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

Waseem Ahmad Khan; Oğuz Yağcı; Khidir Shaib Mohamed; Mona A. Mohamed; Naglaa Mohammed; Waseem Ahmad Khan; Oğuz Yağcı; Khidir Shaib Mohamed; Mona A. Mohamed; Naglaa Mohammed

doi:10.3934/math.2026628

AIMS Mathematics

2026, Volume 11, Issue 5: 15277-15301. doi: 10.3934/math.2026628

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Research article

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

1.
Department of Electrical Engineering, Prince Mohammad Bin Fahd University, P.O. Box 1664, Al Khobar 31952, Saudi Arabia
2.
Department of Mathematics, Kırıkkale University, Kırıkkale 71450, Türkiye
3.
Department of Mathematics, College of Science, Qassim University, Buraydah 51452, Saudi Arabia

Received: 10 March 2026 Revised: 30 April 2026 Accepted: 12 May 2026 Published: 29 May 2026
MSC : Primary 05A05, 05A19; Secondary 05A30, 11B73, 33D05

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.
Citation: Waseem Ahmad Khan, Oğuz Yağcı, Khidir Shaib Mohamed, Mona A. Mohamed, Naglaa Mohammed. On some properties of degenerate $ \mathfrak{q} $-derangement numbers and polynomials[J]. AIMS Mathematics, 2026, 11(5): 15277-15301. doi: 10.3934/math.2026628

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Abstract

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.

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