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Fourth Order Difference Equation for the Associated Classical Discrete Orthogonal Polynomials
(1998)
We derive the fourth order difference equation satisfied by the associated of order $\;r\;$ of the classical orthogonal polynomials of a discrete variable.\\The coefficients of this equation are given in terms of the polynomials $\; \sigma\;$ and $\;\tau\;$ which appear in the discrete Pearson equation $\;\;\Delta(\sigma\;\rho)=\tau\;\rho\;\;$ defining the weight $\;\rho(x)\;$ of the classical discrete orthogonal polynomials.
We derive the fourth order $q$-difference equation satisfied by the first associated of the $q$-classical orthogonal polynomials. The coefficients of this equation are given in terms of the polynomials $\; \sigma\;$ and $\;\tau\;$ which appear in the $q$-Pearson difference equation $\;\; D_q(\sigma\,\rho)=\tau\,\rho\;$ defining the weight $\rho$ of the $q$-classical orthogonal polynomials inside the $q$-Hahn tableau.