A BICOMPLEX HURWITZ ZETA-FUNCTION 1. Introduction

26 A. Javtokasfor all z 1 + z 2 i 2 ∈ X 1 × e X 2 , is T-holomorphic on the domain X 1 × e X 2 , andf ′ (z 1 + z 2 i 2 ) = f ′ e1(z 1 − z 2 i 1 )e 1 + f ′ e2(z 1 + z 2 i 1 )e 2 .for all z 1 + z 2 i 2 ∈ X 1 × e X 2 .Theorem 3. Let X be a domain in T, and let f : X → T be a T-holomorphicfunction on X. Then there exist holomorphic functions f e1 : X 1 → C(i 1 ) andf e2 : X 2 → C(i 1 ) with X 1 = P 1 (X) and X 2 = P 2 (X) such thatf(z 1 + z 2 i 2 ) = f e1 (z 1 − z 2 i 1 )e 1 + f e2 (z 1 + z 2 i 1 )e 2for all z 1 + z 2 i 2 ∈ X. We also note that X 1 and X 2 are also domains ofC(i 1 ).The bicomplex Riemann zeta-function was rst dened in [6]. Let ω =z 1 + z 2 i 2 ∈ T with Re(z 1 ) > 1 and |Im(z 2 )| 1 and xed α, 0 < α 1, is dened byζ(s, α) =∞∑m=01(m + α) s .In this paper, we introduce a Hurwitz zeta-function for bicomplex numbers.More precisely, we obtain a holomorphic Hurwitz zeta-function of two complexvariables satisfying the complexied Cauchy-Riemann equations.2. Bicomplex Hurwitz zeta-functionIn this section, we dene an expression∞∑n=01(n + α) ω ,where ω is a bicomplex number. For this, we need the following denition.