Correction of the precision approximations of the Fermi–Dirac functions of integer index

Fermi–Dirac functions of integer index are widely used in problems of electronic transport in dense substances. Polynomial approximations were constructed for its quick computation. Such coefficients are founded for functions of index $1, 2, 3$, which provide ratio error $2\cdot10^{-16}$ with $9$ free parametrs. In this work we used C++ boost::multiprecision library, which allows to calculate with free number of digits. Precision of previously obtained formulas brought to $\sim 5\cdot10^{-18}$ and the same formula has been built for the index $k=4$. It is also shown that simple global formulas, consisting of small number of parameters, reasonably describe the order of value of the functions for all values of the argument and can be used for estimations.