Spivey-type recurrence relation for fully degenerate Bell polynomials

Core claim

On the paper's own terms, the central discovery is Theorem 2.2: for every $m,n\ge 0$ and $\lambda\ne 0$, $$\mathrm{Bel}_{n+m,\$\lambda$}(t)=\sum_{k=0}^{m}\sum_{l=0}^{n}(1)_{k,\$\lambda$}\binom{m}{k}_{\$\lambda$}\binom{n}{l}$t^{{k}}$$F^{{(k)}}$_{n-l,\$\lambda$}(-\$\lambda$ t,k-m\$\lambda$)\,\mathrm{Bel}_{l,\$\lambda$}(t),$$ where $\mathrm{Bel}_{n,\lambda}(t)$ are the fully degenerate Bell polynomials, $\binom{m}{k}_{\lambda}$ are degenerate Stirling numbers of the second kind, and $F^{(k)}_{n,\lambda}(x,y)$ are two-variable degenerate Fubini polynomials of order $k$. Theorem 2.3 gives the analogous identity $$F_{n+m,\$\lambda$}(t)=\sum_{k=0}^{m}\sum_{l=0}^{n}k!\binom{m}{k}_{\$\lambda$}\binom{n}{l}$t^{{k}}$$F^{{(k)}}$_{n-l,\$\lambda$}(t,k-m\$\lambda$)\,F_{l,\$\lambda$}(t)$$ for degenerate Fubini polynomials. A first pass with $t=1$ yields the number-level identity of Theorem 2.1. Both polynomial identities are meant to hold as formal power series in $t$, and the paper shows that taking $\lambda\to 0$ recovers the classical Spivey-type recurrences for ordinary Bell polynomials and Fubini polynomials, with the Bell limit reading $\varphi_{n+m}(t)=\sum_{k,l}\binom{m}{k}\binom{n}{l}t^k k^{n-l}\varphi_l(t)$.