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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 thethree-term recurrence relation for the orthogonal polynomials are related tothe solutions of the fifth Painlev\'e equation P$_{\textup V}$. Initialconditions for different lattices can be transformed to the classical solutionsof P$_{\textup V}$ with special values of the parameters. We also study oneproperty of the B\"acklund transformation of P$_{\textup V}$.

Recurrence Coefficients of a New Generalization of the Meixner Polynomials