Ordinary Potential Polynomials

I’ve run into potential polynomials a lot lately. Most sources I’ve seen are either unclear on how they are defined or use a notation that doesn’t sit well in my brain. Also, they’ll either give an implicit or explicit definition but not both. Both formulations are important. The implicit formulation suggests how potential polynomials arise in practice. The explicit formulation shows how you’d actually compute them when you need to.

The potential polynomials A_r,n are homogeneous polynomials of degree r in n variables. They are usually defined implicitly by

$\left( 1 + \sum_{n=1}^\infty x_n z^n\right )^r = \sum_{n=0}^\infty A_{r, n}(x_1, x_2, \ldots, x_n)z^n$

The can also be defined explicitly in terms of ordinary Bell polynomials B_n,k by

$A_{r, n}(x_1, x_2, \ldots, x_n) = \sum_{k=1}^n {r \choose k} B_{n, k}(x_1, x_2, \ldots, x_{n-k+1})$

Does anyone know why they’re called “potential” polynomials? Is there some analogy with a physical potential? [1]

By the way, potential polynomials are called “ordinary” in the same sense that Bell polynomials and generating functions are called ordinary: to contrast with exponential forms that insert factorial scaling factors. See the previous post for details.

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[1] When I was learning electricity and magnetism, I was confused by the phrase “potential difference.” Did that mean that there isn’t a difference yet, but there was the potential for a difference? No, it means a difference in electrical potential. There’s still a sense in which “potential” carries with it the idea of possibility. A rock sitting on top of a mountain has a lot of gravitational potential, energy that could be released if the rock started falling.

Potential polynomials could sound like functions that aren’t polynomials yet but have the potential to be, maybe if they just tried harder.