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I'd like to build a family of orthogonal polynomials with respect to the weighting function $\omega(x)=\frac{x}{e^x-1}$ on $[0,+\infty)$, i.e the inner product

$$<P_n|P_m>=\displaystyle\int_{0}^{\infty}\frac{xP_n(x)P_m(x)}{e^x-1}dx$$

And more specifically, I would like to have an explicit expression of these polynomials, of the form $$P_n(x)=\displaystyle\sum\limits_{k=0}^{n}p_{k,n}x^k$$

Considering that we have the identity $\displaystyle\int_{0}^{\infty}\frac{x^n}{e^x-1}dx=n!\zeta(n+1)$, this seems doable.

I tried orthogonalizing $(1,X,X^2...)$ with Gram-Schmidt, but I end with a very bad-looking recurrence relation that is impossible to work with. The thing is, it seems to me that Gram-Schmidt always produces those horrible-looking recurrence relations that can't lead any further.

But every known orthogonal polynomials (Legendre, Chebyshev, Laguerre, Hermite...) have nice explicit expressions, and I've even seen casually thrown in a paper "and by the way, the orthogonal polynomials with respect to this particular weighting function on $(0,4)$ can be expressed as follows" (about the function $x:\to\sqrt{\frac{4+x}{4-x}}$) with an explicit expression.

All of this leads me to believe that their is a generic method to find it. But I could find any references for their derivation, they just seems to be considered as "well-known facts".

I thought that maybe there are derived from their Rodrigues formula, but the Chebyshev polynomials don't have one and still have nice expressions.

Do you have any idea how to find the explicit expression ?

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All known orthogonal polynomials, including those you have mentioned, are solutions of differential equations that arise in various physical phenomena. Legendre developed his poylnomials as coefficients of expansion of a Newtonian potential; Hermite polynomials are eigenstates of the quantum harmonic oscillator; The Legendre polynomials arise naturally when solving the Poisson equation for a system with spherical symmetry (such as the hydrogen atom); In quantum mechanics the Schrödinger equation for the hydrogen-like atom is exactly solvable by separation of variables in spherical coordinates. The radial part of the wave function is a (generalized) Laguerre polynomial.

It is quite clear that there exist infinitely many unrelated systems of orthogonal polynomials with respect to various weight-functions, but is should be equally clear that only a chosen few will satisfy differential equations that have actual physical meaning, or would have "nice" explicit representations. As a result, unless your weight-function represents some physical setup, and unless you expect your polynomials to be solutions of some meaningful differential equations, there should be little hope of finding an explicit representation for them, and the fact that the well-known systems do have such representations should not be taken as evidence for the existence of such a representation, but rather as evidence for the lack of such.

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A little too long for a comment : there is no generic method to find a closed form for orthogonal polynomials wrt to a particular inner-product. It is the opposite, to obtain Hermite,Laguerre,Legendre they started from a closed-form two variable analytic function $$f(x,t) = \sum_n g_n(x) t^n$$ satisfying $$ \int_a^b f(x,t) f(x,u) w(x) dx = h(ut)$$ for some $w,h$, which implies $$ \int_a^b g_n(x)g_m(x)w(x)dx = \cases{ h^{(n)}(0) \ \text{ if } n=m\\ 0 \text{ otherwise}}$$ The triplet $f,w,h$ exists in closed-form only in a very few special cases.

Given the Laguerre closed-form for $w(x) = e^{-x}$ to find the orthogonal polynomials for your $w_2(x)=\frac1{e^x-1}$ inner product you'll need to invert the infinite matrix $M_{nm} = <L_n,L_m>_{w_2}$

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I think you need to use the Szego-Jacobi parameters or something. I remember this from a long time ago. http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.64.1023

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