Some Identities on the High-Order q-Euler Numbers and Polynomials with Weight 0

and Applied Analysis 3 From the left hand side of (23), we have


Introduction
As a well-known definition, the Euler polynomial   () is given by In the special case,  = 0,   (0) =   is the th Euler number.
In [9], Kim derived some identities between the sums of products of Frobenius-Euler polynomials and Frobenius-Euler polynomials of higher order.The main idea is to construct nonlinear ordinary differential equations with respect to  which are closely related to the generating function of Frobenius-Euler polynomial.In [3], Choi considered nonlinear ordinary differential equations with respect to  not .
In this paper, we construct nonlinear ordinary differential equations with respect to .The purpose of this paper is to give some new identities on the high order -Euler numbers and polynomials with weight 0 by using the differential equations of .

Construction of Nonlinear Differential Equations
We define From ( 7) and ( 8), we note that By differentiating (8) with respect to , we get (1) +  =  2 . ( By differentiating (10) with respect to , we get where  () =   /  .By the derivative of (11) with respect to , we have Continuing this process, we get Let us consider the derivative of ( 13) with respect to  to find the coefficient   () in (13).
We know that (28) has a unique solution under some conditions as follows.
Theorem A (see [17, page 65]).Suppose that , , and  are of class  1 in a domain Ω of R 3 containing the point ( 0 ,  0 ,  0 ) and suppose that Then in a neighborhood  of ( 0 ,  0 ) there exists a unique solution of (28) at every point of initial curve contained in .
It is customary to write (27) in the form Since /( − 1) = / is separable, we get 1 is a solution of partial differential equation of (27).From (30), we get the linear equation By the integrating factor method, we have The exponential integral   () is defined by where  is Euler constant. 2 is another solution of partial differential equation of (27), and  1 and  2 are linearly independent.