Symmetry Fermionic p-Adic q-Integral on Z p for Eulerian Polynomials

Kim et al. 2012 introduced an interesting p-adic analogue of the Eulerian polynomials. They studied some identities on the Eulerian polynomials in connection with the Genocchi, Euler, and tangent numbers. In this paper, by applying the symmetry of the fermionic p-adic q-integral on Zp, defined by Kim 2008 , we show a symmetric relation between the q-extension of the alternating sum of integer powers and the Eulerian polynomials.


Introduction
The Eulerian polynomials A n t , n 0, 1, . .., which can be defined by the generating function Let p be a fixed odd prime number.Let Z p , Q p , and C p be the ring of p-adic integers, the field of p-adic numbers, and the completion of the algebraic closure of Q p , respectively.Let | • | p be the p-adic valuation on Q, where |p| p p −1 .The extended valuation on C p is denoted by the same symbol | • | p .Let q be an indeterminate, where |1 − q| p < 1.Then, the q-number is defined by For a uniformly or strictly differentiable function f : Z p → C p see 1, 3-6 , the fermionic p-adic q-integral on Z p is defined by Then, it is easy to see that where f 1 x f x 1 .By using the same method as that described in 1 , and applying 1.5 to f, where f x q 1−ω x e −x 1 q ωt 1.6 for ω ∈ Z >0 , we consider the generalized Eulerian polynomials on Z p by using the fermionic p-adic q-integral on Z p as follows: By expanding the Taylor series on the left-hand side of 1.7 and comparing the coefficients of the terms t n /n!, we get We note that, by substituting ω 1 into 1.8 , x n dμ −1/q x 1.9 is the Witt's formula for the Eulerian polynomials in 1, Theorem 1 .Recently, Kim et al. 1 investigated new properties of the Eulerian polynomials A n −q at q 1 associated with the Genocchi, Euler, and tangent numbers.
Let T k,1/q n denote the q-extension of the alternating sum of integer powers, namely, is the alternating sum of integer powers see 4 .In particular, we have Let ω 1 , ω 2 be any positive odd integers.Our main result of symmetry between the qextension of the alternating sum of integer powers and the Eulerian polynomials is given in the following theorem, which is symmetric in ω 1 and ω 2 .

1.12
Observe that Theorem 1.1 can be obtained by the same method as that described in 4 .If q 1, Theorem 1.1 reduces to the form stated in the remark in 4, page 1275 .
Using 1.11 , if we take ω 2 1 in Theorem 1.1, we obtain the following corollary.
Corollary 1.2.Let ω 1 be any positive odd integer and n ≥ 0.Then, one has

Proof of Theorem 1.1
For the proof of Theorem 1.1, we will need the following two identities see 2.4 and 2.5 related to the Eulerian polynomials and the q-extension of the alternating sum of integer powers.

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Let ω 1 , ω 2 be any positive odd integers.From 1.7 , we obtain This has an interesting p-adic analytic interpretation, which we shall discuss below see Remark 2.1 .It is easy to see that the right-hand side of 2.1 can be written as

2.2
In 1.10 , let q q ω 1 .The left-hand, right-hand side, by definition, becomes A comparison of 2.1 and 2.3 yields the identity By slightly modifying the derivation of 2.4 , we can obtain the following identity: Remark 2.1.The derivations of identities are based on the fermionic p-adic q-integral expression of the generating function for the Eulerian polynomials in 1.7 and the quotient of integrals in 2.4 , 2.5 that can be expressed as the exponential generating function for the q-extension of the alternating sum of integer powers.
Observe that similar identities related to the Eulerian polynomials and the q-extension of the alternating sum of integer powers in 2.4 and 2.5 can be found, for instance, in 3, 1.8 , 4, 21 , and 6, Theorem 4 .
Proof of Theorem 1.1.Let ω 1 , ω 2 be any positive odd integers.Using the iterated fermionic padic q-integral on Z p and 1.7 , we have

2.6
Now, we put From 1.7 and 2.5 , we see that

2.8
On the other hand, from 1.7 and 2.4 , we have

2.9
By comparing the coefficients on both sides of 2.8 and 2.9 , we obtain the result in Theorem 1.1.
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Concluding Remarks
Note that many other interesting symmetric properties for the Euler, Genocchi, and tangent numbers are derivable as corollaries of the results presented herein.For instance, considering 1, 5 , These formulae are valid for any positive odd integers ω 1 , ω International Journal of Mathematics and Mathematical Sciences and they can help simplify the proofs of known properties.For more important properties, see, for instance, 1 or 2 .
where δ n,k is the Kronecker symbol see 1 .Thus far, few recurrences for the Eulerian polynomials other than 1.2 have been reported in the literature.Other recurrences are of importance as they might reveal new aspects and properties of the Eulerian polynomials, 1 where E n denotes the nth Euler number defined by E n : E n 0 , and the Euler polynomials are defined by the generating function between the Genocchi and the Bernoulli numbers.It follows from 3.6 and the Staudt-Clausen theorem that the Genocchi numbers are integers.It is easy to see that It is well known that the tangent coefficients or numbers T n , defined by tan t Ramanujan 7, page 5 observed that 2 n 2 n − 1 B n /n and, therefore, the tangent coefficients, are integers for n ≥ 1.From 3.3 , 3.6 , 3.7 , and 3.10 , the obtained symmetric formulae involve the Bernoulli, Genocchi, and tangent numbers see 1 .