Abstract
The classic definition of binomial numbers involves factorials, making unfeasible their extension for negative integers. The methodology applied in this paper allows to establish several new binomial numbers extensions for the integer domain, reproduces to integer arguments those extensions that are proposed in other works, and also verifies the results of the usual binomial numbers. To do this, the impossibility to compute factorials with negative integer arguments is eliminated by the replacement of the classic binomial definition to a new one, based on operations recently proposed and, until now, referred to as transformations by the successive sum applied on sequences indexed by integers. By particularizing these operations for the sequences formed and indexed by integers, it is possible to redefine the usual binomial numbers to any integer arguments, with the advantage that the values are more easily computed by using successive additions instead of multiplications, divisions or possibly more elaborate combinations of these operators, which could demand more than one or two sentences to their application.
Highlights
Binomial numbers have an essential role in the most varied fields of pure and applied mathematics and occur in many important algebraic developments
The earliest known binomial application example is due to Euclid (325 BC–265 BC) which through geometry concluded that the square area is equal to the sum of the rectangles areas contained in it [2], whereas the representation of binomial constants in the form of a table is usually attributed to the Chinese mathematician Yang Hui (1238 AD–1298 AD), there are some divergences between references
This paper presents in a synthetic form how to define and extend the usual binomial numbers for any integer arguments, according to the proposition made in [9], which changed the approach to the question
Summary
Binomial numbers have an essential role in the most varied fields of pure and applied mathematics and occur in many important algebraic developments. It was not used factorials or some kind of product in the definition of binomials; instead, the so-denominated transformations by the successive sum on sequences indexed by integer numbers defined applying additions have been used Since such transformations can be performed within any arguments in Z, it has become possible to obtain binomial numbers both in the usual natural domain and in the integer one. These results are consistent with those of other authors cited and are obtained considering a set of interest properties that must remain valid to any extension Behind this proposition there is a new methodology for approaching recursive functions, through mathematical operations that transform integer-indexed sequences from the solution of a system of equations composed of a known value and a recursion. As it will be seen in this work, nested summations had not been used in the cases of: unit and null orders, and negative first order
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