Abstract
In a recent article, the first and second kinds of multivariate Chebyshev polynomials of fractional degree, and the relevant integral repesentations, have been studied. In this article, we introduce the first and second kinds of pseudo-Lucas functions of fractional degree, and we show possible applications of these new functions. For the first kind, we compute the fractional Newton sum rules of any orthogonal polynomial set starting from the entries of the Jacobi matrix. For the second kind, the representation formulas for the fractional powers of a r×r matrix, already introduced by using the pseudo-Chebyshev functions, are extended to the Lucas case.
Highlights
Applications of special functions and polynomials can be found in the solution to every problem in mathematical physics, engineering, statistics, biology, and in general, in applied mathematics
The representation formulas for the fractional powers of a r × r matrix, already introduced by using the pseudo-Chebyshev functions, are extended to the Lucas case
The above equation can be obviously generalized to an r × r matrix and to a general fractional exponent, according to the results proven in [22,23] using the SKMP-C functions
Summary
Applications of special functions and polynomials can be found in the solution to every problem in mathematical physics, engineering, statistics, biology, and in general, in applied mathematics. Multivariate Lucas polynomials of the second kind have proved useful for computing integer-exponent powers of a matrix [24] In this regard it will not be useless to remember, once again, that every holomorphic function of a matrix, and in particular the exponential, is nothing but the polynomial interpolating the function on the matrix’s spectrum and it is useless to consider expansions in series of the given matrix. Chebyshev multivariate functions of the second kind with a fractional index have recently been used for computing the roots of nonsingular complex matrices [22], and those of the first kind for computing the moments, with fractional exponent, of the density of zeros of a set of orthogonal polynomials [23] It remains to define the multivariate Lucas polynomials of the first and second kind of fractional index (called for simplicity, pseudo-Lucas functions). The first kind of pseudo-Lucas functions are applied, in what follows, to derive the fractional Newton sum rules for a general set of orthogonal polynomials, defined by a three-term recurrence relation (through its Jacobi matrix), extending the equations proven in [26,28,29,30]
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