Domain Extensions of Binomial Numbers Applying Successive Sums Transformations on Sequences Indexed by Integers - Barroso, Marlo M.; Karam-Filho, José; Giraldi, Gilson A. (Trends in Computational and Applied Mathematics, 2020)
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.