| ||Computer Modeling in Engineering & Sciences || |
On Degenerate Array Type Polynomials
1Key Laboratory of Intelligent Manufacturing Technology, Inner Mongolia Minzu University, Tongliao, 028000, Inner Mongolia, China
2Department of Mathematics, Akdeniz University, Antalya, 07058, Turkey
3Institute of Mathematics, Henan Polytechnic University, Jiaozuo, 454010, China
4School of Mathematical Sciences, Tiangong University, Tianjin, 300387, China
*Corresponding Author: Feng Qi. Email: firstname.lastname@example.org
Dedicated to retired Professor Ji-Shan Tian, former vice president of Henan University, China
Received: 17 August 2021; Accepted: 27 October 2021
Abstract: In the paper, with the help of the Faá di Bruno formula and an identity of the Bell polynomials of the second kind, the authors define degenerate -array type polynomials, establish two explicit formulas, and present several recurrence relations of degenerate -array type polynomials and numbers.
Keywords: Degenerate array polynomial; Stirling number of the second kind; generating function; explicit formula; recurrence relation
In this paper, we use the following notation:
The Stirling numbers of the second kind S(n, m) for can be generated by
and can be computed as
See [, p.206] and the paper .
The -array type polynomials were defined in  by the generating function
See also the papers [4,5]. It is clear that S(n, m;0;1) = S(n, m). In the paper , Simsek obtained and constructed several generating functions and many relations of generalized Stirling type numbers, the array type polynomials, and Eulerian type polynomials. In the paper , Bayad et al. deduced interesting and meaningful identities associated with -array type polynomials, -Stirling numbers of the second kind, and the Apostol–Bernoulli numbers, while they dealt with -array polynomials by applying -delta operator. Readers interested to the Apostol-Bernoulli numbers and polynomials may consult to the papers [8–10] and closely related references therein.
In the paper , Carlitz introduced degenerate Bernoulli and Euler polynomials and by
respectively. For x = 0, the quantities and are called as degenerate Bernoulli and Euler numbers. Since , Eqs. (3) and (4) reduce to the generating functions for classical Bernoulli and Euler polynomials, respectively.
We now define degenerate -array type polynomials by
It is easy to see that
which is defined by (2). When x = 0, we call the quantities degenerate -array type numbers.
In this paper, utilizing the Faá di Bruno formula and an identity of the Bell polynomials of the second kind, we establish several explicit formulas and recurrence relations of (degenerate) -array type numbers and polynomials.
Let us notice that the Faá di Bruno formula, which can be viewed as an extension of chain rule to higher derivatives, has been applied to establish explicit and closed-form formulas of many important numbers and polynomials in analytic and combinatorial number theory. For more details, please refer to, for example, the papers [12–18] and closely related references therein.
2 Some Identities of the Bell Polynomials of the Second Kind
The Bell polynomials of the second kind for can be defined by
See [, p. 134]. For , the Faà di Bruno formula is described in [, p.139] in terms of the Bell polynomials of the second kind by
has been applied and reviewed in [, Lemma 2.2], [, Remark 6.1], and [, Section 1.3]. The explicit formula (8) is equivalent to
which was presented in [, Theorems 2.1 and 4.1], where the falling factorial is defined for by
When , the explicit formulas (8) and (9) can be rearranged as
where extended binomial coefficient is defined by
and the classical Euler gamma function can be defined by
For new results and applications about the Bell polynomials‘ of the second kind Bn, k, please refer to the papers [13,19,21–23] and closely related references therein.
3 Explicit Formulas of Degenerate -Array Type Polynomials
In this section, we establish two explicit formulas for degenerate -array type numbers and polynomials, respectively.
Theorem 3.1. For , degenerate -array type numbers can be computed by
Proof. Making use of and as in the Faá di Bruno formula (6) and applying (10) result in
as . Considering the generating function in (5) for x = 0, we proved the explicit formula (12). The proof of Theorem 3.1 is complete.
Remark 3.1. From (5), it follows immediately that
By virtue of the explicit formula (12), we obtain the first few values of degenerate -array type numbers for as follows:
Remark 3.2. The explicit formula (12) in Theorem 3.1 and seven concrete values listed in Remark 3.1 reveal that degenerate -array type numbers are polynomials of and with degrees m and , respectively.
Theorem 3.2. For , degenerate -array type polynomials can be computed by
Proof. For , it is easy to see that
as . Letting as and making use of the Faà di Bruno formula (7) give
as . Considering the generating function in (5), we proved the explicit formula (13). The proof of Theorem 3.2 is complete.
Remark 3.3. From the generating function (5), we can easily obtain
By virtue of the explicit formula (13), we can calculate the first few values of degenerate -array type polynomials for as follows:
Remark 3.4. From the explicit formula (13) in Theorem 3.2 and the four concrete values in Remark 3.3, we conclude that degenerate -array type polynomials are polynomials of x, , and of degrees n, m, and , respectively.
Remark 3.5. When x = 0 in Theorem 3.2, the explicit formula (13) becomes (12) in Theorem 3.1.
Remark 3.6. For further better understanding degenerate -array type polynomials , we demonstrate two angles of the graph of for and in Fig. 1. The blue plane in Fig. 1 is the -plane.
Figure 1: Two angles of the graph of for and , plotted by Wolfram Mathematica 12.0
Remark 3.7. For further better understanding degenerate -array type polynomials , we demonstrate two angles of the graph of for 0 < x < 9 and in Fig. 2. The blue plane in Fig. 2 is the -plane.
Figure 2: Two angles of the graph of for 0 < x < 9 and , plotted by Wolfram Mathematica 12.0
Remark 3.8. For further better understanding degenerate -array type polynomials , we demonstrate two angles of the graph of for 0 < x < 9 and in Fig. 3. The blue plane in Fig. 3 is the -plane.
Figure 3: Two angles of the graph of for 0 < x < 9 and , plotted by Wolfram Mathematica 12.0
4 Recurrence Relations of Degenerate -Array Type Polynomials
In this section, we establish several recurrence relations of degenerate -array type polynomials.
Theorem 4.1. Degenerate -array type polynomials satisfy the recurrence relation
Proof. From Eq. (5), it follows that
Comparing coefficients of the terms on both sides concludes (14). The proof of Theorem 4.1 is complete.
Theorem 4.2. Degenerate -array type polynomials satisfy the recurrence relation
Consequently, we have
Proof. Differentiating with respect to t yields
where, on the other hand,
Further replacing x by and simplifying lead to (15).
Taking in (15) and considering (6) give (16). The proof of Theorem 4.2 is complete.
Remark 4.1. One of anonymous referees commented that the -array type polynomials are related to numbers considered in the papers [24–26].
In this paper, with the help of the Faá di Bruno formula (7) and the identity (10) for the Bell polynomials of the second kind Bn, k, we define degenerate -array type polynomials by (5), establish two explicit formulas (12) and (13) in Theorems 3.1 and 3.2, and present several recurrence relations (14), (15), and (16) of degenerate -array type polynomials and numbers and in Theorems 4.1 and 4.2.
Acknowledgement: The authors thank the editors and anonymous referees for their careful corrections to, valuable comments on, and helpful suggestions to the original version of this paper.
Funding Statement: The first two authors, Mrs. Lan Wu and Xue-Yan Chen, were partially supported by the College Scientific Research Project of Inner Mongolia (Grant No. NJZY19156 and Grant No. NJZZ19144), by the Natural Science Foundation Project of Inner Mongolia (Grant No. 2021LHMS05030), and by the Development Plan for Young Technological Talents in Colleges and Universities of Inner Mongolia (Grant No. NJYT22051) 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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