ISSN 1061-9208, Russian Journal of Mathematical Physics, Vol. 15, No. 1, 2008, pp. 58-65. c  Pleiades Publishing, Ltd., 2008. Analytic Continuation of the Multiple Daehee q -l -Functions Associated with Daehee Numbers T. Kim* and Y. Simsek** *EECS, Kyungpook National University, Taegu 702-701, S. Korea E-mail: tkim@knu.ac.kr, tkim64@hanmail.net **Akdeniz University, Faculty of Science, Department of Mathematics, 07058, Antalya, Turkey E-mail: yilmazsimsek@hotmail.com, simsekyil63@yahoo.com, ysimsek@akdeniz.edu.tr Received November 30, 2005 Abstract. In this paper, we present a new generating function which is related to Daehee numbers. By using the Mellin transformation formula and the Cauchy theorem for this gen- erating function, we define multiple Daehee q-l-functions and q-zeta functions. We also give the values of this function at negative integers. DOI: 10.1134/S106192080801007X 1. INTRODUCTION, DEFINITIONS, AND NOTATION Multiple zeta functions and related numbers and functions occur within the context of knot theory and quantum field theory ([19, 21–23, 28]). There are many conjectures related to these sums, and their incompletion shows that both the communities of mathematicians and physicists do not completely understand the field at present. The Barnes multiple zeta functions and gamma functions were encountered by Shintani within the context of analytic number theory. The special values of multiple zeta functions at positive integers have come to the foreground in recent years both in connection with theoretical physics (Feynman diagrams) and the theory of mixed Tate motives. Historically, Euler already investigated the double zeta values in the XVIIIth century ([45, 3, 4, 42, 43, 14, 48, 57, 51]). In [23], T. Kim defined the analytic continuation of multiple zeta functions (the Euler–Barnes multiple zeta functions) depending on parameters a 1 ,a 2 ,...,a r taking positive values in the com- plex number field, ζ r (s, w, u|a 1 ,...,a r )= ∞  m 1 ,...,m r =0 u −(m 1 +···+m r ) (w + m 1 a 1 + ··· + m r a r ) s , where Re(w) > 0 and u ∈ C with |u| > 1. He studied some interesting properties of the Euler– Barnes multiple zeta functions at negative integers. He also constructed p-adic Euler integrals used in the proof of Witt-type formulas for the Barnes type multiple Frobenius–Euler numbers. Further, he tried to include simple numerical calculations and experimentations for the aficionados to discover their own conjectures and problems. Recently, many problems of non-Archimedean fundamental analysis were developed in mathe- matical physics and mathematics (for example, in measure theory, operator theory, and the theory of function spaces [16, 20, 37, 57, 4, 47, 7]). In particular, the physical interpretation of the spectrum of the p-adic position operator seems to be important for all of p-adic mathematical physics. The classical Euler numbers were defined by the rule 2 e t +1 = ∞  n=0 E n n! t n , |t| < 2π ([19, 23, 49, 29]). For u ∈ C with |u| > 1, the Frobenius–Euler polynomials are defined by the formula 1 − u e t − u e xt = e H(x,u)t = ∞  n=0 H n (x, u) t n n! , where we use the notation by symbolically replacing H m (x, u) by H m (x, u). In the case of x = 0, the Frobenius–Euler polynomials are called Frobenius–Euler numbers. Write H m (u)= H m (0,u) ([19, 23, 50]). In this paper, we denote by N the set of positive integers. Let χ be a primitive 58