On the special functions higher than the multiple gamma-functions (Q5930107)

Barnes' multiple zeta function is defined by NEWLINE\[NEWLINE\zeta_r(s;w;\omega_1,\cdots,\omega_r)= \sum_{m_1,\cdots,m_r=0}^\infty(w+m_1\omega_1+\cdots+m_r\omega_r)^{-s}.NEWLINE\]NEWLINE The author gives a formula representing \(\zeta_{r+1}''(0;w;\omega_1,\cdots,\omega_r,\omega_{r+1})\) by \(\zeta_r'' (0;w;\omega_1,\cdots,\omega_r)\) and Barnes' multiple Bernoulli polynomials. Here \('\) means the differentiation with respect to \(s\). This is a generalization of a well-known formula for the Hurwitz zeta-function. To this end the author studies special functions essentially equal to the contour integrals NEWLINE\[NEWLINE\int_I \frac{e^{-wt}}{\prod_{i=1}^r(1-e^{-\omega_it})}\frac{\log^2t}{t} dtNEWLINE\]NEWLINE and NEWLINE\[NEWLINE\int_I\frac{t}{1-e^{-t}}\frac{e^{-zt}}{t^{m+1}}\log^2t dt,NEWLINE\]NEWLINE where \(I\) is a contour that starts from \(+\infty\) on the real axis, encircles the origin once counter-clockwise and returns to \(+\infty\).