  We investigate new generalizations of the Meixner polynomials on the lattice
$\mathbb{N}$, on the shifted lattice $\mathbb{N}+1-\beta$ and on the bi-lattice
$\mathbb{N}\cup (\mathbb{N}+1-\beta)$. We show that the coefficients of the
three-term recurrence relation for the orthogonal polynomials are related to
the solutions of the fifth Painlev\'e equation P$_{\textup V}$. Initial
conditions for different lattices can be transformed to the classical solutions
of P$_{\textup V}$ with special values of the parameters. We also study one
property of the B\"acklund transformation of P$_{\textup V}$.
