A Curious Trigonometric Infinite Product in Context

Abstract

By treating the multiple argument identity of the logarithm of the Gamma function as a functional equation, we obtain a curious infinite product representation of the \(sinc\) function in terms of the cotangent function. This result is believed to be new. It is then shown how to convert the infinite product to a finite product, which turns out to be a simple telescoping of the double angle \(sin\) function. In general, this result unifies known infinite product identities involving various trigonometric functions when the product term index appears as an exponent. In one unusual case, what appears to be a straightforward limit, suggests a counterexample to Weierstrass' factor theorem. A resolution is offered. An Appendix presents the general solution to a simple functional equation. This work is motivated by its educational interest.