Про деякі дискретні квантові розподіли

The article investigates the discrete \(q\)-distributions – the \(q\)-Binomial, \(q\)-Negative Binomial, and \(q\)-Poisson – which play an important role in quantum calculus, \(q\)-combinatorics, and the theory of special functions. Based on the application of \(q\)-derivatives and the \(q\)-Taylor formula, a unified approach is proposed for constructing recurrence relations for the initial and central moments of these distributions. Explicit formulas for low-order moments are obtained, generalizing classical results and correctly reducing to them when \(q = 1\). The proposed method allows for the systematization of known fragmentary results and provides a basis for further research into \(q\)-probabilistic models, particularly in connection with \(q\)-orthogonal polynomials and stochastic processes.