Recurrence for probabilistic extension of Dowling polynomials

Abstract Spivey found a remarkable recurrence relation for Bell numbers, which was generalized to that for Bell polynomials by Gould-Quaintance. The aim of this article is to generalize their recurrence relation for Bell polynomials to that for the probabilistic Dowling polynomials associated with <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>Y</m:mi> </m:math> Y and also that for the probabilistic <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>r</m:mi> </m:math> r -Dowling polynomials associated with <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>Y</m:mi> </m:math> Y . Here <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>Y</m:mi> </m:math> Y is a random variable whose moment generating function exists in a neighborhood of the origin.