On some properties of degenerate $\mathfrak{q}$-derangement numbers and polynomials

Using a Carlitz-type degenerate $\mathfrak{q}$-exponential kernel together with the $\lambda$-falling factorial $(\mu {)}_{\zeta ,\lambda }$, we introduce a $\lambda$-degenerate $\mathfrak{q}$-analog of the derangement family. The associated exponential generating function defines the degenerate $\mathfrak{q}$-derangement polynomials ${\mathfrak{d}}_{\zeta ,\mathfrak{q}}(\mu ;\lambda )$ and yields explicit coefficient formulas, recurrence relations, convolution identities, and determinant representations. The main structural point is that these polynomials are governed by a lower triangular transform in the $\mathfrak{q}$-factorial basis; this transform has a two-term inverse and organizes the connections with degenerate $\mathfrak{q}$-Stirling, $\mathfrak{q}$-Bell, and $\mathfrak{q}$-Fubini polynomials. We also show that the same mechanism is stable under higher-order kernels and under a degenerate $(\mathfrak{p},\mathfrak{q})$-extension. The limiting regimes $\lambda \to 0$ and $\mathfrak{q}\to 1$ recover, respectively, the standard $\mathfrak{q}$-derangements and the classical derangement polynomials.