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.