The $p$-numerical semigroup of the triple of arithmetic progressions

For given positive integers $a_1,a_2,\dots,a_k$ with $\gcd(a_1,a_2,\dots,a_k)=1$, the denumerant $d(n)=d(n;a_1,a_2,\dots,a_k)$ is the number of nonnegative solutions $(x_1,x_2,\dots,x_k)$ of the linear equation $a_1 x_1+a_2 x_2+\dots+a_k x_k=n$ for a positive integer $n$. For a given nonnegative integer $p$, let $S_p=S_p(a_1,a_2,\dots,a_k)$ be the set of all nonnegative integers $n$'s such that $d(n)>p$. In this paper, we are interested in the $p$-Frobenius number, which is the maximum of the set of gaps $\mathbb N_0\backslash S_p$. Here $\mathbb N_0$ denotes the set of nonnegative integers. When $p=0$, $S=S_0$ is the original numerical semigroup, and the $0$-Frobenius number is the original Frobenius number. The explicit formula for two variables is known not only for $p=0$ but also for $p>0$, but when there are three or more variables, it is difficult even in the special case of $p=0$. For $p>0$, it is not only more difficult, but no explicit formula had been found. In this paper, explicit formulas of the $p$-Frobenius number and related values are given for the triple of arithmetic progressions. The main tool is to determine the elements of the $p$-Ap\'ery set.