The polar complex exponential transform is given by:
$M_{nl} = {1\over \pi } \int_{0}^{2\pi }\int_{0}^{1}\left[ H_{nl}( r,\theta) \right]^{\ast} f(r,\theta) rdrd\theta$
with $n$ and $l$ such that $|n|=|l| = 0,1,....\infty$
The polar cosine transform is given by:
$M^{{\rm C}}_{nl} = \Omega_{n}\int_{0}^{2\pi }\int_{0}^{1}\left[ H^{{\rm C}}_{nl}( r,\theta) \right]^{\ast} f(r,\theta) rdrd\theta$
with $n$ and $l$ such that $n, |l| = 0,1,....\infty$
The polar sine transform is given by:
$M^{{\rm S}}_{nl} = \Omega_{n}\int_{0}^{2\pi }\int_{0}^{1}\left[ H^{{\rm S}}_{nl}( r,\theta) \right]^{\ast} f(r,\theta) rdrd\theta$
with $n$ and $l$ such that $n = 1,....\infty$, $|l| = 0,1,....\infty$
Now, the equation $M$ is not my focus here but rather the range of $n$ and $l$ which differs for each equation. Could anyone please explain how are $n$ and $l$ chosen based on the range it falls into?
For the first equation, i assume $n$ must equate to $l$. For example:
$n = 0,1,2$
$l = 0,1,2$
Hence the possible combinations are [0,0],[0,1],[0,2],[1,0],[1,1],[1,2],[2,0],[2,1],[2,2].
However, i do not quite get the other two equations with their respective values for n and l. Any help is much appreciated.