Properties of -extensions of Euler numbers and polynomials which generalize those satisfied by and are used to construct -extensions of -adic Euler measures and define -adic -
 -series which interpolate -Euler numbers at negative integers. Finally, we give Kummer Congruence for the -extension of ordinary Euler numbers.

/pdf/a-note-on-the-euler-measures-4eb73kda0s.pdf

A Note on the -Euler Measures

In this paper, we construct trigonometric functions in the form of a sum T
 p
 (h, k) which is referred to as a Dedekind-type DC-(Dahee and Changhee) sum. We establish analytic properties of this sum, find its trigonometric representations, and prove a reciprocity theorem for these sums. Furthermore, we obtain relationships between the Clausen functions, polylogarithm function, Hurwitz zeta function, generalized Lambert series (G-series), Hardy-Berndt sums, and the sum T
 p
 (h, k). We also give some applications related to these sums and functions.

Special functions related to Dedekind-type DC-sums and their applications

The purpose of this paper is to present a systemic study of some families of multiple q-Euler numbers and polynomials and we construct multiple q-zeta functions which interpolate multiple q-Euler numbers at a negative integer. This is a partial answer to the open question in a previous publication (see Kim et al 2001 J. Phys. A: Math. Gen. 34 7633–8).

http://iopscience.iop.org/article/10.1088/1751-8113/43/25/255201/pdf

Barnes-type multiple q-zeta functions and q-Euler polynomials

In this paper we construct trigonometric functions of the sum T_{p}(h,k), which is called Dedekind type DC-(Dahee and Changhee) sums. We establish analytic properties of this sum. We find trigonometric representations of this sum. We prove reciprocity theorem of this sums. Furthermore, we obtain relations between the Clausen functions, Polylogarithm function, Hurwitz zeta function, generalized Lambert series (G-series), Hardy-Berndt sums and the sum T_{p}(h,k). We also give some applications related to these sums and functions.

Special functions related to dedekind type dc-sums and their applications

In this paper, we present new generating functions related to q-Euler numbers and polynomials of higher order. Using those generating functions, we present new identities involving q-Euler numbers and polynomials of higher order.

New approach to q-Euler polynomials of higher order

In the recent paper the interesting q-Euler numbers and polynomials introduced in JMAA. The purpose of this paper is to construct the modified q-Euler numbers and polynomiasl. Finally we will give the interesting many identities related to these numbers and polynomials.

The Modified q-Euler numbers and polynomials

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The Modified q-Euler numbers and polynomials