Relations of the hilbert matrices and the hankel determinants with some special polynomials and numbers

In this thesis, relations of the Hilbert matrices and the Hankel determinants with some families of special polynomials and numbers have been studied. Using these matrices and determinants, the basic properties of the related special polynomials and numbers, including the generating functions, have been investigated and also some properties of some orthogonal polynomials have been given. Recurrence formulas are also given for some orthogonal polynomials with the help of some properties of the Hilbert matrices and the Hankel determinants. Relationships between some special orthogonal polynomials and Laplace distribution have been investigated and some important results of this distribution have been given. With the help of the given results and relations, identities and relations including Laplace distribution and Bernoulli numbers, Bernoulli polynomials and second type Euler numbers have been obtained. The results obtained in this thesis study have the potential to be applied both in the basic theory of orthogonal polynomials and in the fields of probability and statistics.