|2k−r+2|2+2)[∏j=r2k1Γ(ν+2)]Dr−1=(2k−1)!!2kΓ(ν+k+1)[Γ(ν+2)]2k−∑ℓ=1k(2k2ℓ−2)(2k−2ℓ+1)!!2k−ℓ+1Γ(ν+k−ℓ+3)[Γ(ν+2)]2k−2ℓ+1D2ℓ−1=(2k−1)!!2kΓ(ν+k+1)[Γ(ν+2)]2k−(2k−1)!!2kΓ(ν+k+2)[Γ(ν+2)]2k−1D1−∑ℓ=2k(2k2ℓ−2)(2k−2ℓ+1)!!2k−ℓ+1Γ(ν+k−ℓ+3)[Γ(ν+2)]2k−2ℓ+1D2ℓ−1=(2k−1)!!2kΓ(ν+k+1)[Γ(ν+2)]2k−(2k−1)!!2kΓ(ν+k+2)[Γ(ν+2)]2k−1Γ(ν+1)−∑ℓ=2k(2k2ℓ−2)(2k−2ℓ+1)!!2k−ℓ+1Γ(ν+k−ℓ+3)[Γ(ν+2)]2k−2ℓ+1D2ℓ−1=(2k−1)!!k2kΓ(ν+k+2)[Γ(ν+2)]2k−∑ℓ=2k(2k2ℓ−2)(2k−2ℓ+1)!!2k−ℓ+1Γ(ν+k−ℓ+3)[Γ(ν+2)]2k−2ℓ+1D2ℓ−1

for k≥2. Further setting

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

for k∈N produces

(2k−1)!!(2k)!!1(ν+k+1k)ν+1[Γ(ν+2)]2k+1bk+1(ν)=(2k−1)!!k2kΓ(ν+k+2)[Γ(ν+2)]2k−∑ℓ=2k(2k2ℓ−2)(2k−2ℓ+1)!!2k−ℓ+1Γ(ν+k−ℓ+3)[Γ(ν+2)]2k−2ℓ+1(2ℓ−3)!!(2ℓ−2)!!1(ν+ℓℓ−1)ν+1[Γ(ν+2)]2ℓ−1bℓ(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+1−Γ(ν+k+2)(2k−1)!!∑ℓ=2k(2k2ℓ−2)(2k−2ℓ+1)!!(2ℓ−3)!!Γ(ν+k−ℓ+3)(−1)ℓ(ℓ−1)!ν+122ℓ−2ζν+1(2ℓ−2)

which can be rearranged as

(−1)k+12kζν+1(2k)=k(2k)!!Γ(ν+2)Γ(ν+k+2)−Γ(ν+2)∑ℓ=2k(−1)ℓ2ℓ−1ζν+1(2ℓ−2)(2k−2ℓ+2)!!Γ(ν+k−ℓ+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 k∈N, 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.

References

 1.  Temme, N. M. (1996). Special functions: An introduction to classical functions of mathematical physics. New York: A Wiley-Interscience Publication, John Wiley & Sons, Inc. DOI 10.1002/9781118032572.

 2.  Qi, F., Guo, B. N. (2021). From inequalities involving exponential functions and sums to logarithmically complete monotonicity of ratios of gamma functions. Journal of Mathematical Analysis and Applications, 493(1), 19. DOI 10.1016/j.jmaa.2020.124478.

 3.  Qi, F., Li, W. H., Yu, S. B., Du, X. Y., Guo, B. N. (2021). A ratio of finitely many gamma functions and its properties with applications. Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales Serie A Matemáticas, 115(2), 14. DOI 10.1007/s13398-020-00988-z.

 4.  Abramowitz, M., Stegun, I. A. (1972). Handbook of mathematical functions with formulas, graphs, and mathematical tables. National bureau of standards, applied mathematics series, vol. 55. Mineola, New York: Dover Publications.

 5.  Bender, C. M., Brody, D. C., Meister, B. K. (2003). On powers of Bessel functions. Journal of Mathematical Physics, 44(1), 309–314. DOI 10.1063/1.1526940.

