Abstract
Here, in this paper, the second-kind multivariate pseudo-Chebyshev functions of fractional degree are introduced by using the Dunford–Taylor integral. As an application, the problem of finding matrix roots for a wide class of non-singular complex matrices has been considered. The principal value of the fixed matrix root is determined. In general, by changing the determinations of the numerical roots involved, we could find n r roots for the n-th root of an r × r matrix. The exceptional cases for which there are infinitely many roots, or no roots at all, are obviously excluded.
Highlights
Special functions and polynomials are used in many applications of physics, engineering, and applied mathematics, such as electrodynamics, classical and modern physics, quantum mechanics, classical mechanics, and, more recently, in biological sciences and many other fields
Many multivariate generalizations of hypergeometric functions have been studied in the literature on Special Functions, even through an extension of the Pochhammer symbol [4,5,6,7,8]
In the authors’ opinion, the second-kind pseudo-Chebyshev functions seem to be naturally connected with the problem of computing matrix roots, as the multivariate Chebyshev polynomials are with regard to the representation of integer powers of matrices
Summary
Special functions and polynomials are used in many applications of physics, engineering, and applied mathematics, such as electrodynamics, classical and modern physics, quantum mechanics, classical mechanics, and, more recently, in biological sciences and many other fields. Explicit equations for computing matrix powers were considered in connection with the introduction of multivariate second-kind Chebyshev polynomials (see, for example, [9,10]). Since these articles were written in Italian, they were mostly ignored by the mathematical community. In the authors’ opinion, the second-kind pseudo-Chebyshev functions seem to be naturally connected with the problem of computing matrix roots, as the multivariate Chebyshev polynomials are with regard to the representation of integer powers of matrices.
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