- Research Article
- Open Access
© Leechae Jang et al. 2009
- Received: 30 August 2009
- Accepted: 28 September 2009
- Published: 8 October 2009
- Integral Equation
- Inversion Formula
- Euler Polynomial
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:
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
The Barnes' type Euler polynomials are considered as follows:
where (cf. ).
By (1.9), we see that
From (1.10), we note that
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.
Therefore, we obtain the following theorem.
By (1.7), we easily see that
From (1.7), we can derive
By (2.6), one has
Hence we obtain the following theorem.
It is not difficult to show that the following integral equation is satisfied:
By (2.1), (2.3), and (2.9), we obtain the following corollary.
From (2.3), we note that
This paper was supported by Konkuk University (2009).
- Acikgoz M, Simsek Y: On multiple interpolation functions of the Nörlund-type -Euler polynomials. Abstract and Applied Analysis 2009, Article ID 382574, vol. 2009:-14 Pages.Google Scholar
- Cangul IN, Kurt V, Ozden H, Simsek Y: On the higher-order - -genocchi numbers. Advanced Studies in Contemporary Mathematics 2009,vol. 19 no.(1):pp.39–57.MathSciNetMATHGoogle Scholar
- Govil NK, Gupta V: Convergence of -Meyer-König-Zeller-Durrmeyer operators. Advanced Studies in Contemporary Mathematics 2009, vol 19: pp. 97–108.MATHMathSciNetGoogle Scholar
- Kim T: -Volkenborn integration. Russian Journal of Mathematical Physics 2002,vol. 9 no.(3):pp. 288–299.MATHMathSciNetGoogle Scholar
- Kim T: Some identities on the -Euler polynomials of higher order and -stirling numbers by the fermionic -adic integral on . Russian Journal of Mathematical Physics 2010., vol 17:Google Scholar
- Kim Y-H, Kim W, Ryoo CS: On the twisted -Euler zeta function associated with twisted -Euler numbers. Proceedings of the Jangjeon Mathematical Society 2009,vol. 12no. (1):pp. 93–100.MATHMathSciNetGoogle Scholar
- Simsek Y, Kim T, Pyung IS: Barnes' type multiple Changhee -zeta functions. Advanced Studies in Contemporary Mathematics 2005,vol. 10no. (2):pp. 121–129.MATHMathSciNetGoogle Scholar
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