A Bivariate Finite-Difference Structure of the Power Matrix

Source abstract

Let F(n, k) = nk for nonnegative integers n and k. We study the bivariate finite-difference array generated by the commuting operators Δn and Δk. The n-difference layers form a finite truncated structure whose support is a stepped array, whose terminal cell is a factorial, and whose boundary values reproduce the Stirling numbers of the second kind scaled by r!. In the k-direction, the same matrix has an eigenfunction structure: Δsk F(n, k) = (n − 1)snk. We define the truncated difference pyramid precisely, derive an explicit falling-factorial expansion for every layer, determine its degree and leading coefficient, and introduce mixed differences Δrn ΔskF that connect the two regimes, for which we give an exact closed form. The work is primarily structural and computational. Its contribution is the definition and systematic analysis of a single bivariate object that unifies two classical one dimensional difference regimes, together with an exact computational protocol that verifies three representations of every cell: the two-parent recursion, the alternating-sum formula, and the Stirling expansion. We report explicit verification for N = 6 and N = 8 with the number of tested cells. The construction extends to any polynomial sequence with a known falling-factorial expansion; further extensions (Sheffer sequences, fractional differences) are outlined but not developed.