Stieltjes polynomials

A system of polynomials which satisfy the orthogonality condition where the weight function satisfies , , with finite moments . is the system of orthogonal polynomials associated with . The degree of is equal to its index . The orthogonality conditions define up to a multiplicative constant, but the conditions for given above are not sufficient for to have real zeros in . However, several special cases and classes of weight functions are known for which the zeros of the corresponding Stieltjes polynomials do not only have this property, but also interlace with the zeros of . A simple example is , , , the Chebyshev polynomial of the second kind (cf. Chebyshev polynomials), where , the Chebyshev polynomial of the first kind. For , and . A generalization are the Bernstein–Szegö weight functions where is a polynomial of degree that is positive in [a6], [a7]. Weight functions for which , and for are another class for which the above properties are known to hold asymptotically under certain additional conditions on [a8].

The classical case originally considered by Th.J. Stieltjes is the Legendre weight function , . For this case G. Szegö [a9] proved that all zeros are real, belong to the open interval and interlace with the zeros of the Legendre polynomials . Szegö extended his proof to the ultraspherical, or Gegenbauer, weight function, , , , cf. also Gegenbauer polynomials; Ultraspherical polynomials. For the more general Jacobi weight , results of existence and non-existence can be found in [a4]. Comparatively little is known for unbounded intervals. Numerical results reported in [a5] show that complex zeros arise for the Laguerre weight , , and the Hermite weight , .

An important fact for the analysis of the Stieltjes polynomials is their close connection with the functions of the second kind associated with and . Stieltjes [a1] proved that is precisely the polynomial part of the Laurent expansion of . Szegö's work in [a9] and subsequent investigations are based on this connection.

Several asymptotic representations are available. A simple formula for the Legendre weight function is  uniformly for , see [a2]. Inequalities for Stieltjes polynomials in the case can be found in [a3].

The zeros of Stieltjes polynomials are used for quadrature and for interpolation. In particular, the often-used Gauss–Kronrod quadrature formulas (cf. Gauss–Kronrod quadrature formula) are based on the union of the zeros of and and enable an efficient estimation for the Gauss quadrature formula based on the zeros of . This idea has been carried over to extended interpolation processes (cf. Extended interpolation process). For , adding the zeros of improves the interpolation process based on the zeros of to an optimal-order interpolation process [a3] (see also Extended interpolation process).