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Symmetry 2018, 10(6), 219; https://doi.org/10.3390/sym10060219

Article
Symmetric Identities for Fubini Polynomials
1
Department of Mathematics, College of Science, Tianjin Polytechnic University, Tianjin 300160, China
2
Department of Mathematics, Kwangwoon University, Seoul 139-701, Korea
3
Department of Mathematics, Sogang University, Seoul 121-742, Korea
4
Department of Mathematics Education and ERI, Gyeongsang National University, Jinju, Gyeongsangnamdo 52828, Korea
*
Author to whom correspondence should be addressed.
Received: 20 April 2018 / Accepted: 13 June 2018 / Published: 14 June 2018

Abstract

:
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.
Keywords:
Fubini polynomials; w-torsion Fubini polynomials; fermionic p-adic integrals; symmetric identities

1. 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
Z p f ( x ) d μ 1 ( x ) = lim N x = 0 p N 1 f ( x ) ( 1 ) x ,
where μ 1 ( x + p N Z p ) = ( 1 ) x .
We can easily deduce from (1) that (see [2,3])
Z p f ( x + 1 ) d μ 1 ( x ) + Z p f ( x ) d μ 1 ( x ) = 2 f ( 0 ) .
By invoking (2), we easily get (see [2,4])
Z p e ( x + y ) t d μ 1 ( y ) = 2 e t + 1 e x t = n = 0 E n ( x ) t n n ! ,
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])
n = 0 F n ( x , y ) t n n ! = 1 1 y ( e t 1 ) e x t .
When x = 0 , F n ( y ) = F n ( 0 , y ) , ( n 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 ( x ) , ( n 0 ) . By (4), we easily get (see [6]),
F n ( y ) = k = 0 n S 2 ( n , k ) k ! y k , ( n 0 ) ,
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
1 1 y w ( e t 1 ) w e x t = n = 0 F n , w ( x , y ) t n n ! .
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 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 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.

