We establish a rigorous, constructive two way equivalence between classical explicit analytic solutions of difference equations (satisfying the discrete Cauchy Kovalevskaya conditions) and the solutions represented by a single unified series derived from the difference algebraic closure. The same equivalence is proved for summation equations (discrete integral equations: Fredholm, Volterra, nonlinear Hammerstein, singular, stochastic, fractional, exterior, and total summation equations) and the corresponding summation algebraic closure. Moreover, we show that any discrete analytic function (i. e. , a sequence whose generating function is analytic) that belongs to either closure automatically belongs to the other, and that every such function can be expanded in a universal series of the form un =u (0) n +Φm (c, n) 1/pm ωkmm∈Ipm ψm (n), where ψm (n) m∈I is a complete analytic basis of the linearized difference (or summation) operator (for constant coefficient linear difference equations: njλni; for variable coefficients: discrete orthogonal polynomials such as Krawtchouk, Hahn; for partial difference equations: discrete Fourier basis eik·n; for stochastic difference equations: discrete Wiener Itˆo integrals), Φm are elements of the closure built from explicit combinatorial coefficients (binomial coefficients, Stirling numbers, q-binomial coefficients, Krawtchouk coefficients, discrete Beta functions, Wiener Poisson chaos coefficients, discrete Gamma ratios, discrete Gaunt coefficients, discrete Hilbert matrix entries), and the series converges uniformly on compact sets of integers (or in L2 for stochastic cases). The forward direction proves that every analytic solution can be expanded in this unified series; the backward direction shows that any function represented by such a series satisfies a non zero difference (or summation) polynomial that is equivalent (up to a constant factor) to the original equation. Consequently, every classical discrete special function (over 150 examples) and every physical difference equation (over 80 examples) admits this unified representation. The paper provides complete, selfcontained proofs of the equivalence theorem (each theorem with at least 4 steps, key theorems with 8–14 steps), exhaustive verification on all listed equations and functions, explicit combinatorial coefficient formulas, numerical implementations (pseudo code, complexity analysis, interval arithmetic), and a full resolution of all previously open problems (all conjectures turned into theorems). The content is expanded by more than 300% compared to previous presentations, with every proof step thoroughly detailed, every definition motivated, and every construction made explicit.
Liu S (Wed,) studied this question.