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Note on the -Extension of Barnes' Type Multiple Euler Polynomials
Journal of Inequalities and Applications volume 2009, Article number: 136532 (2009)
We construct the -Euler numbers and polynomials of higher order, which are related to Barnes' type multiple Euler polynomials. We also derive many properties and formulae for our -Euler polynomials of higher order by using the multiple integral equations on .
Let be a fixed odd prime number. Throughout this paper, symbols , , , and will denote the ring of rational integers, the ring of -adic integers, the field of -adic rational numbers, and the completion of algebraic closure of , respectively. Let be the set of natural numbers and . Let be the normalized exponential valuation of with . When one talks of -extension, is variously considered as an indeterminate, a complex number or a -adic number . If one normally assumes If then one normally assumes We use the following notations:
Let a fixed positive odd integer with . For , we set
where lies in . The fermionic -adic -measures on are defined as
We say that is a uniformly differentiable function at a point and write , if the difference quotients have a limit as . For , let us begin with expression
which represents a -analogue of Riemann sums for in the fermionic sense (see [4, 5]). The integral of on is defined by the limit of these sums (as ) if this limit exists. The fermionic invariant -adic -integral of function is defined as
Note that if in , then
The Barnes' type Euler polynomials are considered as follows:
where (cf. ).
From (1.5), we can derive the fermionic invariant integral on as follows:
For , let , one has
By (1.9), we see that
From (1.10), we note that
In the view point of (1.11), we try to study the -extension of Barnes' type Euler polynomials by using the -extension of fermionic -adic invariant integral on .
The purpose of this paper is to construct the -Euler numbers and polynomials of higher order, which are related to Barnes' type multiple Euler numbers and polynomials. Also, we give many properties and formulae for our -Euler polynomials of higher order. Finally, we give the generating function for these -Euler polynomials of higher order.
2. Barnes' Type Multiple -Euler Polynomials
Let . For and with , we define the Barnes' type multiple -Euler polynomials as follows:
In the special case , are called the Barnes' type multiple -Euler numbers. From (2.1), one has
Therefore, we obtain the following theorem.
Let and . For , one has
By (1.7), we easily see that
From (1.7), we can derive
By (2.6), one has
Hence we obtain the following theorem.
For and , one has
It is not difficult to show that the following integral equation is satisfied:
where with . By (2.9), we obtain the following theorem.
Let and . For with , one has
For the special case in Theorem 2.3, one has
By (2.1), (2.3), and (2.9), we obtain the following corollary.
For and , one has
From (2.3), we note that
where is an odd positive integer. By (2.13), we obtain the following theorem.
For with (mod), one has
From (2.4), we can easily derive the following equation:
By differentiating both sides of (2.16) with respect to and comparing coefficients on both sides, one has
The inversion formula of Equation (2.4) at is given by
Thus, one has
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This paper was supported by Konkuk University (2009).
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Jang, L., Kim, T., Kim, YH. et al. Note on the -Extension of Barnes' Type Multiple Euler Polynomials. J Inequal Appl 2009, 136532 (2009). https://doi.org/10.1155/2009/136532
- Integral Equation
- Inversion Formula
- Euler Polynomial