Generalization of Lambert W function, Bessel polynomials and transcendental equations

2014/12/31 by Giorgio Mugnaini, Mugnaini, Giorgio
Engineering · Mathematics · Physics and Astronomy · #Classical Analysis and ODEs (math.CA) #Experimental and Theoretical Physics Studies #FOS: Mathematics #Sports Dynamics and Biomechanics #math.CA

paper · pdf · doi:10.48550/arxiv.1501.00138

openalex publication_date 2014/12/31 · arxiv created 2015/04/27 · arxiv updated 2015/04/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 : (x²D)ⁿ=xⁿ⁺¹Dⁿx^n-1

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