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

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

1  Introduction and Motivations

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

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

Concretely and explicitly, the power series expansion

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:

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

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

In [8], Theorem 2.3, alternative recursive relations

and

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

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

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

and

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

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

This implies that

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

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

Let

and

Then

and

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

Accordingly, we acquire

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

and

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

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

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)