Two variable higher-order central Fubini polynomials

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.

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.

The ordered central Bell numbers of order r are defined by the generating function

By (14), we get where δ is the central difference operator given by Therefore, by (20), we obtain the following theorem.

From (14), we can derive the following equation (22): Comparing the coefficients on both sides of (22), the following theorem is obtained.

From \(r\in \mathbb{N}\) with \(r\geq 2\), we have Therefore, by (14) and (24), we obtain the following convolution formula.

Now, we observe that From (14) and (25), we have Therefore, by (26), we obtain the following theorem.

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.

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.

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.

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.

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.

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.

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,

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.