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...

Bethe ansatz solution of the two-axis two-spin Hamiltonian is derived based on Jordan-Schwinger boson realization SU(2) algebra. It shown that equations can be obtained as zeros related extended Heine-Stieltjes polynomials. Symmetry properties excited levels system and polynomials are discussed. As an example application theory, two equal spin case studied in detail, which demonstrates each ban...

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.

Let G be a locally compact σ-compact group. Motivated by an earlier notion for discrete groups due to Effros and Ruan, we introduce the multidimensional Fourier algebra A(G) of G. We characterise the completely bounded multidimensional multipliers associated with A(G) in several equivalent ways. In particular, we establish a completely isometric embedding of the space of all n-dimensional compl...

We study a generalisation of operator spaces modelled on Lp spaces, instead of Hilbert spaces, using the notion of p-complete boundedness, as studied by Pisier and Le Merdy. We show that the Figà-Talamanca-Herz Algebras Ap(G) becomes quantised Banach algebras in this framework, and that the cohomological notion of amenability of these algebras corresponds to amenability of the locally compact g...

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...