 6.  Guo, B. N., Lim, D., Qi, F. (2021). Series expansions of powers of arcsine, closed forms for special values of Bell polynomials, and series representations of generalized logsine functions. AIMS Mathematics, 6(7), 7494–7517. DOI 10.3934/math.2021438.

 7.  Baricz, Á. (2010). Powers of modified Bessel functions of the first kind. Applied Mathematics Letters, 23(6), 722–724. DOI 10.1016/j.aml.2010.02.015.

 8.  Moll, V. H., Vignat, C. (2014). On polynomials connected to powers of Bessel functions. International Journal of Number Theory, 10(5), 1245–1257. DOI 10.1142/S1793042114500249.

 9.  Comtet, L. (1974). Advanced combinatorics: The art of finite and infinite expansions, Revised and Enlarged Edition. Dordrecht, Netherlands: Reidel Publishing Co. DOI 10.1007/978-94-010-2196-8.

10. Qi, F., Niu, D. W., Lim, D., Yao, Y. H. (2020). Special values of the Bell polynomials of the second kind for some sequences and functions. Journal of Mathematical Analysis and Applications, 491(2), 31. DOI 10.1016/j.jmaa.2020.124382.

11. Kishore, N. (1964). The Rayleigh polynomial. Proceedings of the American Mathematical Society, 15(6), 911–917. DOI 10.1090/S0002-9939-1964-0168823-2.

12. Kishore, N. (1964). A structure of the Rayleigh polynomial. Duke Mathematical Journal, 31(3), 513–518. DOI 10.1215/S0012-7094-64-03150-3.

13. Kishore, N. (1965). Binary property of the Rayleigh polynomial. Duke Mathematical Journal, 32(3), 429–435. DOI 10.1215/S0012-7094-65-03243-6.

14. Kishore, N. (1968). Congruence properties of the Rayleigh functions and polynomials. Duke Mathematical Journal, 35, 557–562. DOI 10.1215/S0012-7094-68-03557-6.

15. Bakker, M., Temme, N. M. (1984). Sum rule for products of Bessel functions: Comments on a paper by Newberger. Journal of Mathematical Physics, 25(5), 1266–1267. DOI 10.1063/1.526282.

16. Newberger, B. S. (1983). Erratum: New sum rule for products of Bessel functions with application to plasma physics. Journal of Mathematical Physics, 24(8), 2250. 10.1063/1.525940.

17. Newberger, B. S. (1982). New sum rule for products of Bessel functions with application to plasma physics. Journal of Mathematical Physics, 23(7), 1278–1281. DOI 10.1063/1.525510.

18. Howard, F. T. (1985). Integers related to the Bessel function J1(z). Fibonacci Quarterly, 23(3), 249–257.

19. Bourbaki, N. (2004). Elements of mathematics: Functions of a real variable: Elementary theory, translated from the 1976 french original by Philip Spain. Elements of mathematics (Berlin). Berlin: Springer-Verlag. DOI 10.1007/978-3-642-59315-4.

20. Qi, F., Huang, C. J. (2020). Computing sums in terms of beta, polygamma, and Gauss hypergeometric functions. Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales Serie A Matemáticas, 114(1), 191. DOI 10.1007/s13398-020-00927-y.

21. Qi, F., Kouba, O., Kaddoura, I. (2020). Computation of several Hessenberg determinants. Mathematica Slovaca, 70(6), 1521–1537. DOI 10.1515/ms-2017-0445.

22. Qi, F., Niu, D. W., Lim, D., Guo, B. N. (2020). Closed formulas and identities for the Bell polynomials and falling factorials. Contributions to Discrete Mathematics, 15(1), 163–174. DOI 10.11575/cdm.v15i1.68111.

23. Cahill, N. D., D’Errico, J. R., Narayan, D. A., Narayan, J. Y. (2002). Fibonacci determinants. The College Mathematics Journal, 33(3), 221–225. DOI 10.1080/07468342.2002.11921945.

images
This work is licensed under a Creative Commons Attribution 4.0 International License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.