Analytic continuation of multiple \(q\)-zeta functions and their values at negative integers. (English) Zbl 1115.11068

Summary: In the paper we define the analytic continuation of the multiple \(q\)-zeta functions (the \(q\)-analog of Barnes’ multiple zeta function) in the complex number field:
\[ \zeta_{r,q}(s,x)=\sum^{\infty}_{n_1,\dots,n_r=0} \frac{q^x+\sum^r_{i=1}n_i}{[x+\sum^r_{i=1}n_i]^s_q} \] if \(\operatorname{Re}(x)>0\) and \(|q|<1\), where \([k]_q=(1-q^k)/(1-q)\). We also study their behavior near the poles, present a partial answer to the open question recalled in [ibid. 10, No. 1, 91–98 (2003; Zbl 1072.11090)], and finally give the corresponding functional equations. Asymptotic formulas for some relevant \(q\)-functions are discussed.


11S80 Other analytic theory (analogues of beta and gamma functions, \(p\)-adic integration, etc.)
33D15 Basic hypergeometric functions in one variable, \({}_r\phi_s\)
11M38 Zeta and \(L\)-functions in characteristic \(p\)


Zbl 1072.11090