Hankelformen

Hankelformen are a type of quadratic form, named after the German mathematician Hermann Hankel, that arise in various areas of mathematics, particularly in the study of linear recurrence relations and orthogonal polynomials. A Hankel form is characterized by its constant entries along the anti-diagonals. Specifically, for a matrix $H$ of size $n \times n$, it is a Hankel form if $H_{i,j} = H_{i-1, j+1}$ for all valid indices $i$ and $j$. This means that $H_{1,1} = H_{2,2} = \dots = H_{n,n}$, $H_{1,2} = H_{2,3} = \dots = H_{n-1,n}$, and so on. The elements of a Hankel form are thus determined by the first row and the first column of the matrix.

The properties of Hankel forms are significant in several mathematical contexts. For instance, the determinant of a Hankel form is related to the coefficients of a polynomial, and its properties can reveal information about the roots of that polynomial. In the theory of orthogonal polynomials, Hankel matrices play a role in defining the recurrence relations that govern these polynomials. Furthermore, Hankel forms appear in signal processing and control theory, where they are used to represent certain types of systems and to analyze their stability. Their structure allows for efficient computational algorithms, making them useful in practical applications. The study of Hankel forms often involves investigating their invertibility, eigenvalues, and their connection to continued fractions.