Research in Mathematics

In this Article Collection, we aim to promote the study of special functions and, in particular of orthogonal polynomials and their applications, not only in the traditional field of mathematical physics equations and in detail of the Fredholm, Volterra, and Mellin-type integral equations and integral differential equations, but also in those of the combinatorial theory, the search for optimal interpolatory processes, analytical theory of numbers and linear analysis. Many articles have appeared, including recent ones. Mindfully, on special sequences of numbers or polynomials in the context of the analytical theory of numbers.

Approximation is the basis of all mathematical procedures since the exact solutions to the problems of applied sciences are limited to a negligible number of cases. The same real numbers, not rational ones, can be conceived only for the possibility of assigning them approximations by default and by excess, with a consequent estimate of the error. Archimedes had already highlighted this problem in the determination of π, a pure symbol, without the method introduced to construct its limitations. Then Euler himself (who was the first to use pi), in representing the sum of the reciprocals of the squares of integers in the form π2/6 introduces an approximation since this necessarily happens in all infinite procedures, from the sum of series to integration, from the search for the zeros of a function to the solution of differential equations. In the framework of this Collection, the orthogonality property is certainly the most important one to use in approximation problems, since it allows us to avoid the numerical instability that occurs when using the Hilbert matrix.

There is no need to remember the importance of the numerous families of polynomials and special functions that are used in the solutions of the most diverse mathematical physics problems. Among these, they have great importance in Bessel functions, recently linked to Legendre polynomials. The analysis of fractional calculus through the concepts and formalism of some classes of orthogonal polynomials (in particular Hermite polynomials) has had an important development, also in relation to the large interest that this research sector has had in recent years.

The analysis of fractional calculus through the concepts and the formalism of some classes of orthogonal polynomials (in particular the Hermite polynomials) and a further research area such as the study of extensions to the case of the fractional index of Chebyshev polynomials, even in the multidimensional case of pseudo-Chebyshev and pseudo-Lucas polynomials. Further investigations are aimed at the generalizations of numbers of Bernoulli, Euler, Hahn, Bell, et al., also through expressions of polynomials in the form of determinants. Finally, the relations of the multidimensional orthogonal polynomials (in particular Lucas polynomials) and their relative applications to the study of linear dynamic systems are now well known and therefore they allow us to broaden the knowledge in the above disciplinary areas considered.

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