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Inequalities for ultraspherical polynomials and the gamma function. (English) Zbl 0532.33007
Der Verf. beweißt mit elementaren Mitteln für die Gegenbauerschen Polynome $P\sb n\sp{(\lambda)}$ die Abschätzung $$ (\sin \theta)\sp{\lambda}\vert P\sb n\sp{(\lambda)}(\cos \theta)\vert<2\sp{1- \lambda}/\Gamma(\lambda)(n+\lambda)\sp{1-\lambda},$$ $$0\le \theta \le \pi,\quad 0<\lambda<1,\quad n\in N.$$ Diese Abschätzung ist besser als {\it G. Szegö’s} Ungleichung [Orthogonal polynomials (1959; Zbl 0089.275)], wo $n\sp{1-\lambda}$ statt $(n+\lambda)\sp{1-\lambda}$ steht. Einige Abschätzungen für $\Gamma(n+\lambda)/\Gamma(n+1)$ werden betrachtet.
Reviewer: E.Riekstiņš

MSC:
33C55Spherical harmonics
33B15Gamma, beta and polygamma functions
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References:
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