Given a sequence g[subscript k] greater than 0, the "g-factorial" product [big product][superscript k] [subscript i=1] g[subscript i] is extended from integer k to real x by generalizing properties of the gamma function [Gamma](x). The Euler-Mascheroni constant [gamma] and the beta and zeta functions are also generalized. Specific examples include…

Based on the generating functions, for any positive integers "n" and "k", identities are established and the explicit formula for a[subscript i](k) in terms of Fibonomial coefficients are presented. The corresponding results are extended to some other famous sequences including Lucas and Pell sequences.

The function 1 divided by "x"[superscript 2] minus "e"[superscript"-x"] divided by (1 minus "e"[superscript"-x"])[superscript 2] for "x" greater than 0 is proved to be strictly decreasing. As an application of this monotonicity, the logarithmic concavity of the function "t" divided by "e"[superscript "at"] minus "e"[superscript"(a-1)""t"] for "a"…