Symmetric Identities for Fubini Polynomials

Abstract

We represent the generating function of w-torsion Fubini polynomials by means of a fermionic p-adic integral on ${\mathbb{Z}}_p$. Then we investigate a quotient of such p-adic integrals on ${\mathbb{Z}}_p$, representing generating functions of three w-torsion Fubini polynomials and derive some new symmetric identities for the w-torsion Fubini and two variable w-torsion Fubini polynomials.

1. Introduction and Preliminaries

In recent years, various p-adic integrals on ${\mathbb{Z}}_p$ have been used in order to find many interesting symmetric identities related to some special polynomials and numbers. The relevant p-adic integrals are the Volkenborn, fermionic, q-Volkenborn, and q-fermionic integrals of which the last three were discovered by the first author T. Kim (see [1,2,3]). They have been used by a good number of researchers in various contexts and especially in unfolding new interesting symmetric identities. This verifies the usefulness of such p-adic integrals. Moreover, we can expect that people will find some further applications of these p-adic integrals in the years to come. The present paper is an effort in this direction. Assume that p is any fixed odd prime number. Throughout our discussion, we will use the standard notations ${\mathbb{Z}}_p$, ${\mathbb{Q}}_p$, and ${\mathbb{C}}_p$ to denote the ring of p-adic integers, the field of p-adic rational numbers and the completion of the algebraic closure of ${\mathbb{Q}}_p$, respectively. The p-adic norm ${|·|}_p$ is normalized as ${\left|p\right|}_p=\frac{1}{p}$. Assume that $f\left(x\right)$ is a continuous function on ${\mathbb{Z}}_p$. Then the fermionic p-adic integral of $f\left(x\right)$ on ${\mathbb{Z}}_p$ was introduced by Kim (see [2]) as

$$\begin{array}{c}\hfill {\int }_{{\mathbb{Z}}_p}f\left(x\right)d{\mu }_{-1}\left(x\right)=\underset{N\to \infty }{lim}\sum _{x=0}^{p^N-1}f\left(x\right){(-1)}^x,\end{array}$$

(1)

where ${\mu }_{-1}(x+p^N{\mathbb{Z}}_p)={(-1)}^x$.

We can easily deduce from (1) that (see [2,3])

$$\begin{array}{c}\hfill {\int }_{{\mathbb{Z}}_p}f(x+1)d{\mu }_{-1}\left(x\right)+{\int }_{{\mathbb{Z}}_p}f\left(x\right)d{\mu }_{-1}\left(x\right)=2f\left(0\right).\end{array}$$

(2)

By invoking (2), we easily get (see [2,4])

$$\begin{array}{c}\hfill {\int }_{{\mathbb{Z}}_p}{e}^{(x+y)t}d{\mu }_{-1}\left(y\right)=\frac{2}{e^t+1}{e}^{xt}=\sum _{n=0}^{\infty }{E}_n\left(x\right)\frac{t^n}{n!},\end{array}$$

(3)

where $E_n\left(x\right)$ are the usual Euler polynomials.

As is known, the two variable Fubini polynomials are defined by means of the following (see [5,6])

$$\begin{array}{c}\hfill \sum _{n=0}^{\infty }{F}_n(x,y)\frac{t^n}{n!}=\frac{1}{1-y(e^t-1)}{e}^{xt}.\end{array}$$

(4)

When $x=0$, $F_n\left(y\right)=F_n(0,y)$, $(n\ge 0$), are called Fubini polynomials. Further, if $y=1$, then $O{b}_n=F_n(0,1)$ are the ordered Bell numbers (also called Frobenius numbers). They first appeared in Cayley’s work on a combinatorial counting problem in 1859 and have many different combinatorial interpretations. For example, the ordered Bell numbers count the possible outcomes of a multi-candidate election. From (3) and (4), we note that $F_n(x,-1/2)=E_n\left(x\right)$, $(n\ge 0)$. By (4), we easily get (see [6]),

$$\begin{array}{c}\hfill F_n\left(y\right)=\sum _{k=0}^n{S}_2(n,k)k!y^k,(n\ge 0),\end{array}$$

(5)

where $S_2(n,k)$ are the Stirling numbers of the second kind.

