Related papers: Arnol'd's limit and the Lagrange inversion

We give a survey of the Lagrange inversion formula, including different versions and proofs, with applications to combinatorial and formal power series identities.

As an application of Cauchy's Theorem we prove that $\int_0^1\arctan\left({\arctanh x-\arctan x\over \pi+\arctanh x-\arctan x}\right) {dx\over x}= {\pi\over 8}\log{\pi^2\over 8}$ answering a question first posted in Mathematics Stack…

A general integral expression to transform power series is applied to $\arcsin{x}$ and its positive integer powers. We concentrate on the first to the fourth powers and obtain infinite classes of new power series involving central binomial…

We present a simple inductive proof of the Lagrange Inversion Formula.

The goal of the paper is to present two simple proofs of the Lagrange Inversion Formula for formal power series. Both proofs are non-external in the sense that they use concepts that do not go beyond the scope of formal power series…

Leopold Vietoris and Guido Hoheisel showed how the existence of $\lim_{x\to 0}\frac{\sin x}{x}$ can be derived from the trigonometric addition formulas. In this article two new proofs for this result are given. In addition it is discussed…

We prove an analogue of the Lagrange Inversion Theorem for Dirichlet series. The proof is based on studying properties of Dirichlet convolution polynomials, which are analogues of convolution polynomials introduced by Knuth in [4].

We prove central limit theorem for linear eigenvalue statistics of orthogonally invariant ensembles of random matrices with one interval limiting spectrum. We consider ensembles with real analytic potentials and test functions with two…

We consider the integral $\int_0^\infty\left(\frac{\sin x}{x}\right)^n\;dx$ as a function of the positive integer $n$. We show that there exists an asymptotic series in $\frac{1}{n}$ and compute the first terms of this series together with…

If $f\in L^1({\mathbb R})$ it is proved that $\lim_{S\to\infty}\lVert f-f\ast D_S\rVert=0$, where $D_S(x)=\sin(Sx)/(\pi x)$ is the Dirichlet kernel and $\lVert f\rVert = \sup_{\alpha<\beta}|\int_{\alpha}^{\beta}f(x)\,dx|$ is the Alexiewicz…

Given a stationary first-order autoregressive process X_t (with lag-one correlation rho satisfying |rho|<1), we examine the Central Limit Theorem for (1/n)*ln |X_1...X_n| and compute variances to high precision. Given a nonstationary…

In this paper, we prove that for fixed $k\geq 1$, the Wilker type inequality {equation*} \frac{2}{k+2}(\frac{\sin x}{x}) ^{kp}+\frac{k}{k+2}(\frac{% \tan x}{x})^{p}>1 {equation*}% holds for $x\in (0,\pi /2) $ if and only if $p>0$ or $p\leq…

In this paper we prove that inf_{|z_k| => 1} max_{v=1,...,n^2} |sum_{k=1}^n z_k^v| = sqrt n+O(n^{0.2625+epsilon}). This improves on the bound O(sqrt (n log n)) of Erdos and Renyi. In the special case of $n+1$ being a prime we have…

We consider whether L = limsup_{n to infty} n ||T^{n+1}-T^n|| < infty implies that the operator T is power bounded. We show that this is so if L<1/e, but it does not necessarily hold if L=1/e. As part of our methods, we improve a result of…

We prove a Santal\'{o} and a reverse Santal\'{o} inequality for the polarity transform, which was recently re-discovered by Artstein-Avidan and Milman, in the class consisting of (even) log-concave functions attaining their maximal value 1…

Let $R$ be an associative ring with an identity and suppose that $a,b,c,d \in R$ satisfy $bdb = bac,dbd = acd$. If $ac$ has generalized Drazin ( respectively, pseudo Drazin, Drazin) inverse, we prove that $bd$ has generalized Drazin…

We prove a central limit theorem for the length of the longest subsequence of a random permutation which follows one of a class of repeating patterns. This class includes every fixed pattern of ups and downs having at least one of each,…

Watson proved Kirkman's hypothesis (partially solved by Cayley). Using Lagrange Inversion, we drastically shorten Watson's computations and generalize his results at the same time.

For martingales with a wide range of integrability, we will quantify the rate of convergence of the central limit theorem via Wasserstein distances of order $r$, $1\le r\le 3$. Our bounds are in terms of Lyapunov's coefficients and the…

Using appropriate power series evaluations, we determine all moments of arbitrary positive powers of the arcsine. As consequences we evaluate several doubly infinite classes of power series involving central binomial coefficients and…