Interlacing of zeros of orthogonal polynomials under modification of the measure

Autores: Dimitar K. Dimitrov, Mourad E. H. Ismail, Fernando R. Rafaeli
Localización: Journal of approximation theory, ISSN 0021-9045, Vol. 175, Nº 1, 2013, págs. 64-76
Idioma: inglés
DOI: 10.1016/j.jat.2013.07.007
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Resumen
- We investigate the mutual location of the zeros of two families of orthogonal polynomials. One of the families is orthogonal with respect to the measure d�Ê(x), supported on the interval (a, b) and the other with respect to the measure |x .c|�Ñ |x .d|�Á d�Ê(x), where c and d are outside (a, b). We prove that the zeros of these polynomials, if they are of equal or consecutive degrees, interlace when either 0 < �Ñ, �Á . 1 or �Á = 0 and 0 < �Ñ . 2. This result is inspired by an open question of Richard Askey and it generalizes recent results on some families of orthogonal polynomials. Moreover, we obtain further statements on interlacing of zeros of specific orthogonal polynomials, such as the Askey.Wilson ones.