Abstract
Special bilinear functions (SBF) proved to be applicable in many situations and for a good number of problems. Hence it is important to generalize them to a higher degree by expanding previous work. In the beginning, we give a quick review of SBF [or quacroms of second degree and dimension 2 x n]; then we give a few applications based on previously published research concentrating on their use in evaluating some special functions and where we present the evaluation of Chebyshev polynomials as a new work. Following that, we define special trilinear functions (STF) of three n-tuples vectors, which are the generalization of SBF. Finally, a few applications, such as taking the product of three polynomials of degree n, are given stressing the fact that the process of taking the product of three integers using STF techniques, practically, takes place in a very efficient way and with no mentioned effort. A short discussion on the future of the subject constitutes the conclusion of our article.
Highlights
Special bilinear functions (SBF) of two real vectors were introduced, with the name “quacroms of dimension 2 × n ” [1]
We give a quick review of SBF [or quacroms of second degree and dimension 2 × n ]; we give a few applications based on previously published research concentrating on their use in evaluating some special functions and where we present the evaluation of Chebyshev polynomials as a new work
We define special trilinear functions (STF) of three n-tuples vectors, which are the generalization of SBF
Summary
Special bilinear functions (SBF) of two real vectors were introduced, with the name “quacroms of dimension 2 × n ” [1]. More applications were found for them, especially after introducing linear quacrom equations (LQE) along with their solutions, where new algorithms were designed to compute known special functions, or polynomials, such as Bernoulli. A generalization to quacroms of dimension 3× n was introduced and studied with few applications [5]. We give a quick review of SBF, adding an extra example of its use to calculate the first few Chebyshev polynomials. In the section to follow, we present special trilinear functions (STF) (or quacroms of dimension 3× n ) but with a new face, expanding the subject so as to clarify the concepts and to get a deeper insight.
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