An algorithm to obtain global solutions of the double confluent Heun equation

A procedure is proposed to construct solutions of the double confluent Heun equation with a determinate behaviour at the singular points. The connection factors are expressed as quotients of Wronskians of the involved solutions. Asymptotic expansions are used in the computation of those Wronskians. The feasibility of the method is shown in an example, namely, the Schroedinger equation with a quasi-exactly-solvable potential

On reducing the Heun equation to the hypergeometric equation

The reductions of the Heun equation to the hypergeometric equation by polynomial transformations of its independent variable are enumerated and classified. Heun-to-hypergeometric reductions are similar to classical hypergeometric identities, but the conditions for the existence of a reduction involve features of the Heun equation that the hypergeometric equation does not possess; namely, its cross-ratio and accessory parameters. The reductions include quadratic and cubic transformations, which may be performed only if the singular points of the Heun equation form a harmonic or an equianharmonic quadruple, respectively; and several higher-degree transformations. This result corrects and extends a theorem in a previous paper, which found only the quadratic transformations. [See K. Kuiken, "Heun's equation and the hypergeometric equation", SIAM Journal on Mathematical Analysis 10:3 (1979), 655-657.]Comment: 36 pages, a few additional misprints correcte

On asymptotic constants related to products of Bernoulli numbers and factorials

We discuss the asymptotic expansions of certain products of Bernoulli numbers and factorials, e.g., $$\prod_{\nu=1}^n |B_{2\nu}| \quad \text{and} \quad \prod_{\nu=1}^n (k \nu)!^{\nu^r} \quad \text{as} \quad n \to \infty$$ for integers $k \geq 1$ and $r \geq 0$. Our main interest is to determine exact expressions, in terms of known constants, for the asymptotic constants of these expansions and to show some relations among them.Comment: 22 pages, final revised versio

The Legal Profession in the Age of Digitalization:An outline of Three Potential Transformations in Legal Education

There is consensus within the legal profession that it needs to adapt to the on-going digitalisation of the legal market and the changing means of production of the legal commodity. This adaptation will also necessitate a transformation of legal education to assimilate the changes that the legal profession will undergo. The question is, however, how might the legal profession adapt to its digitalisation? In this article, we will describe three possible pathways that the legal profession might follow. These are based on synchronous sociological models of the dynamics of the legal profession and the legal market as well as diachronous sociological descriptions of the history of the legal profession over the past century. In order to concretise these hypotheses, we will focus on the legal profession in three similar countries between which there is some level of comparability: Denmark, Germany and the Netherlands. The three hypothetical pathways are understood to be non-mutually exclusive. We will then answer our core question: how must legal education be transformed to take into consideration the digitalisation of the legal profession? To answer this question, we will describe three possible transformations in legal education that would consider the pathways that the legal profession might pursue to adapt to the digitalisation of its market and the production of its commodity