  Inspired by J. Novak's works on the asymptotic behavior of the BGW and the
HCIZ matrix integrals \cite{[N0]} and by the algebraic and geometric properties
of the Hurwitz numbers \cite{[IP]}, \cite{[LZZ]}, \cite{[LR]}, \cite{[OP]},
\cite{[Z1]}, and by the symplectic surgery theory of the relative GW-invariants
\cite{[IP]}, \cite{[LR]}, using the elements of the transform matrix from the
integral Macdonald function with two parameters to the homogeneous symmetric
power sum functions \cite{[M]}, we have constructed the Macdonald-Hurwitz
numbers. As an application, we have constructed a series of new genus-expanded
cut-and-join differential operators, which can be thought of as the
generalization of the Laplace-Beltrami operators and have the genus-expanded
integral Macdonald functions as their common eigenfunctions. We have also
obtained some generating wave functions of the same degree, which are generated
by the Macdonald-Hurwitz numbers and can be expressed in terms of the new
cut-and-join differential operators and the initial values. Another application
is that we have constructed a new commutative associative algebra
$(C(\mathbb{F}[S_{d}]),\circ_{q,t})$ (referring to the last section (6)).
  By taking the limit along a special path $\eta(A|B)$ (referring to the
formulas (140), (141)), we specialize $(C(\mathbb{F}[S_{d}]),\circ_{q,t})$ to
be a commutative associative algebra
$(C(\hat{\mathbb{F}}[S_{d}]),\circ_{A|B})$, which will be proven to be
isomorphic to
  the middle-dimensional $\mathbb{\mathbb{C}^*}$-equivalent cohomological rings
via the Jack functions over the Hilbert scheme points of $\mathbb{C}^2$
constructed by W. Li, Z. Qin, and W. Wang in \cite{[LQW2]}.}
