q-Poisson, q-Dobinski, q-Rota and q-coherent states

The q-Dobinski formula may be interpreted as the average of powers of random variable X q with the q-Poisson distribution. Forty years ago Rota G. C. [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) Then Bell numbers (see: formula (4) in [1]) are defined by L(X n) = B n , n ≥ 0 (3) The above formula is exactly the Dobinski formula [2] if L is interpreted as the average functional for the random variable X with the Poisson distribution with L(X) = 1. It is Blissard calculus inspired umbral formula [1]. Recently an interest to Stirling numbers and consequently to Bell numbers was revived among " q-coherent states physicists " [3, 4, 5]. Namely the expectation value with respect to coherent state |γ > with |γ| = 1 of the 1