Hahn's generalized problem and corresponding Appell sequences

This thesis is devoted to some aspects of the theory of orthogonal polynomials, paying a special attention to the classical ones (Hermite, Laguerre, Bessel and Jacobi). The elements of a classical sequence are eigenfunctions of a second order linear differential operator with polynomial coefficients $\mathcal{L}$ known as the Bochner's operator. In an algebraic manner, a classical sequence is also caracterised through the so-called Hahn's property, which states that an orthogonal polynomial sequence is classical if and only if the sequence of its (normalised) derivatives is also orthogonal.

In the present work we show that an orthogonal polynomial sequence is classical if and only if any of its polynomials fulfils a certain differential equation of order $2k$, for some positive integer $k$. We thoroughly reveal the structure of such differential equation and, for each classical family, we explicitly present the corresponding $2k$-order differential operator $\mathcal{L}_{k}$. When we consider $k=1$, we recover the Bochner's differential operator: $\mathcal{L}_{1} = \mathcal{L}$. On the other hand, as a consequence of Bochner's result, any element of a classical sequence must be an eigenfunction of a polynomial with constant coefficients in powers of $\mathcal{L}$. As a result of the introduction of the so-called $A$-modified Stirling numbers (where $A$ indicates a complex parameter), we are able to establish inverse relations between the powers of the Bochner operator $\mathcal{L}$ and $\mathcal{L}_{k}$.

Afterwards, we proceed to the quadratic decomposition of an Appell sequence. The four polynomial sequences obtained by this approach are also Appell sequences but with respect to another lowering differential operator, denoted $\mathcal{F}_{\varepsilon}$, where $\varepsilon$ is either 1 or -1. Thus, we introduce and develop the concept of Appell sequences with respect to the operator $\mathcal{F}_{\varepsilon}$ (where, more generally, $\varepsilon$ denotes a complex parameter): the $\mathcal{F}_{\varepsilon}$-Appell sequences. Subsequently, we seek to find all the orthogonal polynomial sequences that are also $\mathcal{F}_{\varepsilon}$-Appell, which are, indeed, the $\mathcal{F}_{\varepsilon}$-Appell sequences that satisfy Hahn's property respect to $\mathcal{F}_{\varepsilon}$. This latter consists of the Laguerre sequences of parameter $\varepsilon/2$, up to a linear change of variable. Inspired by this problem, we characterise all the $\mathcal{F}_{\varepsilon}$-classical sequences.

While ferreting out the all $\mathcal{F}_{\varepsilon}$-classical sequences, apart from the Laguerre sequence, we find certain Jacobi sequences (with two parameters).

The quadratic decomposition of Appell sequences with respect to other lowering operators is also considered and the results obtained are akin to the aforementioned ones attained in the analogous problem.