A probabilistic generalization of the Bell polynomials

Soni, R.; Vellaisamy, P.; Pathak, A. K.

doi:10.1007/s41478-023-00642-y

Cited by 7 publications

(3 citation statements)

References 26 publications

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“…Assume that (Y j ) j≥1 is a sequence of mutually independent copies of Y , and that [2,3,14,31]). The probabilistic Stirling numbers of the second kind associated with random variable Y are defined by [2,3,14,22,32]).…”

Section: Introductionmentioning

confidence: 99%

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ON q-DERANGEMENT NUMBERS AND POLYNOMIALS

2022

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The derangement number [Formula: see text] is the number of fixed point free permutations on a set of [Formula: see text] elements and the derangement polynomial [Formula: see text]) is a natural extension of the derangement number [Formula: see text]. The aim of this paper is to introduce [Formula: see text]-derangement numbers and polynomials, which are [Formula: see text]-analogs of the derangement numbers and polynomials, and to investigate their connection with some other [Formula: see text]-special numbers and polynomials. In more detail, we derive explicit expressions and recurrence relations for the [Formula: see text]-derangement numbers and polynomials. Further, we obtain some identities involving such polynomials and numbers and other special [Formula: see text]-polynomials and numbers, which include [Formula: see text]-Bell polynomials, [Formula: see text]-analogs of Fubini polynomials and [Formula: see text]-Stirling numbers of the second kind.

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Section: Introductionmentioning

confidence: 99%

“…The probabilistic Bell polynomials associated with random variable Y are defined by (13) φ [3,14,32]).…”

Section: Introductionmentioning

confidence: 99%

ON q-DERANGEMENT NUMBERS AND POLYNOMIALS

2022

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“…t n n! , (|t| < r) exists for some r > 0, where E stands for the mathematical expectation (see [3,4,16,24]). The aim of this paper is to study probabilistic versions of the degenerate Whitney numbers of the second kind and those of the degenerate Dowling polynomials, namely the probabilistic degenerate Whitney numbers of the second kind associated with Y , W Y m,λ (n, k) and the probabilistic degenerate Dowling polynomials associated with Y , D Y m,λ (n, x).…”

Section: Introductionmentioning

confidence: 99%

Fully degenerate Bell polynomials associated with degenerate Poisson random variables

Kim

2021

Open Mathematics

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Many mathematicians have studied degenerate versions of quite a few special polynomials and numbers since Carlitz’s work (Utilitas Math. 15 (1979), 51–88). Recently, Kim et al. studied the degenerate gamma random variables, discrete degenerate random variables and two-variable degenerate Bell polynomials associated with Poisson degenerate central moments, etc. This paper is divided into two parts. In the first part, we introduce a new type of degenerate Bell polynomials associated with degenerate Poisson random variables with parameter α > 0 \alpha \hspace{-0.15em}\gt \hspace{-0.15em}0 , called the fully degenerate Bell polynomials. We derive some combinatorial identities for the fully degenerate Bell polynomials related to the n n th moment of the degenerate Poisson random variable, special numbers and polynomials. In the second part, we consider the fully degenerate Bell polynomials associated with degenerate Poisson random variables with two parameters α > 0 \alpha \gt 0 and β > 0 \beta \hspace{-0.15em}\gt \hspace{-0.15em}0 , called the two-variable fully degenerate Bell polynomials. We show their connection with the degenerate Poisson central moments, special numbers and polynomials.

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