Abstract
Recently, the central Fubini polynomials were introduced in connection with central factorial numbers of the second kind. In this paper, we consider two variable higher-order central Fubini polynomials as a ‘central analogue’ of two variable higher-order Fubini polynomials. We investigate some properties, identities, and recurrence relations for these polynomials by making use of generating functions and umbral calculus. In particular, we obtain various expressions for the two variable higher-order central Fubini polynomials and express them in terms of some families of special polynomials and vice versa.
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1 Introduction
For \(n\in \mathbb{N}\cup \{0\}\), the central factorial \(x^{[n]}\) is defined as
As is well known, the central factorial numbers of the second kind \(T(n,k)\) (\(n,k\geq 0\)) are defined by
From (2), we can derive the following generating function for \(T(n,k)\):
It is known that the two variable Fubini polynomials \(F_{n}^{(r)} (x;y)\) of order r are defined by
where r is a positive integer.
In particular, if \(r=1\), then \(F_{n} (x;y)=F_{n}^{(1)}(x;y) \) are called two variable Fubini polynomials. For \(x=0\), \(F_{n}^{(r)}(y)=F_{n}^{(r)}(0;y)\) and \(F_{n}^{(r)}=F_{n}^{(r)}(1)=F _{n}^{(r)}(0;1)\) are respectively called the Fubini polynomials of order r and the Fubini numbers of order r. Further, in the special case of \(y=1\), \(F_{n}^{(r)}(x;1)\) are the ordered Bell polynomials of order r. Recently, the central Fubini polynomials have been defined by
From (5), one can see that
Next, we will quickly review very basics of umbral calculus. Let \(\mathbb{C}\) be the field of complex numbers, and let
We set \(\mathbb{P}=\mathbb{C}[x]\) and define \(\mathbb{P}^{*}\) by the vector space of all linear functionals on \(\mathbb{P}\). For any given \(L\in \mathbb{P}^{*}\) and \(p(x)\in \mathbb{P}\), throughout the paper, \(\langle L\mid p(x)\rangle \) represents the action of the linear functional L on \(p(x)\). For \(f(t)=\sum_{k=0}^{\infty }a_{k} \frac{t^{k}}{k!} \in \mathcal{F}\), \(\langle f(t)\mid \cdot \rangle \) denotes the linear functional on \(\mathbb{P}\) given by
For \(L\in \mathbb{P}^{*}\), let \(f_{L}(t)=\sum_{k=0}^{\infty }\langle L\mid x ^{k}\rangle \frac{t^{k}}{k!} \in \mathcal{F}\). Then we have \(\langle f_{L}(t)\mid x ^{n}\rangle =\langle L\mid x^{n}\rangle \), \((n\geq 0)\), and the map \(L\longmapsto f_{L}(t)\) is a vector space isomorphism from \(\mathbb{P}^{*}\) to \(\mathcal{F}\). Thus, \(\mathcal{F}\) may be viewed as the vector space of all linear functionals on \(\mathbb{P}\) as well as the algebra of formal power series in t. Hence, an element \(f(t)\in \mathcal{F}\) will be thought of as both a formal power series and a linear functional on \(\mathbb{P}\). \(\mathcal {F}\) is called the umbral algebra, the study of which is the umbral calculus.
The order \({O}(f(t))\) of \(f(t)(\neq 0)\in \mathcal {F}\) is the smallest integer k such that the coefficient of \(t^{k}\) does not vanish. For \(f(t), g(t)\in \mathcal {F}\) with \({O}(g(t))=0\), \({ O}(f(t))=1\), there exists a unique sequence of polynomials \(S_{n}(x)\) (\(\deg S_{n}(x)=n\)) such that
where \(\delta _{n,k}\) is the Kronecker symbol. Such a sequence is called the Sheffer sequence for \((g(t), f(t))\), which is denoted by \(S_{n}(x)\sim (g(t), f(t))\). It is known that \(S_{n}(x)\sim (g(t), f(t))\) if and only if
where \(\bar{f}(t)\) is the compositional inverse of \(f(t)\) satisfying \(f(\bar{f}(t))=\bar{f}(f(t))=t\). For \(S_{n}(x)\sim (g(t), f(t))\), we have the following Sheffer identity:
where \(P_{n}(x)=g(t)S_{n}(x)\sim (1,f(t))\) (see [15]).
For any \(h(t)\in \mathcal {F}\), \(p(x)\in \mathbb {P}\), we have
Let \(S_{n}(x)\sim (g(t),f(t))\) and \(r_{n}(x)\sim (h(t),l(t))\). Then we have
where
In the forthcoming section, we will consider two variable higher-order central Fubini polynomials as a ‘central analogue’ of two variable higher-order Fubini polynomials. We introduce some properties and present several identities and recurrence relations for these polynomials by making use of generating functions and umbral calculus. Further, we show various expressions for the two variable higher-order central Fubini polynomials and express them in terms of some families of special polynomials and vice versa in the following section.
2 Two variable higher-order central Fubini polynomials
In view of (4), we consider the two variable higher-order central Fubini polynomials which are given by
where r is a positive integer. Here, in this paper, y will be an arbitrary but fixed real number so that \(F_{n,c}^{(r)}(x;y)\) are polynomials in x for each fixed y.
