A link \(L\) in \(S^3\) is called a \(p\)-periodic link (\(p\geq 2\) an integer) if there exists an orientation preserving auto-homeomorphism \(h\) of \(S^3\) such that \(h(L)=L\), \(h\) is of order \(p\) and the set \(Fix(h)\) of fixed points of \(h\) is a circle disjoint from \(L\). In this case the link \(L/ \langle h \rangle\cup Fix(h)\) in the orbit space \(S^3/ \langle h \rangle\cong S^3\) is called the quotient link of \(L\). A link \(L\) in \(S^3\) is called a \(p\)-periodic link with rational quotient if it is a \(p\)-periodic link whose quotient is a 2-bridge link. In this paper the authors give a formula for the Casson invariant of a \(p\)-period knot in \(S^3\) whose quotient link is a 2-bridge link with Conway normal form \(C(2, 2n_1,-2,2n_2,\dots,2n_{2m},2)\) via the integers \(p,n_1,n_2,\dots,n_{2m}\) (\(p\geq2\) and \(m\geq1\)). (The Casson invariant can be defined as half of the value of the second derivative of the Alexander polynomial \(\Delta_K(t)\) at \(t=1\).) If the integers \(n_1,n_2,\dots,n_{2m}\) have the same sign the \(\Delta\)-unknotting number of the \(p\)-periodic knot is given in terms of the integers \(p,n_1,n_2,\dots,n_{2m}\). Finally, a recurrence formula for the calculation of the Alexander polynomial of the 2-bridge knot with Conway normal form \(C(2n_1,2n_2,\dots,2n_m)\) is given.