A Method for Solving and Simplifying a Class of Radical Infinite Products

In this paper, the author conducts an in-depth study of a radical infinite product of the form \ (₊=₀^f (k) ^2^{-k}=f (a) f (a+1) {f (a+2) } \). The convergence of this expression is briefly discussed. A method for simplifying such infinite products is proposed. By exploring cases where \ (f (x) \) represents hyperbolic functions, trigonometric functions, and complex-valued functions, the Dobinski identity is reproduced and generalized. Furthermore, leveraging Weierstrass's theorem and special functions, a relationship is established between this form of infinite nested radical and general infinite products.