Generalized Delta-coherent pairs
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초록
A pair of quasi-definite linear functionals {u_0,u_1}is a generalized Delta-coherent pair if monic orthogonalpolynomials {P_n(x)}_{n=0}^infty and{R_n(x)}_{n=0}^infty relative to u_0 and u_1,respectively, satisfy a relation R_n(x) = frac{1}{n+1}Delta P_{n+1}(x)-frac{sigma_n}{n}Delta P_n(x)-frac{tau_{n-1}}{n-1}Delta P_{n-1}(x), ngeq 2, wheresigma_n and tau_n are arbitrary constants and Deltap=p(x+1)-p(x) is the difference operator.We show that if {u_0,u_1} is a generalized Delta-coherentpair, then u_0 and u_1 must be discrete-semiclassical linearfunctionals. We also find conditions under which either u_0 oru_1 is discrete-classical.
Abstract
A pair of quasi-definite linear functionals {u_0,u_1}is a generalized Delta-coherent pair if monic orthogonalpolynomials {P_n(x)}_{n=0}^infty and{R_n(x)}_{n=0}^infty relative to u_0 and u_1,respectively, satisfy a relation R_n(x) = frac{1}{n+1}Delta P_{n+1}(x)-frac{sigma_n}{n}Delta P_n(x)-frac{tau_{n-1}}{n-1}Delta P_{n-1}(x), ngeq 2, wheresigma_n and tau_n are arbitrary constants and Deltap=p(x+1)-p(x) is the difference operator.We show that if {u_0,u_1} is a generalized Delta-coherentpair, then u_0 and u_1 must be discrete-semiclassical linearfunctionals. We also find conditions under which either u_0 oru_1 is discrete-classical.
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