Each of parameters A-D is a polynomial function of t:
$$A = \text{"m"} + \text{"mt"}.t + \text{"mt2"}.t^2 + \text{"mt3"}.t^3 + \text{" mt4"}.t^4$$
$$B = \text{"m3/2“} + \text{"m3/2t"}.t + \text{"m3/2t2"}.t^2$$
$$C = \text{"m2“} + \text{"m2t"}.t$$
$$D= \text{"m5/2"}$$
The terms in quote means the equally denoted numerical values from the provided table.
The complete final function $1000 \cdot \Delta \rho=f(m,t)$ has for each term $a_{b,c} \cdot m^a \cdot t^b$ the value for $a_{b,c}$ equal to the particular table value named "$m^bt^c$".
So one can the complicated function use in 2 ways:
- The full blown function $1000 \cdot \Delta \rho=f(m,t) = \sum {\left( a_{b,c} \cdot m^a \cdot t^b \right)}$ without the need of coefficients A-D.
- Temperature specific $1000 \cdot \Delta \rho=A \cdot m + B \cdot m^{3/2} + C \cdot m^2 + D \cdot m^{5/2}$ with precalculated A-D values for given temperature.
BTW, it looks like a very interesting numerical approximation. Maybe even better could be rational functions like Padé approximant, that have smaller deviations and better extrapolation behaviour, compared to polynomials.
Feedback for the Padé approximation:
For approximating of analytical functions, see e.g Math SE site - how-to-compute-the-pade-approximation. For approximation of experimental discrete data, the best approach is probably numerical mathematics, the least square method combined with function minimum search algorithm. They probably used the same technique for their numerical model with half-integer powers.
Using half-integer power model is unusual. It may work better than polynomials in many cases, but just for positive numbers. Fortunately, they do not calculate with negative Celsius temperature :-) ) Note that Math SE site has even tag pade-approximation . Also, it is included in MATLAB libraries, if applies. See Octave as both online/offline free functional clone of MATLAB.
See also my answer in this Q: Is there a quick way to approximate a logarithm in Nernst equation?.
