@article{Khan2026, 
author = {Waseem Ahmad Khan and Oğuz Yağcı and Khidir Shaib Mohamed and Mona A. Mohamed and Naglaa Mohammed},
title = {On some properties of degenerate        q  -derangement numbers and polynomials},
year = {2026},
journal = {AIMS Mathematics},
volume = {11},
number = {5},
pages = {15277-15301},
keywords = {degenerate q-derangement numbers and polynomials, degenerate (p, q)-derangement polynomials, degenerate q-Stirling numbers, degenerate q-Bell and Fubini polynomials, generating functions, numerical methods, (p, q)-calculus},
url = {https://www.sciopen.com/article/10.3934/math.2026628},
doi = {10.3934/math.2026628},
abstract = {Using a Carlitz-type degenerate        q  -exponential kernel together with the    λ-falling factorial    (  μ      )          ζ      ,      λ      , we introduce a    λ-degenerate        q  -analog of the derangement family. The associated exponential generating function defines the degenerate        q  -derangement polynomials              d              ζ      ,              q              (  μ          ;    λ    ) 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        q  -factorial basis; this transform has a two-term inverse and organizes the connections with degenerate        q  -Stirling,        q  -Bell, and        q  -Fubini polynomials. We also show that the same mechanism is stable under higher-order kernels and under a degenerate    (      p    ,      q    )-extension. The limiting regimes    λ  →  0 and        q    →  1 recover, respectively, the standard        q  -derangements and the classical derangement polynomials.}
}