$ψ$-Poisson, $q$-Cigler, $ψ$-Dobinski, $ψ$-Rota and $ψ$-coherent states

Abstract

Cigler simple derivation of usual and extended Dobinski formula is recalled and it is noted that both may be interpreted as averages of powers of random variables with the corresponding usual or extended Poisson distributions. In parallel more general formulas of extended operator umbral calculi origin are revealed . The formulas encompass both earlier cases as very specific ones.