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Probabilistic degenerate derangement polynomials.

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Title: Probabilistic degenerate derangement polynomials.
Authors: Kim, Taekyun1 (AUTHOR) tkkim@kw.ac.kr, Kim, Dae San2 (AUTHOR), Dolgy, Dmitry V.3 (AUTHOR) d_dol@mail.ru
Source: Mathematical & Computer Modelling of Dynamical Systems. Dec2025, Vol. 31 Issue 1, p1-14. 14p.
Subjects: Combinatorics, Permutations, Random variables, Characteristic functions, Polynomials, Stochastic processes, Identities (Mathematics), Recursive sequences (Mathematics)
Abstract: In combinatorics, a derangement is a permutation of the elements of a set, such that no element appears in its original position. The number of derangements of an $n$ n -element set is called the $n$ n th derangement number. Recently, the degenerate derangement numbers and polynomials have been studied as degenerate versions. Let $Y$ Y be a random variable whose moment generating function exists in a neighbourhood of the origin. In this paper, we study probabilistic extension of the degenerate derangement numbers and polynomials, namely the probabilistic degenerate derangement numbers and polynomials associated with $Y$ Y. In addition, we consider the probabilistic degenerate $r$ r -derangement numbers associated with $Y$ Y and the probabilistic degenerate derangement polynomials of the second kind associated with $Y$ Y. We derive some properties, explicit expressions, certain identities and recurrence relations for those polynomials and numbers. [ABSTRACT FROM AUTHOR]
Copyright of Mathematical & Computer Modelling of Dynamical Systems is the property of Taylor & Francis Ltd and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)
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  Data: Probabilistic degenerate derangement polynomials.
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  Data: <searchLink fieldCode="AR" term="%22Kim%2C+Taekyun%22">Kim, Taekyun</searchLink><relatesTo>1</relatesTo> (AUTHOR)<i> tkkim@kw.ac.kr</i><br /><searchLink fieldCode="AR" term="%22Kim%2C+Dae+San%22">Kim, Dae San</searchLink><relatesTo>2</relatesTo> (AUTHOR)<br /><searchLink fieldCode="AR" term="%22Dolgy%2C+Dmitry+V%2E%22">Dolgy, Dmitry V.</searchLink><relatesTo>3</relatesTo> (AUTHOR)<i> d_dol@mail.ru</i>
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  Data: <searchLink fieldCode="JN" term="%22Mathematical+%26+Computer+Modelling+of+Dynamical+Systems%22">Mathematical & Computer Modelling of Dynamical Systems</searchLink>. Dec2025, Vol. 31 Issue 1, p1-14. 14p.
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  Data: <searchLink fieldCode="DE" term="%22Combinatorics%22">Combinatorics</searchLink><br /><searchLink fieldCode="DE" term="%22Permutations%22">Permutations</searchLink><br /><searchLink fieldCode="DE" term="%22Random+variables%22">Random variables</searchLink><br /><searchLink fieldCode="DE" term="%22Characteristic+functions%22">Characteristic functions</searchLink><br /><searchLink fieldCode="DE" term="%22Polynomials%22">Polynomials</searchLink><br /><searchLink fieldCode="DE" term="%22Stochastic+processes%22">Stochastic processes</searchLink><br /><searchLink fieldCode="DE" term="%22Identities+%28Mathematics%29%22">Identities (Mathematics)</searchLink><br /><searchLink fieldCode="DE" term="%22Recursive+sequences+%28Mathematics%29%22">Recursive sequences (Mathematics)</searchLink>
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  Data: In combinatorics, a derangement is a permutation of the elements of a set, such that no element appears in its original position. The number of derangements of an $n$ n -element set is called the $n$ n th derangement number. Recently, the degenerate derangement numbers and polynomials have been studied as degenerate versions. Let $Y$ Y be a random variable whose moment generating function exists in a neighbourhood of the origin. In this paper, we study probabilistic extension of the degenerate derangement numbers and polynomials, namely the probabilistic degenerate derangement numbers and polynomials associated with $Y$ Y. In addition, we consider the probabilistic degenerate $r$ r -derangement numbers associated with $Y$ Y and the probabilistic degenerate derangement polynomials of the second kind associated with $Y$ Y. We derive some properties, explicit expressions, certain identities and recurrence relations for those polynomials and numbers. [ABSTRACT FROM AUTHOR]
– Name: AbstractSuppliedCopyright
  Label:
  Group: Ab
  Data: <i>Copyright of Mathematical & Computer Modelling of Dynamical Systems is the property of Taylor & Francis Ltd and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract.</i> (Copyright applies to all Abstracts.)
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RecordInfo BibRecord:
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    Identifiers:
      – Type: doi
        Value: 10.1080/13873954.2025.2529188
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      – Code: eng
        Text: English
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        PageCount: 14
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      – SubjectFull: Combinatorics
        Type: general
      – SubjectFull: Permutations
        Type: general
      – SubjectFull: Random variables
        Type: general
      – SubjectFull: Characteristic functions
        Type: general
      – SubjectFull: Polynomials
        Type: general
      – SubjectFull: Stochastic processes
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      – SubjectFull: Identities (Mathematics)
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      – SubjectFull: Recursive sequences (Mathematics)
        Type: general
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      – TitleFull: Probabilistic degenerate derangement polynomials.
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            NameFull: Kim, Taekyun
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            NameFull: Kim, Dae San
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            NameFull: Dolgy, Dmitry V.
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            – D: 01
              M: 12
              Text: Dec2025
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              Y: 2025
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