In this paper, we give convenient formulas in order to obtain explicit expressions of a generalized Frobenius number called the p-Frobenius number as well as its related values. Here, for a non-negative integer p, the p-Frobenius number is the largest integer whose number of solutions of the linear diophantine equation in terms of positive integers \(a_1,a_2,\ldots ,a_k\) with \(\gcd (a_1,a_2,\ldots ,a_k)=1\) is at most p. When \(p=0\) , the problem is reduced to the famous and classical linear Diophantine problem of Frobenius. 0-Frobenius number is the classical Frobenius number. Our formula is not only a natural extension of the existing classical formulas, but also has the great advantage that the explicit expressions of values such as the p-Frobenius and related numbers can be obtained systematically. The concept and formula of the weighted sum has been given recently. We also give a p-generalized formula for such weighted sums. The central role is the p-Apéry set, which is a generalization of the classical Apéry set.
