Polynomial regression with two variables. How can I find expressions to describe the coefficients?

I'm not sure if this is an appropriate place for this question, so please feel free to redirect me if it is not. I just moved it from Super User, where it seemed like there weren't many similar questions. Please also feel free to suggest tags.

I'm trying to modify part of an old code. It uses regression to describe the relationship between two variables (described as a fourth order power series in X and y).

I know very little about statistics, but can figure out how to get a regression polynomial with constant coefficients for a relationship like in the figure below. It would be something like:

Phi=β0+β1y+β2y2+β3y3+β4y4

My problem is that the code does not use simple constant coefficients (β in the expression above). It does this (below, where now the coefficients are ki to avoid confusion and are described themselves in terms of coefficients, ai).

I need to know how to get the ai values so that I can change a parameter that the code currently uses to suit my own data

In the code, there is a list of the ai values and the only comment is that they are based on numerical fitting of measured data. I have my own unique data that I'm trying to fit though. Does anyone know what I should do to get them or even what to call that process (so I can search online for hints)? Any tips would be much appreciated.