Related papers: Note on the Tau Function

We work with differential expressions of the form \begin{align} \tau_{2n+1} y &=(-1)^ni \{(q_{0}y^{(n+1)})^{(n)}+(q_{0}y^{(n)})^{(n+1)}\}+ \sum\limits_{k=0}^{n}(-1)^{n+k}(p^{(k)}_ky^{(n-k)})^{(n-k)} \\…

In this paper, we generalize the partial fraction decomposition which is fundamental in the theory of multiple zeta values, and prove a relation between Tornheim's double zeta functions of three complex variables. As applications, we give…

In this paper we develop a technique of computation of correlation functions in theories with action being cubic or higher degree form in terms of discriminants of corresponding tensors. These are analogues of formula $\int \exp…

The Riemann-zeta function regularization procedure has been studied intensively as a good method in the computation of the determinant for pseudo-diferential operator. In this paper we propose a different approach for the computation of the…

This short note investigates a number of index integrals of products of the Lommel functions $s_{\mu,\nu}(a)$ and uncovers an integral relationship. between this function and the Tchebyshev polynomials $T_{2n}(x)$.

Let $T$ be a square matrix with a real spectrum, and let $f$ be an analytic function. The problem of the approximate calculation of $f(T)$ is discussed. Applying the Schur triangular decomposition and the reordering, one can assume that $T$…

We present a unified description of birational representation of Weyl groups associated with T-shaped Dynkin diagrams, by using a particular configuration of points in the projective plane. A geometric formulation of tau-functions is given…

Let $f$ be a holomorphic or Maass cusp forms for $ \rm SL_2(\mathbb{Z})$ with normalized Fourier coefficients $\lambda_f(n)$ and \bna r_{\ell}(n)=\#\left\{(n_1,\cdots,n_{\ell})\in \mathbb{Z}^2:n_1^2+\cdots+n_{\ell}^2=n\right\}. \ena Let…

We give a valuative criterion for when a smooth algebraic stack with a separated good moduli space is the quotient of a separated Deligne-Mumford stack by a torus. For doing so, we introduce a new class of morphisms, the so-called effective…

It is shown that if a function defined on the segment [-1,1] has sufficiently good approximation by partial sums of the Legendre polynomial expansion, then, given the function's Fourier coefficients $c_n$ for some subset of $n\in[n_1,n_2]$,…

Certain objects of conformal field theory, for example partition functions on the rectangle and the torus, and one-point functions on the torus, are either invariant or transform simply under the modular group, properties which should be…

We use a q-series identity by Ramanujan to give a combinatorial interpretation of Ramanujan's tau function which involves t-cores and a new class of partitions which we call (m,k)-capsids. The same method can be applied in conjunction with…

Conical functions appear in a large number of applications in physics and engineering. In this paper we describe an extension of our module CONICAL for the computation of conical functions. Specifically, the module includes now a routine…

An infinite class of relations between modular forms is constructed that generalizes evaluations of the Dirichlet beta function at odd positive integers. The work is motivated by a base case appearing in Ramanujan's Notebooks and a parallel…

We give a brief introduction to Drinfeld modular forms, concentrating on the many equivalent constructions of the form h of weight q+1 and type 1, to which we contribute some new characterizations involving Moore determinants, and an…

We fix motivic data $(K/F, E)$ consisting of a Galois extension $K/F$ of characteristic $p$ global fields with arbitrary abelian Galois group $G$ and an ableian $t$-module $E$, defined over a certain Dedekind subring of $F$. For this data,…

This note presents selected values of definite integrals whose integrand contains a power of the Dedekind function having imaginary argument.

In this paper we study sums of Dirichlet series whose coefficients are terms of the Thue-Morse sequence and variations thereof. We find closed-form expressions for such sums in terms of known constants and functions including the Riemann…

We study (support) $\tau$-tilting modules over the trivial extensions of finite dimensional algebras. More precisely, we construct two classes of (support)$\tau$-tilting modules in terms of the adjoint functors which extend and generalize…

Let $\tau$ denote the divisor function, and $f$ be any multiplicative function that satisfies some mild hypotheses. We establish the asymptotic formula or non-trivial upper bound for the shifted convolution sum $\sum_{n \leq…