The Calculus of Finite Differences

a₁ absolutely convergent arbitrary periodic arguments B₁ B₂ Bernoulli's polynomials c₁ C₂ consider continued fraction convergence abscissa corresponding denote difference equation difference quotients differential divided differences Example expression factorial series finite follows fundamental system half-plane Hence holomorphic indicial equation integral interpolation formula interval k₁ Laplace's linear difference equation linear equation method multipliers Newton's series Nörlund notation obtain operator P₁ P₁(N partial fractions particular solution points polynomial polynomial of degree positive integer principal solution Prove R₁ rational function reciprocal differences remainder term result roots satisfy series converges shew Stirling's formula summation suppose Theorem U₁ u₁(x U₂ unity Ux+1 v₂ values vanish variable w₁ w₂ whence write x₁ zero α₁ α₂