  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.
