The Bloch-Grüneisen integrals. Some analytic expressions (Q1805910)

The authors give analytic expressions of the Bloch-Grüneisen integrals \(I_n(x)\) [\textit{M. Deutsch}, J. Phys. A: Math. Gen. 20, L811--L813 (1987)] and of similar integrals \(K_n(x)\) defined by \[ I_n(x)= \int^x_0 {t^ne^t \over(e^t-1)^2} dt;\quad K_n(x)= \int^x_0 {t^ne^t\over e^{2t}-1} dt. \] The result include the power series expansion in terms of Bernoulli numbers and an expression as a finite sum of polylogarithms. The numerical accuracy of the formulas is discussed in detail.