For $w\in \mathbb{N}$, we define the two variable w-torsion Fubini polynomials given by

$$\begin{array}{c}\hfill \frac{1}{1-y^w{(e^t-1)}^w}{e}^{xt}=\sum _{n=0}^{\infty }{F}_{n,w}(x,y)\frac{t^n}{n!}.\end{array}$$

(6)

In particular, for $x=0$, $F_{n,w}\left(y\right)=F_{n,w}(0,y)$ are called the w-torsion Fubini polynomials. It is obvious that $F_{n,1}(x,y)=F_n(x,y)$.

We represent the generating function of w-torsion Fubini polynomials by means of a fermionic p-adic integral on ${\mathbb{Z}}_p$. Then we investigate a quotient of such p-adic integrals on ${\mathbb{Z}}_p$, representing generating functions of three w-torsion Fubini polynomials and derive some new symmetric identities for the w-torsion Fubini and two variable w-torsion Fubini polynomials. Recently, a number of researchers have studied symmetric identities for some special polynomials. The reader may refer to [7,8,9,10,11] as an introduction to this active area of research. Some symmetric identities for q-special polynomials and numbers were treated in [12,13,14,15], including q-Bernoulli, q-Euler, and q-Genocchi numbers and polynomials. While some identities of symmetry for degenerate special polynomials were discussed in the more recent papers [6,16,17]. Finally, interested readers may want to have a glance at [18,19] as general references on polynomials.

2. Symmetric Identities for $\mathit{w}$-torsion Fubini and Two Variable $\mathit{w}$-torsion Fubini Polynomials

From (2), we note that

$$\begin{array}{c}\hfill {\int }_{{\mathbb{Z}}_p}{(-1)}^x{\left(y(e^t-1)\right)}^{x}d{\mu }_{-1}\left(x\right)=\frac{2}{1-y(e^t-1)}=2\sum _{n=0}^{\infty }{F}_n\left(y\right)\frac{t^n}{n!},\end{array}$$

(7)

and

$$\begin{array}{c}\hfill e^{xt}{\int }_{{\mathbb{Z}}_p}{(-1)}^z{\left(y(e^t-1)\right)}^{z}d{\mu }_{-1}\left(z\right)=\frac{2}{1-y(e^t-1)}{e}^{xt}=2\sum _{n=0}^{\infty }{F}_n(x,y)\frac{t^n}{n!}.\end{array}$$

(8)

From (7) and (8), we note that

$$\begin{array}{cc}\hfill \left(\sum _{l=0}^{\infty }{x}^l\frac{t^l}{l!}\right)\left(\sum _{m=0}^{\infty }2F_m\left(y\right)\frac{t^m}{m!}\right)& =e^{xt}{\int }_{{\mathbb{Z}}_p}{(-1)}^z{\left(y(e^t-1)\right)}^{z}d{\mu }_{-1}\left(z\right)\hfill \\ & =\sum _{n=0}^{\infty }2F_n(x,y)\frac{t^n}{n!}.\hfill \end{array}$$

(9)

Thus, by (9), we easily get

$$\begin{array}{c}\hfill \sum _{l=0}^n\left(\genfrac{}{}{0pt}{}{n}{l}\right)x^l{F}_{n-l}\left(y\right)=F_n(x,y),(n\ge 0).\end{array}$$

(10)

Now, we observe that

$$\begin{array}{cc}\hfill \frac{1-y^k{(e^t-1)}^k}{1-y(e^t-1)}& =\sum _{i=0}^{k-1}{y}^i{(e^t-1)}^i=\sum _{i=0}^{k-1}\sum _{l=0}^i\left(\genfrac{}{}{0pt}{}{i}{l}\right){(-1)}^{i-l}{y}^i{e}^{lt}\hfill \\ & =\sum _{n=0}^{\infty }\left(\sum _{i=0}^{k-1}\sum _{l=0}^i\left(\genfrac{}{}{0pt}{}{i}{l}\right){(-1)}^{i-l}{y}^i{l}^n\right)\frac{t^n}{n!}\hfill \\ & =\sum _{n=0}^{\infty }\left(\sum _{i=0}^{k-1}{y}^i{\mathsf{\Delta }}^i{0}^n\right)\frac{t^n}{n!},\hfill \end{array}$$

(11)

where $\mathsf{\Delta }f\left(x\right)=f(x+1)-f\left(x\right)$.

