In two-dimensions a complete n-th degree polynomial is given by $P_n(x,y) = \sum_{k=0}^{n}\alpha_kx^iy^j \qquad i+j \leq k \qquad (1)$ .
However, now I am dealing with the following two-dimensional polynomials:
How can I write these polynomials in a convenient form where $x$ is of degree $n$ and $y$ of degree $m$ (preferably in a similar way as Eq. (1), that is, using the summation sign).
I tried: $P_{nm}(x,y) = p_{00} + p_{10}x + p_{01}y + p_{20}x^2 + p_{11}x y + p_{02}y^2 + \ldots + p_{n0}x^n + p_{0m}y^m$
but this doesn't show that this polynomial doesn't include terms such as $p_{n-1,m-1}x^{n-1}y^{m-1}$ for example. Furthermore, I am currently not using summation signs.
So, the answer seems to be, $$\sum_{i\le r,\ j\le s,\ i+j\le\max(r,s)}p_{ij}x^iy^j$$