Sobolev orthogonal polynomials in computing of Hankel determinants

In this paper, we study closed form evaluation for some special Hankel determinants arising in combinatorial analysis, especially for the bidirectional number sequences. We show that such problems are directly connected with the theory of quasi-definite discrete Sobolev orthogonal polynomials. It opens a lot of procedural dilemmas which we will try to exceed. A few examples deal with Fibonacci numbers and power sequences will illustrate our considerations. We believe that our usage of Sobolev orthogonal polynomials in Hankel determinant computation is quite new.