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Some open problems of generalised Bessel polynomials. (English) Zbl 0583.33007

Das verallgemeinerte Besselpolynom y n (x;a,b)= 2 F 0 (-n,n+a-1,(-x/b)) vom Grad n ist Polynomlösung der Differentialgleichung

x 2 y '' +(ax+b)y ' -n(n+a-1)y=0(-aN 0 ,-bN 0 )·

Es seien s r die Newtonsche Summe der r-ten Potenzen der Nullstellen von y n (x;a,b) und h r eine in der Kombinatorik auftretende, homogene symmetrische Funktion der Nullstellen von y n (x;a,b) vom Grad r. Für beide werden explizite Darstellungen und Rekursionsformeln hergeleitet. Ferner werden mit Hilfe der Rekursionsformel für die y n (x;a,b) drei verschiedene Summen mit Besselpolynomen in geschlossener Form ausgewertet.

Reviewer: L.Neckermann
MSC:
33C10Bessel and Airy functions, cylinder functions, 0 F 1
33C60Hypergeometric integrals and functions defined by them
33C45Orthogonal polynomials and functions of hypergeometric type
05A19Combinatorial identities, bijective combinatorics