Probabilistic Degenerate Fubini Polynomials Associated with Random Variables
Rongrong Xu, Taekyun Kim, Dae San Kim, Yuankui Ma
Abstract
Abstract Let Y be a random variable such that the moment generating function of Y exists in a neighborhood of the origin. The aim of this paper is to study probabilistic versions of the degenerate Fubini polynomials and the degenerate Fubini polynomials of order r , namely the probabilisitc degenerate Fubini polynomials associated with Y and the probabilistic degenerate Fubini polynomials of order r associated with Y . We derive some properties, explicit expressions, certain identities and recurrence relations for those polynomials. As special cases of Y , we treat the gamma random variable with parameters $$\alpha ,\beta > 0$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>α</mml:mi> <mml:mo>,</mml:mo> <mml:mi>β</mml:mi> <mml:mo>></mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:math> , the Poisson random variable with parameter $$\alpha > 0$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>α</mml:mi> <mml:mo>></mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:math> , and the Bernoulli random variable with probability of success p .