The commutation relation xy=qyx+hf(y) and Newton's binomial formula

Let x and y be two variables satisfying the commutation relation xy=qyx+hf(y), where f(y) is a polynomial. In this paper, using Young diagrams and generating functions techniques, we study the binomial formula (x+y)ⁿ and we present an identity for x^my. The connection to Operator Calculus is discussed and several special cases are treated explicitly.