Newton series expansion of bosonic operator functions

We show how series expansions of functions bosonic number operators are naturally derived from finite-difference calculus. The scheme employs Newton rather than Taylor known differential calculus, and also works in cases where the expansion fails. For a function operators, such an is automatically normal ordered. Applied to Holstein-Primakoff representation spins, yields exact with finite terms and, addition, allows for systematic spin that respects commutation relations within truncated part full Hilbert space. Furthermore, strongly facilitates calculation expectation values respect coherent states. As third example, we factorial moments cumulants arising context photon or electron counting natural consequence expansions. Finally, elucidate connection between ordering, by determining corresponding integral transformation, which related Mellin transform.