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