For $w\in \mathbb{N}$, the w-torsion Fubini polynomials are represented by means of the following fermionic p-adic integral on ${\mathbb{Z}}_p$:

$$\begin{array}{c}\hfill {\int }_{{\mathbb{Z}}_p}{(-y^w{(e^t-1)}^w)}^{x}d{\mu }_{-1}\left(x\right)=\frac{2}{1-y^w{(e^t-1)}^w}=\sum _{n=0}^{\infty }2F_{n,w}\left(y\right)\frac{t^n}{n!},\end{array}$$

(12)

From (7) and (12), we have

$$\begin{array}{cc}\hfill \frac{{\int }_{{\mathbb{Z}}_p}{(-y(e^t-1))}^{x}d{\mu }_{-1}\left(x\right)}{{\int }_{{\mathbb{Z}}_p}{(-y^{w_1}{(e^t-1)}^{w_1})}^{x}d{\mu }_{-1}\left(x\right)}& =\frac{1-y^{w_1}{(e^t-1)}^{w_1}}{1-y(e^t-1)}=\sum _{i=0}^{w_1-1}{y}^i{(e^t-1)}^i\hfill \\ & =\sum _{n=0}^{\infty }\left(\sum _{i=0}^{w_1-1}{y}^i{\mathsf{\Delta }}^i{0}^n\right)\frac{t^n}{n!},(w_1\in \mathbb{N}).\hfill \end{array}$$

(13)

For $w_1,w_2\in \mathbb{N}$, we let

$$\begin{array}{c}\hfill I=\frac{{\int }_{{\mathbb{Z}}_p}{(-y^{w_1}{(e^t-1)}^{w_1})}^{x_1}d{\mu }_{-1}\left(x_1\right){\int }_{{\mathbb{Z}}_p}{(-y^{w_2}{(e^t-1)}^{w_2})}^{x_2}d{\mu }_{-1}\left(x_2\right)}{{\int }_{{\mathbb{Z}}_p}{(-y^{w_1{w}_2}{(e^t-1)}^{w_1{w}_2})}^{x}d{\mu }_{-1}\left(x\right)}.\end{array}$$

(14)

Here it is important to observe that (14) has the built-in symmetry. Namely, it is invariant under the interchange of $w_1$ and $w_2$.

Then, by (14), we get

$$\begin{array}{c}\hfill I=\left({\int }_{{\mathbb{Z}}_p}{(-y^{w_1}{(e^t-1)}^{w_1})}^{x}d{\mu }_{-1}\left(x\right)\right)\times \left(\frac{{\int }_{{\mathbb{Z}}_p}{(-y^{w_2}{(e^t-1)}^{w_2})}^{x}d{\mu }_{-1}\left(x\right)}{{\int }_{{\mathbb{Z}}_p}{(-y^{w_1{w}_2}{(e^t-1)}^{w_1{w}_2})}^{x}d{\mu }_{-1}\left(x\right)}\right).\end{array}$$

(15)

First, we observe that

$$\begin{array}{cc}\hfill \frac{{\int }_{{\mathbb{Z}}_p}{(-y^{w_2}{(e^t-1)}^{w_2})}^{x}d{\mu }_{-1}\left(x\right)}{{\int }_{{\mathbb{Z}}_p}{(-y^{w_1{w}_2}{(e^t-1)}^{w_1{w}_2})}^{x}d{\mu }_{-1}\left(x\right)}& =\frac{1-y^{w_1{w}_2}{(e^t-1)}^{w_1{w}_2}}{1-y^{w_2}{(e^t-1)}^{w_2}}=\sum _{i=0}^{w_1-1}{y}^{w_{2}i}{(e^t-1)}^{w_{2}i}\hfill \\ & =\sum _{i=0}^{w_1-1}{y}^{w_{2}i}\sum _{l=0}^{w_{2}i}\left(\genfrac{}{}{0pt}{}{w_{2}i}{l}\right){(-1)}^{w_{2}i-l}{e}^{lt}\hfill \\ & =\sum _{n=0}^{\infty }\left(\sum _{i=0}^{w_1-1}{y}^{w_{2}i}{\mathsf{\Delta }}^{w_{2}i}{0}^n\right)\frac{t^n}{n!}.\hfill \end{array}$$

