Leibniz-type formulas for iterated symmetric difference quotients via shift operators

Fix a nonzero step size h and define the symmetric (central) difference quotient Sh as the basic discrete analogue of differentiation. Using shift operators, we prove a twisted product rule for Sh and derive an explicit Leibniz-type formula for the r-fold iterate Shr acting on a product of two functions, where the terms involve iterated symmetric differences evaluated at appropriately shifted points. We further extend the result to products of an arbitrary number of functions, obtaining a multinomial expansion with systematic shifts. Finally, we present a compact operator formulation based on shift automorphisms and a coproduct map on tensor products, which incorporates the Leibniz rules for all orders into a single identity.