We will introduce orthogonal polynomials with respect to a Sobolev inner product associated with a vector of measures supported on the real line. Analytic properties of such polynomials will be studied. We will focus the attention on the case of coherent pairs of measures. As applications, we will deal first with boundary value problems. For spectral methods we will show the effectiveness of the approximation to the solutions by using such Sobolev orthogonal polynomials instead of the standard ones. Second, we will analyze the Sobolev type inner products where you have a symmetric polynomial operator that yields recurrence relations. The corresponding banded matrix has LU factorizations and thus a connection with bi-spectral problems will be studied.

The aim of this course is to present some fast and reliable numerical methods for the computation of Orthogonal Polynomials and Special Functions. The following topics will be considered: 

• computation of recurrence relations for orthogonal polynomials;

• computation of the zeros of orthogonal polynomials and associated Gaussian rules;

• computation of product rules and modified moments for some classes of positive weights;

• computation of special functions.

The theory of fractional integrals and fractional derivatives has been initiated in the 17th century, and they have many real-world applications due to their properties since they interpolate their integer versions. Among these operators Riemann-Liouville and the Caputo variants are the best knowns. Many special functions arise in the investigation of the problems based on these operators. Among them, the Mittag-Leffler function and its generalizations are very important since they appear naturally as solutions of fractional differential equations or fractional difference equations. The Mittag-Leffler function was introduced at the early years of the 20th century by Gösta Mittag-Leffler. On one hand it has some interesting properties as a special function, and on the other hand it plays the role of exponential function in the fractional differential equations. Because of this reason, this function sometimes called the “queen function” by several mathematicians. The main lack of this function is that it doesn’t satisfy the addition property that the exponential function satisfies. The main aim of these lectures is to introduce the possible bivariate (general) versions of this function, investigate their properties as a special function and construct a fractional calculus on these functions having a semigroup property. We also demonstrate how these functions and operators arise naturally from some fractional partial integrodifferential equations of Riemann–Liouville type.

Classes of analytic functions appear in many areas of mathematics. Herglotz, Nevanlinna–Pick and Stieltjes functions enter in a fundamental way in e.g., operator theory, orthogonal polynomials and probability theory. After an introduction of these classes, we give examples of their use, combined with real analytic methods, in solving problems for special functions, such as Euler’s gamma function and socalled higher order gamma functions introduced by Barnes.

We introduce the fractional derivative operator using the Euler definition and we consider Taylor expansions in fractional powers. The fractional exponential functions are exploited in order to extend in the fractional framework several special polynomials functions and numbers, as the Hermite and Laguerre polynomials, Appell-type functions, Bernoulli and Euler polynomials and numbers. The Laguerre-type fractional functions are also considered. Applications of fractional exponential functions to population dynamics problems and extensions of the Laplace transform are shown.