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

Abstract

Employing the Lagrange inverting series, a solution of the transcendental equation $(x-a)(x-b)=l{e}^x$, 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 : ${\left(x^{2}D\right)}^n=x^{n+1}{D}^n{x}^{n-1}$

Cited by in corpus (4)

• Inverse Langevin and Brillouin functions: mathematical properties and physical applications
• Siewert solutions of transcendental equations, generalized Lambert functions and physical applications
• Transcendental equations of the running coupling
• Beamforming Design for Multiuser uRLLC with Finite Blocklength Transmission