A Bohr Mollerup Theorem for Interpolating the Triangular Numbers

The Bohr-Mollerup Theorem (1922) provides an elegant criterion under which the gamma function is the unique function interpolating n!. We prove an analogous uniqueness theorem for interpolating the triangular numbers that, like the original, is grounded in the theory of convex functions. We then explore parallels with the class of quasi-gamma functions defined in a recent paper by T. Bermúdez, A. Martinón, and K. Sadarangani ["On quasi-gamma functions", Journal of Convex Analysis 21 (2014) 765–783].