On the explicit representation of polyorthogonal polynomials
In the classical theory of orthogonal polynomials, the representation of the orthogonal polynomial \(Q_n\) in determinant form is well known: \[ Q_n(z)=\left| \begin{array}{cccc}s_0 & s_1 & \cdots & s_n \\ s_1 & s_2 & \cdots & s_{n+1} \\ \cdots & \cdots & \cdots & \cdots \\ s_{n-1} & s_n & \cdots & s_{2n-1}\\ 1 & z & \cdots & z^n\end{array}\right|, \] where \[ s_i=\int_{\triangle}x^id\mu (x)\ (i=0, 1, \ldots ) \] is a sequence of power moments of the measure \(\mu \), whose carrier coincides with the orthogonality segment \(\triangle\). In this paper, the author proves that a similar representation is also valid for a polyorthogonal polynomial if the power moments of the measures \(\mu _j\) are known. In this respect, notice that polyorthogonal polynomials (named also as \lq multiple orthogonal polynomial\rq) satisfy non-standard orthogonality relations and naturally arise as the common denominator of the Padé approximations for the system of Markov functions. Here, the orthogonality relations are distributed among several measures \(\mu _1,...,\mu _k\). Moreover, the author considers the concept of orthogonality in a broader sense, when the orthogonal polynomial is defined for an arbitrary sequence \(\{s_i\}\), which is not bound by any restrictions. In this general situation, necessary and sufficient conditions for the existence and uniqueness of a polyorthogonal polynomial are found, under which it can be represented in determinant form similar to one above. As a consequence, similar representations are obtained for the residual function, the numerator, and the denominator of the corresponding Padé approximations, complementing this way well-known results in the theory of polyorthogonal polynomials and Padé approximations. A fundamental role is played by new concepts which are introduced in the paper, namely, an admissible index and an almost perfect system of functions.
scientific article; zbMATH DE number 7374471
scientific article