Similar to the Bohr-Mollerup Theorem for the gamma function, the log K-function is the unique (up to an additive constant) eventually 2-convex solution to the equation where is the forward difference operator.[3]
Propertiesedit
It can be shown that for α > 0:
This can be shown by defining a function f such that:
Differentiating this identity now with respect to α yields:
1, 4, 108, 27648, 86400000, 4031078400000, 3319766398771200000, ... (sequence A002109 in the OEIS).
Referencesedit
^Adamchik, Victor S. (1998), "PolyGamma Functions of Negative Order", Journal of Computational and Applied Mathematics, 100: 191–199, archived from the original on 2016-03-03
^Espinosa, Olivier; Moll, Victor Hugo, "A Generalized polygamma function" (PDF), Integral Transforms and Special Functions, 15 (2): 101–115, archived (PDF) from the original on 2023-05-14
^"A Generalization of Bohr-Mollerup's Theorem for Higher Order Convex Functions: a Tutorial" (PDF). Bitstream: 14. Archived (PDF) from the original on 2023-04-05.