G.A. Kalugin, Department of Applied Mathematics, The University of Western Ontario, London, Ontario, Canada, N6A 5B7; gkalugin@uwo.ca

D.J. Jeffrey, Department of Applied Mathematics, The University of Western Ontario, London, Ontario, Canada, N6A 5B7; djeffrey@uwo.ca

Abstract/Résumé:

We consider a sequence of polynomials appearing in expressions for the derivatives of the Lambert \(W\) function. The coefficients of each polynomial are shown to form a positive sequence that is log-concave and unimodal. This property implies that the positive real branch of the Lambert \(W\) function is a Bernstein function.

Nous considérons une séquence de polynômes que l’on retrouve dans l’expression des dérivées de la fonction Lambert \(W\). Nous montrons que les coefficients de chaque polynôme forment une séquence positive qui est log-concave et unimodale. Cette propriété implique que la branche réelle positive de la fonction Lambert \(W\) est une fonction de Bernstein.

Keywords: Bernstein function, completely monotonic function, unimodal sequence

AMS Subject Classification: Special sequences and polynomials 11B83

PDF(click to download): Unimodal sequences show that Lambert $W$ is Bernstein