A Generalization of Euler's Reflection Principle

Continuous sums and products in the sense of Müller and Schleicher provide a framework that extends the usual notion of summation and product to real or even complex indexes. From this perspective, several special functions, including the Gamma, Barnes G, q-Gamma, and Hurwitz zeta functions, can be viewed as interpolated finite sums and products. In this article, we study reflection and multiplication phenomena in this general setting. For a class of functions that we call reflective, we prove that the reflected continuous product is divided into a principal factor and a 1-periodic correction, whose logarithm admits a computable Fourier expansion. When a function has a polynomial zero at the origin, the principal factor becomes trigonometric, yielding an Euler-type reflection formula, which recovers Euler’s reflection formula as a limiting case. We also prove a Gauss-type multiplication theorem in the same framework. Finally, we apply the reflection theorem to the q-Gamma function, obtaining the exact Fourier expansion for its reflection, together with other new identities.