Integer Points in Simplex

Question

Let $$A_w(d,q):=\left\{{\bf k} \in \mathbb{N}_0^d: \sum_{j=1}^d w_j k_j \leq q\right\}$$ denote the number of non-negative integer points in the $\ell_1$-ellipse with semi-axes of length $\frac{q}{w_j}$ (which is a simplex). Similarly, $$B_w(d,q):=\left\{{\bf k} \in \mathbb{N}_+^d: \sum_{j=1}^d w_j k_j \leq q\right\}$$ denotes the number of positive integer points in this simplex.

We can assume that $0 < w_1 \leq w_2 \ldots \leq w_d$ and $q \geq w_d$.

Is there a relationship between $A_w(d,q)$ and $B_w(d,q)$ in the sense that if I have an exact formula for $A_w(d,q)$ I also have an exact formula for $B_w(d,q)$? For example something like $$B_w(d,q) = A_\tilde{w}(d,\tilde{q}) .$$

Thanks!

Answer 1

I found a solution to my problem in this paper - formulas (1.2) and (1.3)

http://homepages.math.uic.edu/~yau/35%20publications/An%20upper.pdf