Degenerate Derangement Polynomials and Numbers

• Main Authors: Minyoung Ma, Dongkyu Lim
• Format: Article
• Language: English
• Published: MDPI AG 2021-06-01
• Series: Fractal and Fractional
• Subjects: degenerate derangement polynomials; degenerate derangement polynomials of the second kind; degenerate gamma distribution; degenerate Fubini polynomials; fully degenerate Bell polynomials; degenerate Stirling numbers
• Online Access: https://www.mdpi.com/2504-3110/5/3/59

In this paper, we consider a new type of degenerate derangement polynomial and number, which shall be called the degenerate derangement polynomials and numbers of the second kind. These concepts are motivated by Kim et al.’s work on degenerate derangement polynomials and numbers. We investigate some properties of these new degenerate derangement polynomials and numbers and explore their connections with the degenerate gamma distributions for the case <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>λ</mi><mo>∈</mo><mo>(</mo><mo>−</mo><mn>1</mn><mo>,</mo><mn>0</mn><mo>)</mo></mrow></semantics></math></inline-formula>. In more detail, we derive their explicit expressions, recurrence relations, and some identities involving our degenerate derangement polynomials and numbers and other special polynomials and numbers, which include the fully degenerate Bell polynomials, the degenerate Fubini polynomials, and the degenerate Stirling numbers of the first and the second kinds. We also show that those polynomials and numbers are connected with the moments of some variants of the degenerate gamma distributions. Moreover, we compare the degenerate derangement polynomials and numbers of the second kind to those of Kim et al.