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arXiv · 1501.00138

Generalization of Lambert $W$ function, Bessel polynomials and transcendental equations

Giorgio Mugnaini
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Abstract

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 : $\left(x^{2}D\right)^{n}=x^{n+1}D^{n}x^{n-1}$

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Giorgio Mugnaini. 2015-04-27. Generalization of Lambert $W$ function, Bessel polynomials and transcendental equations. https://arxiv.org/abs/1501.00138

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