From (9) and (14), we note that
When \(r=1\), \(F_{n,c}^{(1)}(x;y)=F_{n,c}(x;y)\) are called two variable central Fubini polynomials and \(F_{n,c}^{(r)}(y)=F_{n,c}^{(r)}(0;y)\) are called central Fubini polynomials of order r. When \(r=1\), \(F_{n,c}(y)=F_{n,c}^{(1)}(y)\) are called the central Fubini polynomials. From (14), we note that
Thus, by comparing the coefficients on both sides of (16), we get
By (17), we easily get \(\frac{d}{dx}F_{n,c}^{(r)}(x;y) =nF _{n-1,c}^{(r)}(x;y)\), \((n\in \mathbb{N})\). From (14), we note that
Comparing the coefficients on both sides of (18), we have the following theorem.
Theorem 1
For \(n\geq 0\), we have
The ordered central Bell numbers of order r are defined by the generating function
Corollary 1
For \(n\geq 0\), we have
Remark 1
When \(r=1\), \(b_{n,c}=b_{n,c}^{(1)}\) are called the ordered central Bell numbers. Note that
By (14), we get
where δ is the central difference operator given by
Therefore, by (20), we obtain the following theorem.
Theorem 2
For \(n\geq 0\), we have
where δf is as in (21). In particular,
From (14), we can derive the following equation (22):
Comparing the coefficients on both sides of (22), the following theorem is obtained.
Theorem 3
For \(n\geq 0\), we have
In particular,
Remark 2
From (14), we note that
From \(r\in \mathbb{N}\) with \(r\geq 2\), we have
Therefore, by (14) and (24), we obtain the following convolution formula.
Theorem 4
For \(r\in \mathbb{N}\) with \(r\geq 2\) and \(n\geq 0\), we have
Now, we observe that
Therefore, by (26), we obtain the following theorem.
Theorem 5
For \(n\geq 0\), we have
In particular, \(n\in \mathbb{N}\),
From (4) and (9), we note that
For \(n\in \mathbb{N}\), by (11), we get
It is easy to show that
Clearly, the second term of (28) is \(zF_{n-1,c}^{(r)}(z;y)\).
Therefore, we obtain the following theorem.
Theorem 6
For \(n\geq 0\), we have
and
Now, we will express the two variable higher-order central Fubini polynomials \(F_{n,c}^{(r)} (x;y)\) as linear combinations of some well-known special polynomials.
We first recall that
If we let
where \(S_{n} (x)\sim (h(t), l(t))\). Then, from (12) and (13), we see that
Therefore, the following theorem can be established.
Theorem 7
Let \(S_{n}(x)\sim (h(t), l(t))\) for \(n\geq 0\), then we have
where
Let \(B_{n}(x)\) (\(n\geq 0\)) be the ordinary Bernoulli polynomials given by
Note that \(B_{n}(x)\sim ( \frac{e^{t}-1}{t}, t )\).
Assume that
where
Therefore, by (33), we obtain the following theorem.
Theorem 8
For \(n\geq 0\), we have
The falling factorial sequence is defined by
Note that the generating function of \((x)_{n}\) is given by
By (34), we get \((x)_{n}\sim ( 1, e^{t}-1)\).
Assume that
Then, by Theorem 7, we have
where \(S_{2}(n,k)\) are the numbers of the second kind given by
Therefore, we obtain the following theorem.
Theorem 9
For \(n\geq 0\), we have
It is well known that the Bell polynomials are defined by the generating function
Thus, by (37), we get
Assume that
By Theorem 7, we get
where \(S_{1}(n,k)\) are the Stirling numbers of the first kind defined by
Therefore, we obtain the following theorem.
Theorem 10
For \(n\geq 0\), we have
Let \(p(x)\in \mathbb{C}[x]\) be a polynomial of degree ≤n. Then we can write
We observe that
Thus, by (42), we get
Therefore, we obtain the following theorem.
Theorem 11
For \(p(x)\in \mathbb{C} [x]\) with \(\deg p(x)\leq n\), we have
where
For example, let \(p(x)=B_{n}(x)\) (\(n\geq 0\)). Then we have
where
where \(B_{n}=B_{n}(0)\) are Bernoulli numbers. Thus, by (44) and (45), we get
Assume that
where
Hence,
3 Conclusions
Recently, the two variable Fubini polynomials were introduced by Kargin (see [4]) and the central Fubini polynomials associated with central factorial numbers of the second kind by Kim et al. (see [8]). In this paper, we considered two variable higher-order central Fubini polynomials as a ‘central analogue’ of two variable higher-order Fubini polynomials. We investigated some properties, identities, and recurrence relations for these polynomials by making use of generating functions and umbral calculus. In particular, we obtained various expressions for the two variable higher-order central Fubini polynomials and expressed them in terms of some families of special polynomials and vice versa.
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This article is dedicated to Professor Gradimir V. Milovanovic on the occasion of his 70th anniversary.
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Kim, T., Kim, D.S., Jang, GW. et al. Two variable higher-order central Fubini polynomials. J Inequal Appl 2019, 146 (2019). https://doi.org/10.1186/s13660-019-2100-0
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DOI: https://doi.org/10.1186/s13660-019-2100-0