Related papers: Jacobi matrices generated by ratios of hypergeomet…
The hypergeometric zeta function is defined in terms of the zeros of the Kummer function M(a, a + b; z). It is established that this function is an entire function of order 1. The classical factorization theorem of Hadamard gives an…
Classical hypergeometric functions are well-known to play an important role in arithmetic algebraic geometry. These functions offer solutions to ordinary differential equations, and special cases of such solutions are periods of…
The two most commonly used hypergeometric functions are the confluent hypergeometric function and the Gauss hypergeometric function. We review the available techniques for accurate, fast, and reliable computation of these two hypergeometric…
Recently Bringmann, Raum and Richter generalised the definition of Jacobi forms and Skoruppa's skew-holomorphic Jacobi forms by intertwining with harmonic Maass forms. We prove the isomorphism of the Kohnen's plus space analogue of harmonic…
Hypergeometric class equations are given by second order differential operators in one variable whose coefficient at the second derivative is a polynomial of degree $\leq2$, at the first derivative of degree $\leq1$ and the free term is a…
By applying the Euler--Rayleigh methods to a specific representation of the Jacobi polynomials as hypergeometric functions, we obtain new bounds for their largest zeros. In particular, we derive upper and lower bound for…
Two-term asymptotic formulae for the probability distribution functions for the smallest eigenvalue of the Jacobi $ \beta $-Ensembles are derived for matrices of large size in the r\'egime where $ \beta > 0 $ is arbitrary and one of the…
In 1972 H. L. Montgomery announced a remarkable connection between the distribution of the zeros of the Riemann zeta-function and the distribution of eigenvalues of large random Hermitian matrices. Since then a number of startling…
We present a new family of solutions for the Jackiw-Teitelboim model of two-dimensional gravity with a negative cosmological constant. Here, a metric of constant Ricci scalar curvature is constructed, and explicit linearly independent…
We develop a theory of Jacobi polynomials for parabolic subgroups of finite reflection groups that specializes to the cases studied by Heckman and Opdam in which the whole group and the trivial group are considered. For the intermediate…
We introduce hypergeometric functions related to projective Schur functions $Q_{\lambda}$ and describe their properties. Linear equations, integral representations and Pfaffian representations are obtained. These hypergeometric functions…
We construct a semiclassical expression for the Husimi function of autonomous systems in one degree of freedom, by smoothing with a Gaussian function an expression that captures the essential features of the Wigner function in the…
It is shown that the properties of the Gauss decomposition of quantum groups and the known Jimbo homomorphism permit us to realize these groups as subalgebras of well defined algebras constructed from generators of the corresponding…
We obtain bounds for the spectrum and for the total width of the spectral gaps for Jacobi matrices on $\ell^2(\Z)$ of the form $(H\psi)_n= a_{n-1}\psi_{n-1}+b_n\psi_n+a_n\psi_{n+1}$, where $a_n=a_{n+q}$ and $b_n=b_{n+q}$ are periodic…
The hypergeometric distribution is a popular distribution, whose properties have been extensively investigated. Generating functions of this distribution, such as the probability-generating function, the moment-generating function, and the…
We present a fast Jacobi-like algorithm for computing the eigenvalues, and optionally the eigenvectors, of a real normal matrix. The method gains a computational advantage by using Paardekooper's method for skew-symmetric matrices The…
In the preprint of 1993 the author formulated some conjectures on monotonicity of ratios for exponential series remainders. They are equivalent to conjectures on monotonicity of a ratio of Kummer hypergeometric functions and presumably not…
We provide a complete classification of the algebraicity of (generalized) hypergeometric functions with no restriction on the set of their parameters. Our characterization relies on the interlacing criteria of Christol (1987) and…
In this paper, we apply the ratio conjecture of $L$-functions to derive the lower order terms of the $1$-level density of the low-lying zeros of a family quadratic Hecke $L$-functions in the Gaussian field. Up to the first lower order term,…
We consider positive Jacobi matrices $J$ with compact inverses and consequently with purely discrete spectra. A number of properties of the corresponding sequence of orthogonal polynomials is studied including the convergence of their…