Related papers: Evaluations of some Toeplitz-type determinants

In this paper, we investigate the following elliptic system with Sobolev critical growth $-\Delta u+P(|y'|,y'')u=u^{2^*-1}+\frac{\beta}{2} u^{\frac{2^*}{2}-1}v^{\frac{2^*}{2}},\ y\in R^N$, $-\Delta v+Q(|y'|,y'')v=v^{2^*-1}+\frac{\beta}{2}…

We show that certain weighted average of the alpha-determinant of a $kn$ by $kn$ matrix of the form $A\otimes1_{1,k}$, the Kronecker product of a $kn$ by $n$ matrix $A$ and $1$ by $k$ all one matrix $1_{1,k}$, over permutations of $kn$…

We prove a conjectured determinantal inequality: \frac{\det J}{\prod_{i=1}^nJ_{ii}}\le 2(1-\frac{1}{n-1})^{n-1}, where $J$ is a real $n\times n$ ($n\ge 2$) diagonally balanced symmetric matrix.

For the family $P:=x^n+a_1x^{n-1}+\cdots +a_n$ of complex polynomials in the variable $x$ we study its {\em discriminant} $R:=$Res$(P,P',x)$, $R\in \mathbb{C}[a]$, $a=(a_1,\ldots ,a_n)$. When $R$ is regarded as a polynomial in $a_k$, one…

Various new identities, recurrence relations, integral representations, connection and explicit formulas are established for the Bernoulli, Euler numbers and the values of Riemann's zeta function. To do this, we explore properties of some…

Let $B_{n}$ denote the Bernoulli numbers, and $S(n,k)$ denote the Stirling numbers of the second kind. We prove the following identity $$ B_{m+n}=\sum_{\substack{0\leq k \leq n \\ 0\leq l \leq m}}\frac{(-1)^{k+l}\,k!\, l!\,…

In this paper, we establish Kronecker limit type formulas for the generalized Mordell--Tornheim zeta function $\Theta(r,s,t,x)$ as a function of the third variable, in terms of Riemann-zeta and Gamma values. We also give series evaluations…

Given complex numbers $m_1,l_1$ and positive integers $m_2,l_2$, such that $m_1+m_2=l_1+l_2$, we define $l_2$-dimensional hypergeometric integrals $I_{a,b}(z;m_1,m_2,l_1,l_2)$, $a,b=0,...,\min(m_2,l_2)$, depending on a complex parameter…

For a sequence $\mathbf w = \{w_j\}_{j = 2}^\infty$ of positive real numbers, consider the positive semi-definite kernel $\kappa_{\mathbf w}(s, u) = \sum_{j = 2}^\infty w_j j^{-s - \overline{u}}$ defined on some right-half plane $\mathbb…

The $k$-dimensional generalized Euler function $\varphi_k(n)$ is defined to be the number of ordered $k$-tuples $(a_1,a_2,\ldots, a_k) \in \mathbb{N}^k$ with $1\leq a_1,a_2,\ldots, a_k \leq n$ such that both the product $a_1a_2\cdots a_k$…

Let $\Bigl\langle\matrix{n\cr k}\Bigr\rangle$, $\Bigl\langle\matrix{B_n\cr k}\Bigr\rangle$, and $\Bigl\langle\matrix{D_n\cr k}\Bigr\rangle$ be the Eulerian numbers in the types A, B, and D, respectively -- that is, the number of…

In this work we prove a new combinatorial identity and applying it we establish many finite harmonic sum identities. Among many others, we prove that \begin{equation*}…

The principal minors of the Toeplitz matrix $\left( x_{i-j+1}\right)_{1\le i,j,\le n}$, where $x_0=1, x_k=0$ if $k\le -1$, directly determine an involution of the polynomial ring $R[x_1, ... ,x_n]$ over any commutative ring $R$.

We prove denominator identities for the periplectic Lie superalgebra $\mathfrak{p}(n)$, thereby completing the problem of finding denominator identities for all simple classical finite-dimensional Lie superalgebras.

A classical question for a Toeplitz matrix with given symbol is to compute asymptotics for the determinants of its reductions to finite rank. One can also consider how those asymptotics are affected when shifting an initial set of rows and…

We give alternative proofs to (block case versions of) some formulas for Toeplitz and Fredholm determinants established recently by the authors of the title. Our proof of the Borodin-Okounkov formula is very short and direct. The proof of…

Given a finite poset $P$, its zeta matrix $\mathbf Z$ encode fundamental incidence-theoretic information about the order structure. In this paper we introduce and study the \emph{order-complement matrix} $\overline{\mathbf Z} = \mathbf J -…

In this note we deduce well known modular identities for Jacobi theta functions using the spectral representations associated with the real valued Brownian motion taking values on $[-1,+1]$. We consider two cases: (i) reflection at $-1$ and…

We derive two new identities involving the Bernoulli numbers, the Euler numbers, and the Stirling numbers of the first kind using analytic continuation of a well known identity for the Stirling numbers of the first kind.

In this paper, we consider the following problem $$ -\Delta u -\zeta \frac{u}{|x|^{2}} = \sum_{i=1}^{k} \left( \int_{\mathbb{R}^{N}} \frac{|u|^{2^{*}_{\alpha_{i}}}}{|x-y|^{\alpha_{i}}} \mathrm{d}y \right) |u|^{2^{*}_{\alpha_{i}}-2}u +…