The impact of Stieltjes' work on continued fractions and orthogonal polynomials

Stieltjes' work on continued fractions and the orthogonal polynomials related to continued fraction expansions is summarized and an attempt is made to describe the influence of Stieltjes' ideas and work in research done after his death, with an emphasis on the theory of orthogonal polynomials

A two-variable series for knot complements

The physical 3d $\mathcal{N}=2$ theory T[Y] was previously used to predict the existence of some 3-manifold invariants $\hat{Z}_{a}(q)$ that take the form of power series with integer coefficients, converging in the unit disk. Their radial limits at the roots of unity should recover the Witten-Reshetikhin-Turaev invariants. In this paper we discuss how, for complements of knots in $S^3$, the analogue of the invariants $\hat{Z}_{a}(q)$ should be a two-variable series $F_K(x,q)$ obtained by parametric resurgence from the asymptotic expansion of the colored Jones polynomial. The terms in this series should satisfy a recurrence given by the quantum A-polynomial. Furthermore, there is a formula that relates $F_K(x,q)$ to the invariants $\hat{Z}_{a}(q)$ for Dehn surgeries on the knot. We provide explicit calculations of $F_K(x,q)$ in the case of knots given by negative definite plumbings with an unframed vertex, such as torus knots. We also find numerically the first terms in the series for the figure-eight knot, up to any desired order, and use this to understand $\hat{Z}_a(q)$ for some hyperbolic 3-manifolds.Comment: 79 pages; final version, to appear in Quantum Topolog

Elliptic Hypergeometric Laurent Biorthogonal Polynomials with a Dense Point Spectrum on the Unit Circle

Using the technique of the elliptic Frobenius determinant, we construct new elliptic solutions of the $QD$-algorithm. These solutions can be interpreted as elliptic solutions of the discrete-time Toda chain as well. As a by-product, we obtain new explicit orthogonal and biorthogonal polynomials in terms of the elliptic hypergeometric function ${_3}E_2(z)$. Their recurrence coefficients are expressed in terms of the elliptic functions. In the degenerate case we obtain the Krall-Jacobi polynomials and their biorthogonal analogs

Bi-orthogonal systems on the unit circle, Regular Semi-Classical Weights and Integrable Systems - II

We derive the Christoffel-Geronimus-Uvarov transformations of a system of bi-orthogonal polynomials and associated functions on the unit circle, that is to say the modification of the system corresponding to a rational modification of the weight function. In the specialisation of the weight function to the regular semi-classical case with an arbitrary number of regular singularities $\{z_1, ..., z_M \}$ the bi-orthogonal system is known to be isomonodromy preserving with respect to deformations of the singular points. If the zeros and poles of the Christoffel-Geronimus-Uvarov factors coincide with the singularities then we have the Schlesinger transformations of this isomonodromic system. Compatibility of the Schlesinger transformations with the other structures of the system - the recurrence relations, the spectral derivatives and deformation derivatives is explicitly deduced. Various forms of Hirota-Miwa equations are derived for the $\tau$-functions or equivalently Toeplitz determinants of the system.Comment: to appear J. Approx. Theor

PICES Press, Vol. 16, No. 2, July 2008

The 2008 Inter-Sessional Science Board Meeting (pp.1-2, pdf, 0.1 Mb) FUTURE – From Science Plan to Implementation Plan (pp. 3-4, pdf, 0.1 Mb) CFAME Task Team Workshop – Linking and Visualising (p. 5, pdf, 0.1 Mb) PICES WG 21 Meets in Busan, Korea: The Database Meeting (pp. 6-7, pdf, 0.1 Mb) ICES-PICES-IOC Symposium on Climate Change (pp. 8-12, pdf, 1.2 Mb) Zooplankton and Climate: Response Modes and Linkages (pp. 13-15, pdf, 0.2 Mb) PICES Fishery Science Committee Workshop in Gijón (pp. 16-18, pdf, 0.1 Mb) The North Pacific Continuous Plankton Recorder Survey (pp. 19-21, pdf, 0.4 Mb) PICES Ecosystem Status Report Wins Design Award (p. 21, pdf, 0.4 Mb) Canada’s Three Oceans (C3O): A Canadian Contribution to the International Polar Year (pp. 22-25, pdf, 0.8 Mb) New Surface Mooring at Station Papa Monitors Climate (pp. 26-27, pdf, 0.2 Mb) The State of the Western North Pacific in the Second Half of 2007 (pp. 28-29, pdf, 0.4 Mb) The Bering Sea: Current Status and Recent Events (pp. 30-31, pdf, 0.4 Mb) Recent Trends in Waters of the Subarctic NE Pacific (pp.32-33, pdf, 0.3 Mb) 2009 Vintage of Fraser River Sockeye Salmon: A Complex Full Bodied Redd with Mysterious Bouquet (p. 34, pdf, 0.1 Mb) Pacific Biological Station Celebrates Centennial Anniversary, 1908–2008 (p. 35, pdf, 0.3 Mb) Marine and Coastal Fisheries: American Fisheries Society Open Access E-journal (p. 36, pdf, 0.1 Mb) Latest and Upcoming PICES Publications (p. 36, pdf, 0.1 Mb