The binomial Sheffer sequence of Bell / Touchard / exponential polynomials $\phi_n (x) $, whose coefficients are the Stirling numbers of the second kind, have the representation

$(RL)^n=\phi_n (:RL:) $

where $R $ and $L $ are the raising and lowering operators of any sequence of Sheffer polynomials and $:RL:^n=R^n \; L^n$ by definition (a notational convenience).

Are there other polynomial functions of the ladder operators $K(L,R)$ such that

$ K^n(L,R) = u_n(B.(L,R))$

for $u_n(x)$ a Sheffer sequence and $(B.(L,R))^n = B_n(L,R)$ a sequence of operator polynomials?

(Other than for the trivial case $u_n(x) = x^n$.)

Note that $R^nL^n= (RL)_n = (RL)!/(RL-n)!$, the falling factorial polynomials, whose coefficients are the signed Stirling numbers of the first kind.