Symmetric Identities for Fubini Polynomials

We represent the generating function of w-torsion Fubini polynomials by means of a fermionic p-adic integral on Zp. Then we investigate a quotient of such p-adic integrals on Zp, 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.


Introduction and Preliminaries
In recent years, various p-adic integrals on 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 Z p , Q p , and 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 Q p , respectively.The p-adic norm | • | p is normalized as |p| p = 1 p .Assume that f (x) is a continuous function on Z p .Then the fermionic p-adic integral of f (x) on Z p was introduced by Kim (see [2]) as where We can easily deduce from (1) that (see [2,3]) By invoking (2), we easily get (see [2,4]) where E n (x) are the usual Euler polynomials.
As is known, the two variable Fubini polynomials are defined by means of the following (see [5,6]) When x = 0, F n (y) = F n (0, y), (n ≥ 0), are called Fubini polynomials.Further, if y = 1, then Ob 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 (x), (n ≥ 0).By (4), we easily get (see [6]), where S 2 (n, k) are the Stirling numbers of the second kind.
For w ∈ N, we define the two variable w-torsion Fubini polynomials given by In particular, for x = 0, F n,w (y) = F n,w (0, y) are called the w-torsion Fubini polynomials.It is obvious that We represent the generating function of w-torsion Fubini polynomials by means of a fermionic p-adic integral on Z p .Then we investigate a quotient of such p-adic integrals on 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.

Symmetric Identities for w-torsion Fubini and Two Variable w-torsion Fubini Polynomials
From (2), we note that and From ( 7) and ( 8), we note that Thus, by (9), we easily get Now, we observe that where For w ∈ N, the w-torsion Fubini polynomials are represented by means of the following fermionic p-adic integral on Z p : From ( 7) and ( 12), we have For w 1 , w 2 ∈ N, we let 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 First, we observe that From ( 15) and ( 16), we can derive the following equation.
Interchanging the roles of w 1 and w 2 , by ( 14), we get We note that Thus, by ( 18) and ( 19), we get The following theorem is now obtained by Equations ( 17) and (20).