(16)

From (15) and (16), we can derive the following equation.

$$\begin{array}{cc}\hfill I& =\left({\int }_{{\mathbb{Z}}_p}{(-y^{w_1}{(e^t-1)}^{w_1})}^{x}d{\mu }_{-1}\left(x\right)\right)\times \left(\frac{{\int }_{{\mathbb{Z}}_p}{(-y^{w_2}{(e^t-1)}^{w_2})}^{x}d{\mu }_{-1}\left(x\right)}{{\int }_{{\mathbb{Z}}_p}{(-y^{w_1{w}_2}{(e^t-1)}^{w_1{w}_2})}^{x}d{\mu }_{-1}\left(x\right)}\right)\hfill \\ & =\left(\sum _{m=0}^{\infty }2F_{m,w_1}\left(y\right)\frac{t^m}{m!}\right)\times \left(\sum _{k=0}^{\infty }(\sum _{i=0}^{w_1-1}{y}^{w_{2}i}{\mathsf{\Delta }}^{w_{2}i}{0}^k\right)\frac{t^k}{k!}\hfill \\ & =\sum _{n=0}^{\infty }\left(2\sum _{k=0}^n\sum _{i=0}^{w_1-1}{y}^{w_{2}i}{\mathsf{\Delta }}^{w_{2}i}{0}^k{F}_{n-k,w_1}\left(y\right)\left(\genfrac{}{}{0pt}{}{n}{k}\right)\right)\frac{t^n}{n!}.\hfill \end{array}$$

(17)

Interchanging the roles of $w_1$ and $w_2$, by (14), we get

$$\begin{array}{c}\hfill I=\left({\int }_{{\mathbb{Z}}_p}{(-y^{w_2}{(e^t-1)}^{w_2})}^{x}d{\mu }_{-1}\left(x\right)\right)\times \left(\frac{{\int }_{{\mathbb{Z}}_p}{(-y^{w_1}{(e^t-1)}^{w_1})}^{x}d{\mu }_{-1}\left(x\right)}{{\int }_{{\mathbb{Z}}_p}{(-y^{w_1{w}_2}{(e^t-1)}^{w_1{w}_2})}^{x}d{\mu }_{-1}\left(x\right)}\right).\end{array}$$

(18)

We note that

$$\begin{array}{cc}\hfill \frac{{\int }_{{\mathbb{Z}}_p}{(-y^{w_1}{(e^t-1)}^{w_1})}^{x}d{\mu }_{-1}\left(x\right)}{{\int }_{{\mathbb{Z}}_p}{(-y^{w_1{w}_2}{(e^t-1)}^{w_1{w}_2})}^{x}d{\mu }_{-1}\left(x\right)}& =\frac{1-y^{w_1{w}_2}{(e^t-1)}^{w_1{w}_2}}{1-y^{w_1}{(e^t-1)}^{w_1}}=\sum _{i=0}^{w_2-1}{y}^{w_{1}i}{(e^t-1)}^{w_{1}i}\hfill \\ & =\sum _{n=0}^{\infty }\left(\sum _{i=0}^{w_2-1}{y}^{w_{1}i}{\mathsf{\Delta }}^{w_{1}i}{0}^n\right)\frac{t^n}{n!}.\hfill \end{array}$$

(19)

