Abstract
The aim of this paper is to study the complete and incomplete degenerate Bell polynomials, which are degenerate versions of the complete and incomplete Bell polynomials, and to derive some properties and identities for those polynomials. In particular, we introduce some new polynomials associated with the incomplete degenerate Bell polynomials. In fact, they are the coefficients of the reciprocal of the power series given by 1 plus the one appearing as the exponent of the generating function of the complete degenerate Bell polynomials.
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1 Introduction
In recent years, we have seen that studying degenerate versions of many special polynomials and numbers, which was initiated in [2] by Carlitz, yielded very fruitful and interesting results. Especially, it is amusing to note that these studies are not just restricted to polynomials but also include transcendental functions like gamma functions.
The aim of this paper is to further study the complete and incomplete degenerate Bell polynomials (see (10), (12)) which are degenerate versions of the complete and incomplete Bell polynomials. In more detail, we deduce recurrence relations for them. As a corollary, we get a recurrence relation for the degenerate Stirling numbers of the second kind. We consider the problem of finding the reciprocal power series of the invertible formal power series which is equal to 1 plus the power series appearing as the exponent of the generating function of the complete degenerate Bell polynomials in (10). This leads us to introduce the new polynomials \(T_{n,\lambda }(a_{1},a_{2},\ldots,a_{n})\) (see (21)) that are associated with the incomplete degenerate Bell polynomials. As a corollary, this gives us an expression for the reciprocal of the degenerate exponential function \(e_{\lambda }(at)\). In addition, we obtain some identities regarding the degenerate Stirling numbers. For the rest of this section, we recall the facts that are needed throughout this paper.
For any \(\lambda \in \mathbb{R}\), it is known that the degenerate exponential function is defined as
where \((x)_{0,\lambda }=1\), \((x)_{n,\lambda }=x(x-\lambda )\cdots (x-(n-1) \lambda )\), (\(n\ge 1\)).
In particular, for \(x=1\), we write \(e_{\lambda }(t)=e_{\lambda }^{1}(t)\).
In [10], the degenerate Bell polynomials are given by
When \(x=1\), \(\operatorname{Bel}_{n,\lambda }=\operatorname{Bel}_{n,\lambda }(1)\) are called the degenerate Bell numbers. When \(\lambda =1\), the falling factorial sequence is given by \((x)_{0}=1\), \((x)_{n}=(x)_{n,1}=x(x-1)\cdots (x-n+1)\), (\(n\ge 1\)).
It is well known that the Stirling numbers of the first kind are defined by
As the inversion formula of (3), the Stirling numbers of the second kind are defined as
Recently, the degenerate Stirling numbers of the first kind were given by
As the inversion formula of (5), the degenerate Stirling numbers of the second kind are defined by
Here we recall that the degenerate Stirling numbers of the first kind and those of the second kind satisfy the orthogonality relations. Namely, they are related by the following:
where \(\delta _{n,k}\) is Kronecker’s delta.
As is well known, the complete Bell polynomials are defined by
where \(B_{n}(1,1,\ldots,1)=\operatorname{Bel}_{n}\), (\(n\ge 0\)), are the ordinary Bell numbers given by
For \(k\ge 0\), the incomplete Bell polynomials are given by
where
In [10], the complete degenerate Bell polynomials are constructed by
where
Note that
In the light of (9), the incomplete degenerate Bell polynomials are given by
where
and
From (12), we note that
Now, we observe that
By comparing the coefficients on both sides of (14), we get
2 Complete and incomplete degenerate Bell polynomials
From (10), we note that
Taking the derivative with respect to t on both sides of (16), we have
On the other hand,
Therefore, by (17) and (18), we obtain the following theorem.
Theorem 1
For \(n\ge 0\), we have
For \(k,n\in \mathbb{Z}\) with \(n\ge k\ge 1\), taking the derivative with respect to t on both sides of (12), we get
Therefore, by comparing the coefficients on both sides of (19), we obtain the following theorem.
Theorem 2
For \(k,n\in \mathbb{Z}\) with \(n\ge k\ge 1\), we have
Recalling (13), we obtain the following corollary.
Corollary 3
For \(n,k\in \mathbb{Z}\) with \(n\ge k\ge 1\), we have
We need the following lemma for the next result, which is stated without proof in [3, p. 136].
Lemma 4
For \(n,k\in \mathbb{Z}\) with \(n\ge k\ge 1\), we have
Proof
From (12), we see that
from which the first identity follows.
