[Paper Review] Generalization of Lambert $W$ function, Bessel polynomials and transcendental equations

Giorgio Mugnaini|arXiv (Cornell University)|Dec 31, 2014

Sports Dynamics and Biomechanics10 citations

TL;DR

This paper generalizes the Lambert W function by solving a quadratic-exponential transcendental equation $(x-a)(x-b) = le^x$ using Lagrange inversion series, yielding a solution in terms of Bessel polynomials. The key contribution is a novel Rodrigues-type formula for Bessel polynomials and a new differential identity linking reciprocal derivatives to polynomial derivatives.

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

Motivation & Objective

• To generalize the Lambert W function by solving a quadratic-exponential transcendental equation arising in quantum chemistry.
• To derive a formal series solution for $(x-a)(x-b) = le^x$ using Lagrange inversion.
• To establish a novel Rodrigues-type representation for Bessel polynomials through differential operators.
• To explore the structural link between Lagrange inversion terms and classical orthogonal polynomial formulas.
• To identify a new differential identity involving reciprocal derivatives and polynomial derivatives.

Proposed method

• Apply the Lagrange inversion formula to the equation $x = a + l \frac{e^x}{x-b}$, expressing the solution as a power series in $l$.
• Compute the $n$-th derivative term in the series using the Leibniz rule on $\left(\frac{e^x}{x-b}\right)^n$, leading to a double summation.
• Recognize the resulting structure as involving Bessel polynomials $B_{n-1}(z)$, leading to a compact series form in terms of $B_{n-1}\left(\frac{-2}{n(a-b)}\right)$.
• Derive a new Rodrigues-like formula: $B_{n-1}\left(\frac{-2}{x}\right) = e^{-x}x^n \left(\frac{d}{dx}\right)^{n-1}\left[\frac{e^x}{x^n}\right]$, verified via induction.
• Prove the identity $\left(x^2 \frac{d}{dx}\right)^n = x^{n+1} \left(\frac{d}{dx}\right)^n x^{n-1}$ using commutation rules and mathematical induction.
• Connect the structure of Lagrange series terms to generalized Rodrigues formulas, suggesting a deeper link between orthogonal polynomials and transcendental equation solutions.

Experimental results

Research questions

• RQ1 Can the solution to the transcendental equation $(x-a)(x-b) = le^x$ be expressed in terms of special orthogonal polynomials?
• RQ2 Does the Lagrange inversion series for this equation yield a new representation of Bessel polynomials?
• RQ3 Is there a differential identity connecting $\left(x^2 \frac{d}{dx}\right)^n$ to $x^{n+1} \left(\frac{d}{dx}\right)^n x^{n-1}$?
• RQ4 Can the structure of Lagrange series terms be systematically linked to Rodrigues-type formulas for classical orthogonal polynomials?
• RQ5 What is the significance of the appearance of Bessel polynomials in solutions of mixed exponential-polynomial equations?

Key findings

• The solution to $(x-a)(x-b) = le^x$ is given by $x = a + \sum_{n=1}^{\infty} \frac{1}{n!n} \left(\frac{nle^a}{a-b}\right)^n B_{n-1}\left(\frac{-2}{n(a-b)}\right)$, expressing the root in terms of Bessel polynomials.
• A novel Rodrigues-type formula for Bessel polynomials is derived: $B_{n-1}\left(\frac{-2}{x}\right) = e^{-x}x^n \left(\frac{d}{dx}\right)^{n-1}\left[\frac{e^x}{x^n}\right]$, valid for $n \geq 1$.
• The differential identity $\left(x^2 \frac{d}{dx}\right)^n = x^{n+1} \left(\frac{d}{dx}\right)^n x^{n-1}$ is rigorously proven by induction using commutation rules.
• The solution for the symmetric case is obtained by swapping $a$ and $b$, yielding $x = b + \sum_{n=1}^{\infty} \frac{1}{n!n} \left(-\frac{nle^b}{a-b}\right)^n B_{n-1}\left(\frac{2}{n(a-b)}\right)$.
• The paper establishes a formal link between terms in the Lagrange inversion series and Rodrigues formulas, suggesting a deeper structural connection between orthogonal polynomials and transcendental equation solutions.
• The work opens avenues for investigating hypergeometric-type series generated by Lagrange inversion in solving broader classes of transcendental equations.