Second Order Factorials

The purpose of this research was to present a new type of Factorials, based on Arithmetic and Geometric Progressions of Order 2. As part of the Study, there was made a brief exposition on Arithmetic and Geometric Progressions of Higher Order, where the focus will be more on the Arithmetic and Geometric Progressions of Order 2, through which there shall be structured the objects to be studied.
An imminent result obtained from the present Study is a consistent proposal for the generalization of the Gamma Function, a very important function in the intricacies of classical mathematics, namely in Mathematical Analysis and Statistics. This function was presented to the scientific community by the renowned and consecrated mathematician, Leonhard Euler Leonhard Euler (1707 – 1783), around 1730 and is directly related to Infinitesimal Calculus.
A new Counting Technique is also proposed, similar to the classical Combinations, this time based on Geometric Progressions.
As a result of the introduction of a new Counting Technique, a new type of Arithmetic Triangle is proposed, also analogous to Pascal’s Triangle, the latter based on classical Combinations.