Related papers: An ansatz for the asymptotics of hypergeometric mu…
We propose a new method for obtaining complete asymptotic expansions in a systematic manner, which is suitable for counting sequences of various graph families in dense regime. The core idea is to encode the two-dimensional array of…
We develop a theoretical study of non-terminating hypergeometric summations with one free parameter. Composing various methods in complex and asymptotic analysis, geometry and arithmetic of certain transcendental curves and rational…
Analytic combinatorics studies asymptotic properties of families of combinatorial objects using complex analysis on their generating functions. In their reference book on the subject, Flajolet and Sedgewick describe a general approach that…
We construct several expansions of the solutions of the confluent Heun equation in terms of the incomplete Beta functions and the Appell generalized hypergeometric functions of two variables of the fist kind. The coefficients of different…
Hyperasymptotics is an analytical method that incorporates exponentially small contributions into asymptotic approximations, thereby expanding their domain of validity, improving accuracy, and providing deeper insight into the underlying…
The aim of this work is to analyze general infinite sums containing modified Bessel functions of the second kind. In particular we present a method for the construction of a proper asymptotic expansion for such series valid when one of the…
We conclude our work [arXiv:2403.07628, arXiv:2503.12644] on asymptotic expansions at the soft edge for the classical $n$-dimensional Gaussian and Laguerre ensembles, now studying the gap-probability generating functions. We show that the…
Let $\{a_\rr : \rr \in (\Z^+)^d \}$ be a $d$-dimensional array of numbers, for which the generating function $F(\zz) := \sum_\rr a_\rr \zz^\rr$ is meromorphic in a neighborhood of the origin. For example, $F$ may be a rational multivariate…
Applying the approach based on the equation for the derivative, we construct several expansions of the solutions of the general Heun equation in terms of the incomplete Beta functions. Several expansions in terms of the Appell generalized…
We study the notion of an asymptotically automatic sequence, which generalises the notion of an automatic sequence. While $k$-automatic sequences are characterised by finiteness of $k$-kernels, the $k$-kernels of asymptotically…
Hypergraphs are structures that can be decomposed or described; in other words they are recursively countable. Here, we get exact and asymptotic enumeration results on hypergraphs by means of exponential generating functions. The number of…
We begin by defining general hypergeometric functions over finite fields and obtaining a finite field analogue of a classical symmetry in their complex counterparts. We give a geometric proof for the symmetry by constructing isomorphisms…
We show that in the particular case when a characteristic exponent of the singularity at infinity is zero the solution of the general Heun equation can be expanded in terms of the incomplete Beta functions. By means of termination of the…
We show the existence of a complete, strictly locally convex hypersurface within $\mathbb{H}^{n+1}$ that adheres to a curvature equation applicable to a broad range of curvature functions. This hypersurface possesses a prescribed asymptotic…
The confluent hypergeometric point process represents a universality class which arises in a variety of different but related areas. It particularly describes the local statistics of eigenvalues in the bulk of spectrum near a Fisher-Hartwig…
We study integral representations of the Gevrey series solutions of irregular hypergeometric systems under certain assumptions. We prove that, for such systems, any Gevrey series solution, along a coordinate hyperplane of its singular…
Conjecturing and theorem proving are activities at the center of mathematical practice and are difficult to separate. In this paper, we propose a framework for completing incomplete conjectures and incomplete proofs. The framework can turn…
We explain a discontinuous drop in the exponential growth rate for certain multivariate generating functions at a critical parameter value, in even dimensions d at least 4. This result depends on computations in the homology of the…
The number of independent sets in regular bipartite expander graphs can be efficiently approximated by expressing it as the partition function of a suitable polymer model and truncating its cluster expansion. While this approach has been…
We find an asymptotic enumeration formula for the number of simple $r$-uniform hypergraphs with a given degree sequence, when the number of edges is sufficiently large. The formula is given in terms of the solution of a system of equations.…