$\psi$-Poisson, $q$-Cigler, $\psi$-Dobinski, $\psi$-Rota and $\psi$-coherent states

The Cigler simple derivation of the q-Carlitz-Dobinski formula is recalled and it is noticed that the formula may be interpreted as the average of powers of random variable X q with the q-Poisson distribution. In parallel new q-Cigler-Dobinski and psi-Carlitz-Dobinski formulas are introduced. At first let us anticipate with ψ-remark. ψ denotes an extension of 1 n! n≥0 sequence to quite arbitrary one (" admissible ") and the specific choices are for example : Fibonomialy-extended (F n-Fibonacci sequence) 1 Fn! n≥0 or just " the usual " 1 n! n≥0 or Gauss q-extended 1 nq! n≥0 admissible sequences of extended umbral operator calculus-see more below. With such an extension we may ψ-mnemonic repeat with exactly the same simplicity and beauty what was done by Rota forty years ago. Forty years ago Gian-Carlo Rota [1] proved the exponential generating function for Bell numbers B n to be of the form ∞ n=0 x n n! (B n) = exp(e x − 1) (1) using the linear functional L such that L(X n) = 1, n ≥ 0 (2)