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 \( \displaystyle \prod_{k=a}^{\infty}f(k)^{2^{-k}}=\sqrt{f(a)\sqrt{f(a+1)\sqrt{f(a+2)\cdots}}} \). 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.