Special Functions and Related Topics

Dear Colleagues,

Special functions, including trigonometric functions, have been studied and used for centuries. They present an old branch of mathematics to which many great mathematicians of the past made significant contributions. These include, among others, the following: Bernoulli and Euler numbers and polynomials; Euler's gamma and beta functions; the Digamma (Psi) function; the Pochhammer symbol; Gauss hypergeometric series; Riemann, Hurwitz, and Lerch zeta functions (together with Dirichlet and Mathieu series); Bessel and Struve differential equations; Bessel functions; Fourier-Bessel and Dini series of Bessel functions; Abel's, Jacobi's, and Weierstrass' work on elliptic functions; and the polynomials of Legendre, Jacobi, Laguerre, and Hermite. The need to introduce most of these special functions was for solving specific problems, and they appeared when it became clear that the existing elementary functions were not satisfying enough to describe many unsolved problems in mathematics and physics. So, it was suitable or necessary to present new results as infinite series, integrals, or through solutions of differential equations. F.W. Bessel contributed to the theory of special functions by systematically investigating the functions already considered by Bernoulli, Euler, Lagrange, Fourier, and others, whose research areas were mechanics, astronomy, and heat conduction. Currently, the family of Bessel functions counts many: Bessel functions of the first and second kind, modified Bessel functions of the first and second kind, Struve functions, modified Struve functions, Lommel functions, and others find their way into numerous applications. One topic in the theory of Bessel functions is the functional series of mathematical physics, having great importance in engineering and techniques. The Fourier–Bessel family of infinite series, consisting of Neumann, Kapteyn, Schlömilch, and Dini series, involving Bessel functions of the first kind or some other functions, belong to the hypergeometric representation. These functions appear whenever natural phenomena are studied, in engineering problems, and while performing numerical simulations. They also crop up in statistics, financial models, and economic analysis. Newton and Leibniz leveraged them in the solution of differential equations. Special functions have been continuously developing ever since. There have been many discoveries of several new special functions and their applications in the past thirty years.

Prof. Dr. Slobodan B. Tričković
Prof. Dr. Miomir Stankovic
Guest Editors

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