High precision computation and a new asymptotic formula for the generalized Stieltjes constants

We provide an efficient method to evaluate the generalized Stieltjes constants ${\gamma }_n(a)$ numerically to arbitrary accuracy for large $n$ and $n\gg \mathrm{\mid }a\mathrm{\mid }$ values. The method uses an integral representation for the constants and evaluates the integral by applying the double exponential (DE) quadrature method near the saddle points of the integrands. Further, we provide a highly accurate asymptotic formula for the generalized Stieltjes constants.