Good observation! It is not necessarily part of a power series expansion. The heat capacity data is generally just fit to a nonlinear correlation such as those shown here: trc.nist.gov/TDE/Equations/FEquations.html#Cp-IdealGas The DIPPR database has 2-3 different nonlinear relationships with some that are polynomials while others include terms like sinh or exp(). All of them in the DIPPR database are only a function of temperature and are probably chosen based on which form fits the data best over a specified temperature range. With any nonlinear empirical correlation, it is highly discouraged to use that data outside of the validated temperature range. There are JANNAF correlations for high temperature gasses, if needed.