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We derive a recursive formula for certain relative Gromov-Witten invariants with maximal tangency condition via the Witten-Dijkgraaf-Verlinde-Verlinde equation. For certain relative pairs, we get explicit formulae of invariants using the…
We describe the solutions to a family of rotationally symmetric second order partial differential equations in the complex plane that arises from a four-dimensional complex Lie algebra whose spanning set generates the algebra from which…
The Collatz conjecture is explored using polynomials based on a binary numeral system. It is shown that the degree of the polynomials, on average, decreases after a finite number of steps of the Collatz operation, which provides a weak…
A proof of Lagrange's and Jacobi's four-square theorem due to Hurwitz utilizes orders in a quaternion algebra over the rationals. Seeking a generalization of this technique to orders over number fields, we identify two key components: an…
Exceptional extensions of a class of Laurent biorthogonal polynomials (the so-called Hendriksen-van Rossum polynomials) have been presented by the authors recently. This is achieved through Darboux transformations of generalized eigenvalue…
A recursion due to Kook expresses the Laplacian eigenvalues of a matroid M in terms of the eigenvalues of its deletion M-e and contraction M/e by a fixed element e, and an error term. We show that this error term is given simply by the…
Polynomial multiplication is a fundamental problem in symbolic computation. There are efficient methods for the multiplication of two univariate polynomials. However, there is rarely efficiently nontrivial method for the multiplication of…
Based on a reduction processing, we rewrite a hypergeometric term as the sum of the difference of a hypergeometric term and a reduced hypergeometric term (the reduced part, in short). We show that when the initial hypergeometric term has a…
We derive inversion formulas involving orthogonal polynomials which can be used to find coefficients of differential equations satisfied by certain generalizations of the classical orthogonal polynomials. As an example we consider special…
The theory of orthogonal polynomials on the unit circle is developed for a general class of weights leading to systems of recurrence relations and derivatives of the polynomials and their associated functions, and to functional-difference…
We describe a method to compute Hurwitz-Hodge integrals.
We give an explicit formula for the Hankel transform of a regular sequence in terms of the coefficients of the associated orthogonal polynomials and the sequence itself. We apply this formula to some sequences of combinatorial interest,…
In this paper we find a decomposition of higher order Lipschitz functions into the traces of a polymonogenic function and solve a related Riemann-Hilbert problem. Our approach lies in using a cliffordian Cauchy-type operator, which behaves…
Let $G$ be a complex reductive group, $\theta \colon G \to G$ an involution, and $K = G^\theta$. In arXiv:1206.5547, W. Schmid and the second named author proposed a program to study unitary representations of the corresponding real form…
In this paper the authors show how to use Riemann-Hilbert techniques to prove various results, some old, some new, in the theory of Toeplitz operators and orthogonal polynomials on the unit circle (OPUC's). There are four main results: the…
A proof for a conjecture by Shadrin and Zvonkine, relating the entries of a matrix arising in the study of Hurwitz numbers to a certain sequence of rational numbers, is given. The main tools used are iteration matrices of formal power…
In Lett. Math. Phys. 114, 54 (2024) and 115, 70 (2025), the author introduces what is presented as a novel method for determining whether a sequence of orthogonal polynomials is "classical", based solely on its initial recurrence…
Various recurrence relations between formal orthogonal polynomials can be used to derive Lanczos-type algorithms. In this paper, we consider recurrence relation $A_{12}$ for the choice $U_i(x)=P_i(x)$, where $U_i$ is an auxiliary family of…
If $f\!:\![a,b]\to\R$ such that $f^{(n)}$ is integrable then integration by parts gives the formula \begin{align*} &\intab f(x)\,dx = &\frac{(-1)^n}{n!}\sum_{k=0}^{n-1}(-1)^{n-k-1}\left[ \phi_n^{(n-k-1)}(a)f^{(k)}(a)-…
We determine the skein-valued Gromov-Witten partition function for a single toric Lagrangian brane in $\mathbb{C}^3$ or the resolved conifold. We first show geometrically they must satisfy a certain skein-theoretic recursion, and then solve…