... Ekn,χ,ζx.1.12The purpose of this paper is to present a systemic study of some formulas of thetwisted q-extension of the generalized Euler numbers and polynomials of order k attachedto χ.2. ... q-extension of the generalized Euler numbers and polynomials of order k attached to χ.Advances in Difference Equations 1116 M. Can, M. Cenkci, V. Kurt, and Y. Simsek, “Twisted Dedekind type sums ... ,nr0qrj1h−jnj−ζn1···nr⎛⎝rj1χnj⎞⎠en1···nrxqt∞n0Eh,rn,χ,ζ,qxtnn!.3.3By Laurent series and Cauchy residue theorem in 3.1 and 3.3,weobtainthefollowing theorem.Theorem 3.2. Let χ be Dirichlet’s character with odd conductor d ∈ N, and let ζ e2πi/d....