Upward extension of the Jacobi matrix for orthogonal polynomials

Orthogonal polynomials on the real line always satisfy a three-term recurrence relation. The recurrence coefficients determine a tridiagonal semi-infinite matrix (Jacobi matrix) which uniquely characterizes the orthogonal polynomials. We investigate new orthogonal polynomials by adding to the Jacobi matrix $r$ new rows and columns, so that the original Jacobi matrix is shifted downward. The $r$ new rows and columns contain $2r$ new parameters and the newly obtained orthogonal polynomials thus correspond to an upward extension of the Jacobi matrix. We give an explicit expression of the new orthogonal polynomials in terms of the original orthogonal polynomials, their associated polynomials and the $2r$ new parameters, and we give a fourth order differential equation for these new polynomials when the original orthogonal polynomials are classical. Furthermore we show how the orthogonalizing measure for these new orthogonal polynomials can be obtained and work out the details for a one-parameter family of Jacobi polynomials for which the associated polynomials are again Jacobi polynomials

An atomic and molecular database for analysis of submillimetre line observations

Atomic and molecular data for the transitions of a number of astrophysically interesting species are summarized, including energy levels, statistical weights, Einstein A-coefficients and collisional rate coefficients. Available collisional data from quantum chemical calculations and experiments are extrapolated to higher energies. These data, which are made publically available through the WWW at http://www.strw.leidenuniv.nl/~moldata, are essential input for non-LTE line radiative transfer programs. An online version of a computer program for performing statistical equilibrium calculations is also made available as part of the database. Comparisons of calculated emission lines using different sets of collisional rate coefficients are presented. This database should form an important tool in analyzing observations from current and future (sub)millimetre and infrared telescopes.Comment: Accepted for publication in A&A, 14 pages, 5 figure

Antiperiodic dynamical 6-vertex model I: Complete spectrum by SOV, matrix elements of the identity on separate states and connections to the periodic 8-vertex model

The spin-1/2 highest weight representations of the dynamical 6-vertex and the standard 8-vertex Yang-Baxter algebra on a finite chain are considered in this paper. For the antiperiodic dynamical 6-vertex transfer matrix defined on chains with an odd number of sites, we adapt the Sklyanin's quantum separation of variable (SOV) method and explicitly construct SOV representations from the original space of representations. We provide the complete characterization of eigenvalues and eigenstates proving also the simplicity of its spectrum. Moreover, we characterize the matrix elements of the identity on separated states by determinant formulae. The matrices entering in these determinants have elements given by sums over the SOV spectrum of the product of the coefficients of separate states. This SOV analysis is not reduced to the case of the elliptic roots of unit and the results here derived define the required setup to extend to the dynamical 6-vertex model the approach recently developed in [1]-[5] to compute the form factors of the local operators in the SOV framework, these results will be presented in a future publication. For the periodic 8-vertex transfer matrix, we prove that its eigenvalues have to satisfy a fixed system of equations. In the case of a chain with an odd number of sites, this system of equations is the same entering in the SOV characterization of the antiperiodic dynamical 6-vertex transfer matrix spectrum. This implies that the set of the periodic 8-vertex eigenvalues is contained in the set of the antiperiodic dynamical 6-vertex eigenvalues. A criterion is introduced to find simultaneous eigenvalues of these two transfer matrices and associate to any of such eigenvalues one nonzero eigenstate of the periodic 8-vertex transfer matrix by using the SOV results. Moreover, a preliminary discussion on the degeneracy of the periodic 8-vertex spectrum is also presented.Comment: 36 pages, main modifications in section 3 and one appendix added, no result modified for the dynamical 6-vertex transfer matrix spectrum and the matrix elements of identity on separate states for chains with an odd number of site

Two closed-form evaluations for the generalized hypergeometric function 4F3(116){}_4F_3(\frac1{16})

The objective of this short note is to provide two closed-form evaluations for the generalized hypergeometric function $_4F_3$ of the argument $\frac1{16}$. This is achieved by means of separating a generalized hypergeometric function $_3F_2$ into even and odd components, together with the use of two known results for $_3F_2(\pm\frac14)$ available in the literature. As an application, we obtain an interesting infinite-sum representation for the number $\pi^2$. Certain connections with the work of Ramanujan and other authors are discussed, involving other special functions and binomial sums of different kinds

Lithium: Production and Estimated Consumption. Evidence of Persistence.

Understanding the behavior of the lithium supply and the estimated consumption and flows is important for social and economic development. We focus on estimating persistence and for this purpose, we use techniques based on fractional integration. The empirical results provide evidence of mean reversion for the data corresponding to the global lithium production from 1925 to 2014 but not for U.S. lithium-related series such as production (1900 – 2008), estimated consumption (1900 – 2014), imports (1960 – 2015), and exports (1971 – 2015).pre-print525 K