Starting from a second-order Fuchsian differential equation having five regular singular points, an equation obeyed by a function proportional to the first derivative of the solution of the Heun equation, we construct several expansions of the solutions of the general Heun equation in terms of Appell generalized hypergeometric functions of two variables of the first kind. Several cases when the expansions reduce to those written in terms of simpler mathematical functions such as the incomplete Beta function or the Gauss hypergeometric function are identified. The conditions for deriving finite-sum solutions via termination of the series are discussed. In general, the coefficients of the expansions obey four-term recurrence relations; however, there exist certain choices of parameters for which the recurrence relations involve only two terms, though not necessarily successive. For such cases, the coefficients of the expansions are explicitly calculated and the general solution of the Heun equation is constructed in terms of the Gauss hypergeometric functions.

We consider particle systems (also known as point processes) on the line and in the plane and are particularly interested in “hole” events, when there are no particles in a large disk (or some other domain). We survey the extensive work on hole probabilities and the related large deviation principles (LDP), which has been undertaken mostly in the last two decades. We mainly focus on the recent applications of LDP-inspired techniques to the study of hole probabilities and the determination of the most likely configurations of particles that have large holes. As an application of this approach, we illustrate how one can confirm some of the predictions of Jancovici, Lebowitz, and Manificat for large fluctuation in the number of points for the (two-dimensional) $$\beta $$ -Ginibre ensembles. We also discuss some possible directions for future investigations.

We introduce and investigate a class of complex semi-infinite banded Toeplitz matrices satisfying the condition that the spectra of their principal submatrices accumulate onto a real interval when the size of the submatrix grows to ∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\infty $$\\end{document}. We prove that a banded Toeplitz matrix belongs to this class if and only if its symbol has real values on a Jordan curve located in C\\{0}\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\mathbb {C}}}{\\setminus }\\{0\\}$$\\end{document}. Surprisingly, it turns out that, if such a Jordan curve is present, the spectra of all the principal submatrices have to be real. The latter claim is also proved for matrices given by a more general symbol. The special role of the Jordan curve is further demonstrated by a new formula for the limiting density of the asymptotic eigenvalue distribution for banded Toeplitz matrices from the studied class. Certain connections between the problem under investigation, Jacobi operators, and the Hamburger moment problem are also discussed. The main results are illustrated by several concrete examples; some of them allow an explicit analytic treatment, while some are only treated numerically.

Trigonometric formulas are derived for certain families of associated Legendre functions of fractional degree and order, for use in approximation theory. These functions are algebraic, and when viewed as Gauss hypergeometric functions, belong to types classified by Schwarz, with dihedral, tetrahedral, or octahedral monodromy. The dihedral Legendre functions are expressed in terms of Jacobi polynomials. For the last two monodromy types, an underlying ‘octahedral’ polynomial, indexed by the degree and order and having a nonclassical kind of orthogonality, is identified, and recurrences for it are worked out. It is a (generalized) Heun polynomial, not a hypergeometric one. For each of these families of algebraic associated Legendre functions, a representation of the rank-2 Lie algebra mathfrak {so}(5,mathbb {C}) is generated by the ladder operators that shift the degree and order of the corresponding solid harmonics. All such representations of mathfrak {so}(5,mathbb {C}) are shown to have a common value for each of its two Casimir invariants. The Dirac singleton representations of mathfrak {so}(3,2) are included.

It is shown that the difference equation 1 $$\begin{aligned} ({\varDelta }f(z))^2=A(z)(f(z)f(z+1)-B(z)), \end{aligned}$$ where A(z) and B(z) are meromorphic functions, possesses a continuous limit to the differential equation 2 $$\begin{aligned} (w')^2=A(z)(w^2-1), \end{aligned}$$ which extends to solutions in certain cases. In addition, if (1) possesses two distinct transcendental meromorphic solutions, it is shown that these solutions satisfy an algebraic relation, and that their growth behaviors are almost the same in the sense of Nevanlinna under some conditions. Examples are given to discuss the sharpness of the results obtained. These properties are counterparts of the corresponding results on the algebraic differential equation (2).

We obtain several estimates for the $$L^p$$ operator norms of the Bergman and Cauchy–Szego projections over the the Siegel upper half-space. As a by-product, we also determine the precise value of the $$L^p$$ operator norm of a family of integral operators over the Siegel upper half-space.

We compute the expected Riesz energy of random points on flat tori drawn from certain translation invariant determinantal processes and determine the process in the family providing the optimal asymptotic expected Riesz energy.

We consider general linear approximation spaces $$X^b_q$$ based on a quasi-Banach space X, and we analyze the degree of compactness of the embedding $$X^b_q \hookrightarrow X$$ . Applications are given to periodic Besov spaces on the d-torus, including spaces of generalized and logarithmic smoothness. In particular, we obtain the exact asymptotic behavior of approximation and entropy numbers of embeddings of such Besov spaces in Lebesgue spaces and in Besov spaces of logarithmic smoothness.

This paper investigates the zero locus of a sequence of polynomials generated by a bivariate rational function with a denominator of the form $$G(z,t)=P(t)+zt^{r}$$ , where the zeros of P are positive and real. We show that every member of a family of such generating functions—parametrized by the degree of P and r—gives rise to a sequence of polynomials $$\{H_{m}(z)\}_{m=0}^{\infty }$$ , whose terms are eventually hyperbolic. We also identify the real interval $$I \subset \mathbb {R}^+$$ in which the collection of zeros $$\cup _{m \gg 1} \mathcal {Z}(H_m(z))$$ forms a dense set.

Linear second-order differential equations having a large real parameter and turning point in the complex plane are considered. Classical asymptotic expansions for solutions involve the Airy function and its derivative, along with two infinite series, the coefficients of which are usually difficult to compute. By considering the series as asymptotic expansions for two explicitly defined analytic functions, Cauchy’s integral formula is employed to compute the coefficient functions to a high order of accuracy. The method employs a certain exponential form of Liouville–Green expansions for solutions of the differential equation, as well as for the Airy function. We illustrate the use of the method with the high accuracy computation of Airy-type expansions of Bessel functions of complex argument.