Factorial Series and Partial Sums Satisfying the Same Difference Equations

Abstract

Let y be a factorial series and y_n be its n the partial sum. If y satisfies a linear difference equation with polynomial coefficients, then 1) three exists a linear difference equation of order independent of n, but with coefficients depending on n,for which both y and y_n are solutions, and 2) three exists an algebraic difference equation, entirely independent of n, which is satisfied by y and every y_n.