Volume 13, 2007, Number 1

This paper establishes and characterizes the theorem on rhotrix exponent rule, and presents the theory to stimulate systematization of expressing some special series and polynomial equations in terms of the relatively new method of representing arrays of real numbers.

Odd integers raised to an odd power n can equal a sum of squares only if the integers are in Class ̅1₄ of the Modular Ring Z₄. The primes raised to an odd power are unique in that they only have (n − 1) square couples. These couples depend on the single couple (a₁, b₁). A general equation is given for predicting the square couples of pⁿ and odd-powered composites. Even integers raised to an odd power have no primitive solutions for such square couples, because the sum of two odd squares falls in Class ̅2₄ where there are no powers at all.

This paper considers some properties of generalized binomial coefficients which are defined in terms of rising factorial coefficients. Analogies with classical results in number theory and some generalized special functions are highlighted.

In a series of papers we shall discuss new types of Pascal’s like triangles. Triangles from the present form, but not with the present sense, are described in different publications, e.g.  [1,2,3], but at least the author had not found a research with similar idea. In the first part of our research we shall study properties of \standard” sequences, while in the next parts the objects of our interest will be more nonstandard sequences.