We prove a remarkable property of the binomial transform - it converts multiplication by the discrete variable into a certain difference operator. We also consider the case of dividing by the discrete variable.

The properties presented here are used to compute various binomial transform formulas involving Harmonic numbers, Fibonacci numbers, Stirling numbers of the second kind, and Laguerre polynomials. Several new identities are proved and some known results are given new short proofs.

binomial transform, backward difference operator, harmonic numbers, Stirling numbers of the second kind, Fibonacci numbers, recurrence relations, Euler’s series transformation.