How many lattice paths are there from an arbitrary point $(a,b)$ to another point $(c,d)$ that stay weakly (i.e. it can touch the line) under a slope of the form $y = \mu x$, with $\mu \in \mathbb{N}$?
I managed to find results for the slope $y = x$ from chapter 10 of "Bóna, M. (Ed.). (2015). Handbook of enumerative combinatorics. CRC Press.": $${c+d-a-b \choose c-a} - {c+d-a-b \choose c-b+1}$$
From the same source, I also have results for a slope $y = \frac{1}{\mu} x$: $${c+d-a-b \choose c-a} - \sum_{i = \lfloor \frac{a}{\mu} \rfloor +1}^d {i(\mu+1)-a-b-1 \choose i - b} \frac{c-\mu d+1}{c+d-i(\mu +1)+1} {c+d-i(\mu +1) +1 \choose d-i}$$
However there does not seem to be existing results for a slope of positive integer. There is an approach for a slope $\mu y = v x$, but seems like overkill for a simple slope $\mu \in \mathbb{N}$. The approach makes use of generating functions which I admit I have a hard time to follow. I also don't think I can use a simple reflection method in this case?