Thus, by (18) and (19), we get

$$\begin{array}{cc}\hfill I& =\left({\int }_{{\mathbb{Z}}_p}{(-y^{w_2}{(e^t-1)}^{w_2})}^{x}d{\mu }_{-1}\left(x\right)\right)\times \left(\frac{{\int }_{{\mathbb{Z}}_p}{(-y^{w_1}{(e^t-1)}^{w_1})}^{x}d{\mu }_{-1}\left(x\right)}{{\int }_{{\mathbb{Z}}_p}{(-y^{w_1{w}_2}{(e^t-1)}^{w_1{w}_2})}^{x}d{\mu }_{-1}\left(x\right)}\right)\hfill \\ & =\left(\sum _{m=0}^{\infty }2F_{m,w_2}\left(y\right)\frac{t^m}{m!}\right)\times \left(\sum _{k=0}^{\infty }(\sum _{i=0}^{w_2-1}{y}^{w_{1}i}{\mathsf{\Delta }}^{w_{1}i}{0}^k\right)\frac{t^k}{k!}\hfill \\ & =\sum _{n=0}^{\infty }\left(2\sum _{k=0}^n\sum _{i=0}^{w_2-1}{y}^{w_{1}i}{\mathsf{\Delta }}^{w_{1}i}{0}^k{F}_{n-k,w_2}\left(y\right)\left(\genfrac{}{}{0pt}{}{n}{k}\right)\right)\frac{t^n}{n!}.\hfill \end{array}$$

(20)

The following theorem is now obtained by Equations (17) and (20).

Theorem 1.

For $w_1,w_2\in \mathbb{N}$ with $w_1\equiv 1$ (mod 2), $w_2\equiv 1$ (mod 2), $n\ge 0$, we have

$$\begin{array}{c}\hfill \sum _{k=0}^n\sum _{i=0}^{w_1-1}\left(\genfrac{}{}{0pt}{}{n}{k}\right)F_{n-k,w_1}\left(y\right)y^{w_{2}i}{\mathsf{\Delta }}^{w_{2}i}{0}^k=\sum _{k=0}^n\sum _{i=0}^{w_2-1}\left(\genfrac{}{}{0pt}{}{n}{k}\right)F_{n-k,w_2}\left(y\right)y^{w_{1}i}{\mathsf{\Delta }}^{w_{1}i}{0}^k.\end{array}$$

(21)

Remark 1.

In particular, for $w_1=1$, we have

$$\begin{array}{c}\hfill F_n\left(y\right)=\sum _{k=0}^n\sum _{i=0}^{w_2-1}\left(\genfrac{}{}{0pt}{}{n}{k}\right)F_{n-k,w_2}\left(y\right)y^i{\mathsf{\Delta }}^i{0}^k.\end{array}$$

(22)

By expressing I in a different way, we have

$$\begin{array}{cc}\hfill I& =\left({\int }_{{\mathbb{Z}}_p}{(-y^{w_1}{(e^t-1)}^{w_1})}^{x}d{\mu }_{-1}\left(x\right)\right)\times \left(\frac{{\int }_{{\mathbb{Z}}_p}{(-y^{w_2}{(e^t-1)}^{w_2})}^{x}d{\mu }_{-1}\left(x\right)}{{\int }_{{\mathbb{Z}}_p}{(-y^{w_1{w}_2}{(e^t-1)}^{w_1{w}_2})}^{x}d{\mu }_{-1}\left(x\right)}\right)\hfill \\ & =\left({\int }_{{\mathbb{Z}}_p}{(-y^{w_1}{(e^t-1)}^{w_1})}^{x}d{\mu }_{-1}\left(x\right)\right)\times \left(\frac{1-y^{w_1{w}_2}{(e^t-1)}^{w_1{w}_2}}{1-y^{w_2}{(e^t-1)}^{w_2}}\right)\hfill \\ & =\left(\sum _{i=0}^{w_1-1}{y}^{w_{2}i}{(e^t-1)}^{w_{2}i}\right)\times \left(\frac{2}{1-y^{w_1}{(e^t-1)}^{w_1}}\right)\hfill \\ & =\sum _{i=0}^{w_1-1}\sum _{l=0}^{w_{2}i}\left(\genfrac{}{}{0pt}{}{w_{2}i}{l}\right)y^{w_{2}i}{(-1)}^l\frac{2}{1-y^{w_1}{(e^t-1)}^{w_1}}{e}^{(w_{2}i-l)t}\hfill \\ & =2\sum _{n=0}^{\infty }\left(\sum _{i=0}^{w_1-1}\sum _{l=0}^{w_{2}i}\left(\genfrac{}{}{0pt}{}{w_{2}i}{l}\right)y^{w_{2}i}{(-1)}^l{F}_{n,w_1}(w_{2}i-l,y)\right)\frac{t^n}{n!}.\hfill \end{array}$$

