We introduce a degenerate $(q,\lambda )$-extension of the two-variable $q$-Legendre-type polynomials by deforming the $q$-Bessel–Tricomi kernel through the degenerate falling-factorial weights $(1{)}_{k,\lambda }$. The resulting family $\{{\mathcal{L}}_{n,q}^{(\lambda )}(x,y){\}}_{n\ge 0}$ is defined by a single generating function and interpolates both the recently studied two-variable $q$-Legendre polynomials and their classical limits as $\lambda \to 0$ and/or $q\to 1$. We derive an explicit finite-sum representation, an operational Rodrigues-type formula, and a quasi-monomial structure that yields raising and lowering operators together with a fundamental $q$-difference equation in the variable $y$. We further compute the degenerate Laplace and Sumudu transforms of the generating kernel and obtain corresponding transform identities for ${\mathcal{L}}_{n,q}^{(\lambda )}$. Several reduction formulas, even/odd subsequence decompositions, and a moment-functional interpretation are presented, along with linearization coefficients, low-degree examples, and numerical table.

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.

Recently, by combining type 2 Bernoulli and Euler polynomials with the central Bell polynomials, the central Bell-based type 2 Bernoulli and Euler polynomials of order $\alpha$ were considered, and many of their properties, formulas, and applications were investigated. The main aim of this work is to consider higher-order central Bell-based type 2 Bernoulli and Euler polynomials of complex variable, by which, both sine and cosine central Bell-based type 2 Bernoulli and Euler polynomials of order $\lambda$ are introduced by treating the imaginary and real components separately. Then, diverse summation formulas, differential formulas, addition formulas, and correlation formulas with new and existing old polynomials and numbers are derived in a systematic way. Also, several intriguing connections of sine and cosine central Bell-based type 2 Bernoulli and Euler polynomials of order $\lambda$ with the bivariate and one-variable central Bell polynomials, and the classical Stirling and central factorial numbers of the second kinds are investigated in detail. Moreover, the first few members of the new polynomials are provided by the lists, and the distributions of zeros of the new polynomials are illustrated by graphical representations, enhancing the understanding of the numerical data and facilitating a more intuitive grasp of the concepts discussed.