The second identity is deduced from the first by replacing l by \(n-1-l\). □
For a given formal power series \(\sum_{l=0}^{\infty }(1)_{l,\lambda }a_{l}\frac{t^{l}}{l!}\), with \(a_{0}=1\), we want to determine the reciprocal power series \(\sum_{m=0}^{\infty }(1)_{m,\lambda }b_{m}\frac{t^{m}}{m!}\) satisfying \(\sum_{l=0}^{\infty }(1)_{l,\lambda }a_{l}\frac{t^{l}}{l!}\sum_{m=0}^{ \infty }(1)_{m,\lambda }b_{m}\frac{t^{m}}{m!}=1\).
Theorem 5
Assume \(\sum_{l=0}^{\infty }(1)_{l,\lambda }a_{l}\frac{t^{l}}{l!}\sum_{m=0}^{ \infty }(1)_{m,\lambda }b_{m}\frac{t^{m}}{m!}=1\), with \(a_{0}=1\). Then we have
In other words, we have
where the new polynomials \(T_{n,\lambda }(a_{1},a_{2},\ldots,a_{n})\) (\(n\ge 0\)), associated with the incomplete degenerate Bell polynomials, are defined by
Proof
We observe first that \(1= \sum_{n=0}^{\infty } (\sum_{l=0}^{n}\binom{n}{l}(1)_{l, \lambda }a_{l}(1)_{n-l,\lambda }b_{n-l} )\frac{t^{n}}{n!}\). Thus we have
We show the identity in (20) by induction on \(n\ge 1\). From (22), we note that
If \(n=1\), we note from (23) that \((1)_{1,\lambda }b_{1}=-(1)_{1,\lambda }a_{1}=\sum_{k=1}^{1}B_{1,k}^{( \lambda )}(a_{1})(-1)^{k}k!\).
Assume that \(n>1\) and that (20) holds for all positive integers smaller than n. From (23) and Lemma 4, we have
□
Letting \(a_{i}=a^{i}\) for all integers \(i\ge 0\), we obtain the following corollary.
Corollary 6
The following identity holds true:
From (6), we can easily derive the following equation:
Now, we observe that
On the other hand
From (25) and (26), we note that
The rising λ-factorial sequence is defined by
Therefore, by (27) and (28), we obtain the following theorem, the second of which follows from the orthogonality relations in (7) for the degenerate Stirling numbers.
Theorem 7
For \(n\in \mathbb{N}\), we have
Note that
Therefore, by (29), we obtain the following corollary.
Corollary 8
For \(n\in \mathbb{N}\), we have
3 Conclusion
In recent years, we have seen that degenerate versions of many special polynomials and numbers were investigated by means of various different tools like generating functions, combinatorial methods, umbral calculus techniques, probability theory, p-adic analysis, special functions, analytic number theory and differential equations. Studying them has been rewarding; yielding many interesting results not only in combinatorics and number theory but also in probability, differential equations and symmetric identities. Moreover, they have potential applications in engineering and the sciences.
In this paper, we studied the complete and incomplete degenerate Bell polynomials which are degenerate versions of the complete and incomplete Bell polynomials and obtained some identities and properties as to such polynomials. Above all, we considered the problem of finding the reciprocal power series of the invertible formal power series which is equal to 1 plus the power series appearing as the exponent of the generating function of the complete degenerate Bell polynomials. This led us to introduce the new polynomials \(T_{n,\lambda }(a_{1},a_{2},\ldots,a_{n})\) that are associated with the incomplete degenerate Bell polynomials. As a corollary, this gave us an expression for the reciprocal of the degenerate exponential function \(e_{\lambda }(at)\).
It is one of our future projects to continue to explore various degenerate versions of some special polynomials and numbers and to find many applications in mathematics, science and engineering.
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This research was supported by the Daegu University Research Grant, 2020.
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TK and DSK conceived of the framework and structured the whole paper; TK and DSK wrote the paper; DSK completed the revision of the article; JWP and SHL checked the errors of the article. All authors have read and agreed to the published version of the manuscript.
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Kim, D.S., Kim, T., Lee, SH. et al. Some new formulas of complete and incomplete degenerate Bell polynomials. Adv Differ Equ 2021, 326 (2021). https://doi.org/10.1186/s13662-021-03479-6
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DOI: https://doi.org/10.1186/s13662-021-03479-6