Bastola's Infinite Derivative Sum Equation

We introduce a novel functional equation: y(x) = Σ from n=1 to ∞ of yⁿ(x), where yⁿ denotes the n-th derivative of y(x). This formulation explores the sum of all derivatives of a function, contrasting with traditional differential equations that only involve a finite number of derivatives. We provide a closed form solution, discuss convergence, and outline potential generalizations. This derivation uses standard tools from symbolic operational calculus, including the shift operator (Taylor/Euler) and algebraic properties of the derivative operator (Heaviside). To the best of the author’s knowledge, the sum of all derivatives equation and its closed form solution are original to the author.