2. Symmetric Identities for w -torsion Fubini and Two Variable w -torsion Fubini Polynomials

From (2), we note that
Z p ( 1 ) x ( y ( e t 1 ) ) x d μ 1 ( x ) = 2 1 y ( e t 1 ) = 2 n = 0 F n ( y ) t n n ! ,
and
e x t Z p ( 1 ) z ( y ( e t 1 ) ) z d μ 1 ( z ) = 2 1 y ( e t 1 ) e x t = 2 n = 0 F n ( x , y ) t n n ! .
From (7) and (8), we note that
l = 0 x l t l l ! m = 0 2 F m ( y ) t m m ! = e x t Z p ( 1 ) z ( y ( e t 1 ) ) z d μ 1 ( z ) = n = 0 2 F n ( x , y ) t n n ! .
Thus, by (9), we easily get
l = 0 n n l x l F n l ( y ) = F n ( x , y ) , ( n 0 ) .
Now, we observe that
1 y k ( e t 1 ) k 1 y ( e t 1 ) = i = 0 k 1 y i ( e t 1 ) i = i = 0 k 1 l = 0 i i l ( 1 ) i l y i e l t = n = 0 i = 0 k 1 l = 0 i i l ( 1 ) i l y i l n t n n ! = n = 0 i = 0 k 1 y i Δ i 0 n t n n ! ,
where Δ f ( x ) = f ( x + 1 ) f ( x ) .
For w N , the w-torsion Fubini polynomials are represented by means of the following fermionic p-adic integral on Z p :
Z p ( y w ( e t 1 ) w ) x d μ 1 ( x ) = 2 1 y w ( e t 1 ) w = n = 0 2 F n , w ( y ) t n n ! ,
From (7) and (12), we have
Z p ( y ( e t 1 ) ) x d μ 1 ( x ) Z p ( y w 1 ( e t 1 ) w 1 ) x d μ 1 ( x ) = 1 y w 1 ( e t 1 ) w 1 1 y ( e t 1 ) = i = 0 w 1 1 y i ( e t 1 ) i = n = 0 i = 0 w 1 1 y i Δ i 0 n t n n ! , ( w 1 N ) .
For w 1 , w 2 N , we let
I = Z p ( y w 1 ( e t 1 ) w 1 ) x 1 d μ 1 ( x 1 ) Z p ( y w 2 ( e t 1 ) w 2 ) x 2 d μ 1 ( x 2 ) Z p ( y w 1 w 2 ( e t 1 ) w 1 w 2 ) x d μ 1 ( x ) .
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
I = Z p ( y w 1 ( e t 1 ) w 1 ) x d μ 1 ( x ) × Z p ( y w 2 ( e t 1 ) w 2 ) x d μ 1 ( x ) Z p ( y w 1 w 2 ( e t 1 ) w 1 w 2 ) x d μ 1 ( x ) .
First, we observe that
Z p ( y w 2 ( e t 1 ) w 2 ) x d μ 1 ( x ) Z p ( y w 1 w 2 ( e t 1 ) w 1 w 2 ) x d μ 1 ( x ) = 1 y w 1 w 2 ( e t 1 ) w 1 w 2 1 y w 2 ( e t 1 ) w 2 = i = 0 w 1 1 y w 2 i ( e t 1 ) w 2 i = i = 0 w 1 1 y w 2 i l = 0 w 2 i w 2 i l ( 1 ) w 2 i l e l t = n = 0 i = 0 w 1 1 y w 2 i Δ w 2 i 0 n t n n ! .
From (15) and (16), we can derive the following equation.
I = Z p ( y w 1 ( e t 1 ) w 1 ) x d μ 1 ( x ) × Z p ( y w 2 ( e t 1 ) w 2 ) x d μ 1 ( x ) Z p ( y w 1 w 2 ( e t 1 ) w 1 w 2 ) x d μ 1 ( x ) = m = 0 2 F m , w 1 ( y ) t m m ! × k = 0 ( i = 0 w 1 1 y w 2 i Δ w 2 i 0 k t k k ! = n = 0 2 k = 0 n i = 0 w 1 1 y w 2 i Δ w 2 i 0 k F n k , w 1 ( y ) n k t n n ! .
Interchanging the roles of w 1 and w 2 , by (14), we get
I = Z p ( y w 2 ( e t 1 ) w 2 ) x d μ 1 ( x ) × Z p ( y w 1 ( e t 1 ) w 1 ) x d μ 1 ( x ) Z p ( y w 1 w 2 ( e t 1 ) w 1 w 2 ) x d μ 1 ( x ) .
We note that
Z p ( y w 1 ( e t 1 ) w 1 ) x d μ 1 ( x ) Z p ( y w 1 w 2 ( e t 1 ) w 1 w 2 ) x d μ 1 ( x ) = 1 y w 1 w 2 ( e t 1 ) w 1 w 2 1 y w 1 ( e t 1 ) w 1 = i = 0 w 2 1 y w 1 i ( e t 1 ) w 1 i = n = 0 i = 0 w 2 1 y w 1 i Δ w 1 i 0 n t n n ! .
Thus, by (18) and (19), we get
I = Z p ( y w 2 ( e t 1 ) w 2 ) x d μ 1 ( x ) × Z p ( y w 1 ( e t 1 ) w 1 ) x d μ 1 ( x ) Z p ( y w 1 w 2 ( e t 1 ) w 1 w 2 ) x d μ 1 ( x ) = m = 0 2 F m , w 2 ( y ) t m m ! × k = 0 ( i = 0 w 2 1 y w 1 i Δ w 1 i 0 k t k k ! = n = 0 2 k = 0 n i = 0 w 2 1 y w 1 i Δ w 1 i 0 k F n k , w 2 ( y ) n k t n n ! .
The following theorem is now obtained by Equations (17) and (20).
Theorem 1.
For w 1 , w 2 N with w 1 1 (mod 2), w 2 1 (mod 2), n 0 , we have
k = 0 n i = 0 w 1 1 n k F n k , w 1 ( y ) y w 2 i Δ w 2 i 0 k = k = 0 n i = 0 w 2 1 n k F n k , w 2 ( y ) y w 1 i Δ w 1 i 0 k .
Remark 1.
In particular, for w 1 = 1 , we have
F n ( y ) = k = 0 n i = 0 w 2 1 n k F n k , w 2 ( y ) y i Δ i 0 k .
By expressing I in a different way, we have
I = Z p ( y w 1 ( e t 1 ) w 1 ) x d μ 1 ( x ) × Z p ( y w 2 ( e t 1 ) w 2 ) x d μ 1 ( x ) Z p ( y w 1 w 2 ( e t 1 ) w 1 w 2 ) x d μ 1 ( x ) = Z p ( y w 1 ( e t 1 ) w 1 ) x d μ 1 ( x ) × 1 y w 1 w 2 ( e t 1 ) w 1 w 2 1 y w 2 ( e t 1 ) w 2 = i = 0 w 1 1 y w 2 i ( e t 1 ) w 2 i × 2 1 y w 1 ( e t 1 ) w 1 = i = 0 w 1 1 l = 0 w 2 i w 2 i l y w 2 i ( 1 ) l 2 1 y w 1 ( e t 1 ) w 1 e ( w 2 i l ) t = 2 n = 0 i = 0 w 1 1 l = 0 w 2 i w 2 i l y w 2 i ( 1 ) l F n , w 1 ( w 2 i l , y ) t n n ! .
Interchanging the roles of w 1 and w 2 , by (14), we get
I = Z p ( y w 2 ( e t 1 ) w 2 ) x d μ 1 ( x ) × Z p ( y w 1 ( e t 1 ) w 1 ) x d μ 1 ( x ) Z p ( y w 1 w 2 ( e t 1 ) w 1 w 2 ) x d μ 1 ( x ) = Z p ( y w 2 ( e t 1 ) w 2 ) x d μ 1 ( x ) × 1 y w 1 w 2 ( e t 1 ) w 1 w 2 1 y w 1 ( e t 1 ) w 1 = i = 0 w 2 1 y w 1 i ( e t 1 ) w 1 i × 2 1 y w 2 ( e t 1 ) w 2 = i = 0 w 2 1 l = 0 w 1 i y w 1 i w 1 i l ( 1 ) l 2 1 y w 2 ( e t 1 ) w 2 e ( w 1 i l ) t = 2 n = 0 i = 0 w 2 1 l = 0 w 1 i y w 1 i w 1 i l ( 1 ) l F n , w 2 ( w 1 i l , y ) t n n ! .
Hence, by Equations (23) and (24), we obtain the following theorem.
Theorem 2.
For w 1 , w 2 N with w 1 1 (mod 2), w 2 1 (mod 2), n 0 , we have
i = 0 w 1 1 l = 0 w 2 i y w 2 i w 2 i l ( 1 ) l F n , w 1 ( w 2 i l , y ) = i = 0 w 2 1 l = 0 w 1 i y w 1 i w 1 i l ( 1 ) l F n , w 2 ( w 1 i l , y ) .
Remark 2.
Especially, if we take w 1 = 1 , then by Theorem 2, we get
F n ( y ) = i = 0 w 2 1 l = 0 i i l y i ( 1 ) l F n , w 2 ( i l , y ) .

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 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 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.

Author Contributions

T.K. and D.S.K. conceived the framework and structured the whole paper; T.K. wrote the paper; G.-W.J. and J.K. checked the results of the paper; D.S.K. and J.K. completed the revision of the article.

Conflicts of Interest

The authors declare no conflict of interest.

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