Multiple gamma function

Remarks

Example: for $N = 1$ and $a_1 = 1$, $\zeta_1(s, w \mid 1)$ is the Hurwitz zeta function $\zeta(s, w)$. Lerch's formula $\frac{\partial}{\partial s}\zeta(s, w)\big|_{s=0} = \log \Gamma(w) - \frac12 \log 2\pi$ gives $\Gamma_1(w \mid 1) = \Gamma(w)/\sqrt{2\pi}$, where $\Gamma$ is the gamma function.

The identity $\zeta_N(s, w \mid a) - \zeta_N(s, w + a_j \mid a) = \zeta_{N-1}(s, w \mid a^{(j)})$, where $a^{(j)}$ omits $a_j$, gives the functional equation $$\Gamma_N(w + a_j \mid a) = \frac{\Gamma_N(w \mid a)}{\Gamma_{N-1}(w \mid a^{(j)})},$$ with the convention $\Gamma_0(w) = 1/w$. For $N = 1$ it reads $\Gamma_1(w + 1 \mid 1) = w\, \Gamma_1(w \mid 1)$, the functional equation of $\Gamma$.

Non-example of a valid parameter: if $a_1 = 0$, each term is repeated for every value of $n_1$, the series diverges for every $s$, and $\Gamma_N$ is not defined.

Several normalisations are in use; they differ by factors of the form $\exp(\text{polynomial in } w)$. The function $1/\Gamma_N(w \mid a)$ extends to an entire function of $w$.