The efficient computation of some generalised exponential integrals

The accurate and efficient computation of the special functions Gk(x) is discussed, whereGk(x)=1(k−1)!∫1∞ exp(−xy)(log y)k−1 dyy.These functions appear in the computation of the derivatives of the L-series of an elliptic curve, and in radiative transfer problems from astrophysics. By dividing (0,∞) into 3 sub-intervals, we derive Chebyshev polynomial expansions for Gk, k=1,…,4 with the coefficients given to an accuracy of 20 decimal places.