Classification of finite irreducible conformal modules over $N=2$ Lie conformal superalgebras of Block type

We introduce the $N=2$ Lie conformal superalgebras $\mathfrak{K}(p)$ of Block type, and classify their finite irreducible conformal modules for any nonzero parameter $p$. where $p$ is a nonzero complex number. In particular, we show that such a conformal module admits a nontrivial extension of a finite conformal module $M$ over $K_2$ if $p=-1$ and $M$ has rank $(2+2)$, where $K_2$ is an $N=2$ conformal subalgebra of $\mathfrak{K}(p)$. As a byproduct, we obtain the classification of finite irreducible conformal modules over a series of finite Lie conformal superalgebras $\mathfrak{k}(n)$ for $n\ge 1$. Composition factors of all the involved reducible conformal modules are also determined.