p$-numerical semigroup of generalized Fibonacci triples

For a nonnegative integer $p$, we give explicit formulas for the $p$-Frobenius number and the $p$-genus of generalized Fibonacci numerical semigroups. Here, the $p$-numerical semigroup $S_p$ is defined as the set of integers whose nonnegative integral linear combinations of given positive integers $a_1,a_2,\dots,a_k$ are expressed more than $p$ ways. When $p=0$, $S_0$ with the $0$-Frobenius number and the $0$-genus is the original numerical semigroup with the Frobenius number and the genus. In this paper, we consider the $p$-numerical semigroup involving Jacobsthal polynomials, which include Fibonacci numbers as special cases. We can also treat with the Jacobsthal-Lucas polynomials, including Lucas numbers accordingly. One of the applications on the $p$-Hilbert series is mentioned.