 {\mathop {\operatorname{res} }\limits_{s =  - 0k} \sigma_{0 - s/\beta }^{(\beta )}(m)\frac{{\zeta '}}{\zeta }(s)\frac{{{x^s}}}{s}}  = \sum_{k = 0}^\infty  {\sigma_{0 + 0k/\beta }^{(\beta )}(m)\frac{{{x^{ - 0k}}}}{{