In the recent new edition of the collected works of T.J. Stieltjes, one of us gave an impression of the impact of Stieltjes' work a century after his death 43]. In this paper we give an update and mention some observations which were missing from 43] and some results which appeared during the last two years and which are directly related to Stieltjes' work. In particular there is a large sectio...

We are interested in the special Heun equation, " special " means that all exponent differences are odd multiples of 1/2. Heun means 4 regular singularities.

The pointwise limit S of a sequence of Stieltjes transforms (Sn) of real Borel probability measures (Pn) is itself the Stieltjes transform of a Borel p.m. P if and only if iy S(iy) → −1 as y → ∞, in which case Pn converges to P in distribution. Applications are given to several problems in mathematical physics.

Polynomial solutions to the generalized Lamé equation, the Stieltjes polynomials, and the associated Van Vleck polynomials have been studied since the 1830’s, beginning with Lamé in his studies of the Laplace equation on an ellipsoid, and in an ever widening variety of applications since. In this paper we show how the zeros of Stieltjes polynomials are distributed and present two new interlacin...

In this paper the discrete Sobolev inner product 〈p, q〉 = ∫ p(x)q(x)dμ+ r ∑ i=0 Mi p (c)q(c) is considered, whereμ is a finite positive Borel measure supported on an infinite subset of the real line, c ∈ R and Mi 0, i = 0, 1, . . . , r. Connection formulas for the orthonormal polynomials associated with 〈., .〉 are obtained. As a consequence, for a wide class of measures μ, we give the Mehler–He...

We explore properties of Cauchy-Stieltjes families that have no counterpart in exponential families. We relate the variance function of the iterated Cauchy-Stieltjes family to the pseudo-variance function of the initial Cauchy-Stieltjes family. We also investigate when the domain of means can be extended beyond the ”natural domain”.

We study the asymptotic behavior of the zeros of polynomial solutions of a class of generalized Lamé differential equations, when their coefficients satisfy certain asymptotic conditions. The limit distribution is described by an equilibrium measure in presence of an external field, generated by charges at the singular points of the equation. Moreover, a case of non-positive charges is consider...

Stieltjes’ Theorem (cf. [11]) proves that if {pn}n=0 is an orthogonal sequence, then between any two consecutive zeros of pk there is at least one zero of pn for all positive integers k, k < n; a property called Stieltjes interlacing. We prove that Stieltjes interlacing extends across different sequences of Laguerre polynomials Ln, α > −1. In particular, we show that Stieltjes interlacing holds...

We present a variety of series representations of the Stieltjes and related constants, the Stieltjes constants being the coefficients of the Laurent expansion of the Hurwitz zeta function ζ(s, a) about s = 1. Additionally we obtain series and integral representations of a sum Sγ(n) formed as an alternating binomial series from the Stieltjes constants. The slowly varying sum Sγ(n) + n is an impo...