Elliptic Curve Digital Signature Algorithm (ECDSA) admits universal forgery (UF) if the Attacker can solve the equation $$z=\frac{f_{k-1}(x,y)f_{k+1}(x,y)}{f_{k}(x,y)^2},$$ where $k$ is unknown, $f_{k}(x,y)$ are Division polynomials and $(x,y)$ are the coordinates of some fixed point $P$ on the elliptic curve $E:y^{2}=x^{3}+Ax+B\;$ (everything is over $\mathbb{F}_p$). This UF is based on the formula for the coordinates of the $k$ th multiple of $P(x,y)$: $$kP=\left(x-\frac{f_{k-1}(x,y)f_{k+1}(x,y)}{f_{k}(x,y)^2},\ldots\right).$$

It is known that division polynomials satisfy the following (elliptic divisibility sequence-) recurrence (see Ch. 2 in *Lang S. Elliptic curves: Diophantine analysis. 1978*)
$$f_{m+n}f_{m-n}=f_{m+1}f_{m-1}f_{n}^2-f_{n+1}f_{n-1}f_{m}^2.$$
In particular
$$f_{m+2}f_{m-2}=\alpha f_{m+1}f_{m-1}-\beta f_{m}^2.$$
It means that $h_k:=\frac{f_{k-1}f_{k+1}}{f_{k}^2}$ (needed for UF)
satisfy the recurrence
$$\tag{$*$}h_{m-1}h_{m}^2h_{m+1}=\alpha h_{m}-\beta.$$

Q:Were the properties of sequences defined by $(*)$ studied before (period lengths, randomness, uniform distribution, ...)? Can $\{h_m\}$ be cryptographically better than $\{f_m\}$?

This is only $\!\!\mod p$-question because over $\mathbb{C}$ such sequences are well-known: $h_n=\wp(z)-\wp(z_0+nz).$ They arise in the study of Somos-4 sequences (see *Hone A. Elliptic curves and quadratic recurrence sequences. Bull. Lond. Math. Soc. 2005*.)

The question Universal forgery based on mathematical problem about another problems needed for UF was asked at Crypto separately.