Finite free perpetuities

We introduce and study finite free perpetuities, defined as monic polynomial solutions of degree $n$ to the affine fixed-point equation $$p(z)=\mathbb{E}\text{\hspace{0.17em}⁣}\left[A^{n}p\text{\hspace{0.17em}⁣}\left(\frac{z-B}{A}\right){\mathbf{1}}_{\{A\mathrm{\ne }0\}}\right]+\mathbb{E}\text{\hspace{0.17em}⁣}[(z-B{)}^n{\mathbf{1}}_{\{A=0\}}],$$ where $A$ and $B$ are complex-valued random variables with finite moments up to order $n$. Equivalently, if $p(z)=\mathbb{E}[(z-X{)}^n]$, then $p$ encodes a truncated moment version of the classical perpetuity equation $XAX+B$ with $X$ and $(A,B)$ independent. This places finite free perpetuities between classical perpetuities and free-probabilistic fixed-point laws. We prove existence and uniqueness under weak conditions, and we identify a broad class of admissible pairs $(A,B)$ for which the resulting polynomial has only real, nonnegative zeros. Our approach uses finite free additive and multiplicative convolutions together with a probabilistic representation via the $U$-transform. As a motivating example, we exhibit an explicit family of finite free perpetuities expressed in terms of Jacobi polynomials and show that their empirical root distributions converge to a free-beta-prime law. More generally, for admissible sequences of parameters, we prove weak convergence of the empirical root distributions of finite free perpetuities to the law of a free perpetuity characterized by the corresponding free fixed-point equation. This yields a finite-degree polynomial model approximating free perpetuities and clarifies the connection between classical affine recursions, finite free convolutions, and free probability.