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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: email@example.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 , and [2,3] that the classical Euler gamma function (z) is defined by
Recall from , that the modified Bessel function of the first kind (z) is represented by
where is said to be the order of (z). Recall from , that the generalized combinatorial number (or say, generalized binomial coefficient) is denoted and defined by
Concretely and explicitly, the power series expansion
was listed , For and , the main result in , reads that
with the convention that the sum is zero if the starting index exceeds the finishing index. By the way, in the paper , there are new conclusions and applications on series expansions of powers of several fundamental elementary functions. In , the first five expressions for Bk(, r) were listed as follows:
In , the recursive relation (3) was simplified as
in which the sequence bk() for is not involved.
In , Theorem 2.3, alternative recursive relations
were derived via a probabilistic interpretation of the series expansion of powers of a general series.
In , the complete Bell polynomials, denoted by , are defined by
By the way, in the article , some new results on the Bell polynomials of the second kind are surveyed and reviewed. Let for denote the zeros of , where (z) is the first kind Bessel function which can be represented as
where is called the order of (z). The Bessel zeta function
for q > 1 was originally introduced and studied in [11–14]. In , there are the following special values:
Theorems 3.1 and 3.2 in  read that
Corollary 4.2 in  confirms that Bk(, r) is a polynomial in r of degree k.
One of the reasons why ones investigated the series expansion (2) and the sequence Bk(, 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() generated in (4) has been studied. In , there are two concrete values
Theorem 5.1 in  reads that
Corollary 5.2 in  asserts that the number bk+1(0) is an integer. Theorem 5.4 in  reads that
and confirms that, due to the second value in (7), the sequence bk() for is not an integer sequence.
In this paper, we will establish determinantal expressions and recursive relations of the sequences bk+1(), bk+1(0), and (2k) for . 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(), bk+1(0), and (2k) for .
Theorem 2.1. For and , the sequence can be determinantally represented as
with ( −1)!! = 1 is a (2k + 1) 1 matrix and, with the convention 0 for n < m,
is a matrix.
Proof. Replacing by t in the power series expansion (4) yields
This implies that
for . From (1), it follows that
Hence, we obtain , which confirms the first value in (7).
as , where and ( −1)!! = 1.
In , there exists a general formula
for . 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 (t) and (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 , the sequence bk+1(0) can be determinantally represented as
with ( −1)!! = 1 and the convention for n ¡ m.
Proof. This can be deduced from taking 0 in Theorem 2.1 and reformulating it for intuitive and visual beauty.
Theorem 2.3. For and , the values at q = 2k of the Bessel zeta function can be determinantally represented as
where the matrices P2k+1, 1() and Q2k+1, 2k() 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 and , the Pochhammer symbol (z)n, or say, the rising factorial (z)n, is defined in [6,22] and , by
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 and , the sequence bk+1() can be determinantally represented as
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 , the sequence bk+1(0) can be determinantally represented as
where ( −1)!! = 1.
Proof. This can be deduced from letting 0 in Theorem 3.1 and reformulating it for intuitive and visual beauty.
Theorem 3.3. For and , the sequence (2k) can be determinantally represented as
Proof. Combining (8) with (17) in Theorem 3.1 results in
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() and ( −1)k+12(2k) for .
Theorem 4.1. For and , the sequence bk+1() has the recursive relation
Consequently, the sequence ( −1)k+12(2k) for and satisfies the recursive relation
Proof. Let D0 = 1 and
for . In , it was proved that the Hessenberg determinant Dn for satisfies D1 = e1, 1 and
for n 1, where the product is 1 if the starting index exceeds the finishing index. Replacing n by 2k + 1 for , letting
for 1 i 2k + 1, and taking
for 1 i 2k + 1 and 1 j 2k in (23) yield
for . Further setting
which can be simplified as (20).
Substituting (8) into (20) produces
which can be rearranged as
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(), bk+1(0), and (2k) for , which are supplements of those listed in (6) and (7), as follows:
We notice that the numerical computation of (4), (6), and (8) here correct corresponding ones listed in (6).
Using the famous software Mathematica version 12.0, we plotted graphs of (2k) for 1 k 4 on the interval ( −1, 9) in Fig. 1.
Figure 1: Graphs of (2k) for 1 k 4 on the interval ( −1, 9)
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() defined by (4) and for the Bessel zeta function (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; firstname.lastname@example.org) for his sending electronic version of the paper  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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