Related papers: On Algebraic Expressions of Sigma Functions for (n…

This survey is devoted to the classical and modern problems related to the entire function ${\sigma({\bf u};\lambda)}$, defined by a family of nonsingular algebraic curves of genus $2$, where ${\bf u} = (u_1,u_3)$ and $\lambda = (\lambda_4,…

In this paper, we give one possible definition for functions of several variables applied to endomorphisms of finite dimensional C-vector spaces. This definition is consistent with the usual notion of a function of a square matrix. Some…

Using a general $q$-series expansion, we derive some nontrivial $q$-formulas involving many infinite products. A multitude of Hecke--type series identities are derived. Some general formulas for sums of any number of squares are given. A…

In this paper the notion of an abstract square function (estimate) is introduced as an operator X to gamma (H; Y), where X, Y are Banach spaces, H is a Hilbert space, and gamma(H; Y) is the space of gamma-radonifying operators. By the…

The asymptotic expansion of $n$-dimensional cyclic integrals was expressed as a series of functionals acting on the symmetric function involved in the cyclic integral. In this article, we give an explicit formula for the action of these…

An identity is derived expressing Schur functions as sums over products of pairs of Schur $Q$-functions, generalizing previously known special cases. This is shown to follow from their representations as vacuum expectation values (VEV's) of…

A recent generalization of the "Kleinian sigma function" involves the choice of a point $P$ of a Riemann surface $X$, namely a "pointed curve" $(X, P)$. This paper concludes our explicit calculation of the sigma function for curves cyclic…

We introduce a generalization of the Stirling numbers via symmetric functions involving two weight functions. The resulting extension unifies previously known Stirling-type sequences with known symmetric function forms, as well as other…

Let $\Sigma$ be a surface with negative Euler characteristic, genus at least one and at most one boundary component. We prove that the skein algebra of $\Sigma$ over the field of rational functions can be algebraically generated by a finite…

A representation for the Riemann zeta function valid for arbitrary complex $s=\sigma+it$ is $\zeta(s)=\sum_{n=0}^\infty A(n,s)$, where \[A(n,s)=\frac{2^{-n-1}}{1-2^{1-s}} \sum_{k=0}^n \left(\!\begin{array}{c}n\\k\end{array}\!\right)…

For any arbitrary algebraic curve, we define an infinite sequence of invariants. We study their properties, in particular their variation under a variation of the curve, and their modular properties. We also study their limits when the…

Specializations of Schur functions are exploited to define and evaluate the Schur functions s_\lambda[\alpha X] and plethysms s_\lambda[\alpha s_\nu(X))] for any \alpha - integer, real or complex. Plethysms are then used to define pairs of…

We describe algebraic curves $ X : F(x, y) = 0 $ defined over $\overline{\mathbb{Q}}$ that satisfy the following property: there exist a number field $k$ and an infinite set $S \subset k$ such that, for every $y \in S$, the roots of the…

For $k\ge 1$, the homogeneous symmetric functions $G(k,m)$ of degree $m$ defined by $\sum_{m\ge 0} G(k,m) z^m=\prod_{i\ge 1} \big(1+x_iz+x^2_iz^2+\cdots+x^{k-1}_iz^{k-1}\big)$ are called \emph{Petrie symmetric functions}. As derived by…

We establish the quaternionic weighted zeta function of a graph and its Study determinant expressions. For a graph with quaternionic weights on arcs, we define a zeta function by using an infinite product which is regarded as the Euler…

We consider new series expansions for variants of the so-termed ordinary geometric square series generating functions originally defined in the recent article titled "Square Series Generating Function Transformations" (arXiv: 1609.02803).…

Let $d,n$ be positive integers and $S$ be an arbitrary set of positive integers. We say that $d$ is an $S$-divisor of $n$ if $d|n$ and gcd $(d,n/d)\in S$. Consider the $S$-convolution of arithmetical functions given by (1.1), where the sum…

By treating the multiple argument identity of the logarithm of the Gamma function as a functional equation, we obtain a curious infinite product representation of the $sinc$ function in terms of the cotangent function. This result is…

The Schur $P$-, $Q$-multiple zeta functions were defined by Nakasuji and Takeda inspired by the tableau representation of Schur $P$-, $Q$-functions. While a product of two Schur $P$-functions expands as a linear combination of Schur…

Let X be a homogeneous tree of degree q+1 (for q between 2 and infinity) and let f be a complex function on X times X for which f(x,y) only depend on the distance between x and y in X. Our main result gives a necessary and sufficient…