(23)

Interchanging the roles of $w_1$ and $w_2$, by (14), we get

$$\begin{array}{cc}\hfill I& =\left({\int }_{{\mathbb{Z}}_p}{(-y^{w_2}{(e^t-1)}^{w_2})}^{x}d{\mu }_{-1}\left(x\right)\right)\times \left(\frac{{\int }_{{\mathbb{Z}}_p}{(-y^{w_1}{(e^t-1)}^{w_1})}^{x}d{\mu }_{-1}\left(x\right)}{{\int }_{{\mathbb{Z}}_p}{(-y^{w_1{w}_2}{(e^t-1)}^{w_1{w}_2})}^{x}d{\mu }_{-1}\left(x\right)}\right)\hfill \\ & =\left({\int }_{{\mathbb{Z}}_p}{(-y^{w_2}{(e^t-1)}^{w_2})}^{x}d{\mu }_{-1}\left(x\right)\right)\times \left(\frac{1-y^{w_1{w}_2}{(e^t-1)}^{w_1{w}_2}}{1-y^{w_1}{(e^t-1)}^{w_1}}\right)\hfill \\ & =\left(\sum _{i=0}^{w_2-1}{y}^{w_{1}i}{(e^t-1)}^{w_{1}i}\right)\times \left(\frac{2}{1-y^{w_2}{(e^t-1)}^{w_2}}\right)\hfill \\ & =\sum _{i=0}^{w_2-1}\sum _{l=0}^{w_{1}i}{y}^{w_{1}i}\left(\genfrac{}{}{0pt}{}{w_{1}i}{l}\right){(-1)}^l\frac{2}{1-y^{w_2}{(e^t-1)}^{w_2}}{e}^{(w_{1}i-l)t}\hfill \\ & =2\sum _{n=0}^{\infty }\left(\sum _{i=0}^{w_2-1}\sum _{l=0}^{w_{1}i}{y}^{w_{1}i}\left(\genfrac{}{}{0pt}{}{w_{1}i}{l}\right){(-1)}^l{F}_{n,w_2}(w_{1}i-l,y)\right)\frac{t^n}{n!}.\hfill \end{array}$$

(24)

Hence, by Equations (23) and (24), we obtain the following theorem.

Theorem 2.

For $w_1,w_2\in \mathbb{N}$ with $w_1\equiv 1$ (mod 2), $w_2\equiv 1$ (mod 2), $n\ge 0$, we have

$$\begin{array}{c}\hfill \sum _{i=0}^{w_1-1}\sum _{l=0}^{w_{2}i}{y}^{w_{2}i}\left(\genfrac{}{}{0pt}{}{w_{2}i}{l}\right){(-1)}^l{F}_{n,w_1}(w_{2}i-l,y)=\sum _{i=0}^{w_2-1}\sum _{l=0}^{w_{1}i}{y}^{w_{1}i}\left(\genfrac{}{}{0pt}{}{w_{1}i}{l}\right){(-1)}^l{F}_{n,w_2}(w_{1}i-l,y).\end{array}$$

(25)

Remark 2.

Especially, if we take $w_1=1$, then by Theorem 2, we get

$$\begin{array}{c}\hfill F_n\left(y\right)=\sum _{i=0}^{w_2-1}\sum _{l=0}^i\left(\genfrac{}{}{0pt}{}{i}{l}\right)y^i{(-1)}^l{F}_{n,w_2}(i-l,y).\end{array}$$

(26)

3. Conclusions

In this paper, we introduced w-torsion Fubini polynomials as a generalization of Fubini polynomials and expressed the generating function of w-torsion Fubini polynomials by means of a fermionic p-adic integral on ${\mathbb{Z}}_p$. Then we derived some new symmetric identities for the w-torsion Fubini and two variable w-torsion Fubini polynomials by investigating a quotient of such p-adic integrals on ${\mathbb{Z}}_p$, representing generating functions of three w-torsion Fubini polynomials. It seems that they are the first double symmetric identities on Fubini polynomials. As was done, for example in [4,20,21], we expect that this result can be extended to the case of triple symmetric identities. That is one of our next projects.