%0 Journal Article
%T Generalization of Lambert $W$ function, Bessel polynomials and transcendental equations
%A Giorgio Mugnaini
%J Mathematics
%D 2014
%I arXiv
%X Employing the Lagrange inverting series, a solution of the transcendental equation $(x-a)(x-b)=le^{x}$, that can be considered a quadratic generalization of the equation defining Lambert $W$ function, has been found in terms of Bessel orthogonal polynomials. Once again a transcendental equation can be formally solved by means of classic orthogonal polynomials, suggesting a link between Rodrigues formulas and the terms of Lagrange series. A novel representation for Bessel polynomials has been found, by means of differential identity : $\left(x^{2}D\right)^{n}=x^{n+1}D^{n}x^{n-1}$
%U http://arxiv.org/abs/1501.00138v3