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Generalized Coherent Pairs

2001; Elsevier BV; Volume: 253; Issue: 2 Linguagem: Inglês

10.1006/jmaa.2000.7157

1096-0813

K.H. Kwon, J.H. Lee, Francisco Marcellán,

Numerical methods in engineering

A pair of quasi-definite moment functionals {u0, u1} is a generalized coherent pair if monic orthogonal polynomials {Pn(x)}∞n = 0 and {Rn(x)}∞n = 0 relative to u0 and u1, respectively, satisfy a relationRnx=1n+1P′n+1x−σnnP′nx−τn−1n−1P′n−1x,n≥2,where σn and τn are arbitrary constants, which may be zero. If {u0, u1} is a generalized coherent pair, then u0 and u1 must be semiclassical. We find conditions under which either u0 or u1 is classical. In such a case, we also determine the types of the “companion” moment functionals. Also some illustrating examples and two ways of generating generalized coherent pairs are given. We also discuss the corresponding Sobolev orthogonal polynomials.