Counting derangements and Nash equilibria

The maximal number of totally mixed Nash equilibria in games of several players equals the number of block derangements, as proved by McKelvey and McLennan.On the other hand, counting the derangements is a well studied problem. The numbers are identified as linearization coefficients for Laguerre polynomials. MacMahon derived a generating function for them as an application of his master theorem. This article relates the algebraic, combinatorial and game-theoretic problems that were not connected before. New recurrence relations, hypergeometric formulas and asymptotics for the derangement counts are derived. An upper bound for the total number of all Nash equilibria is given.Comment: 22 pages, 1 table; Theorem 3.3 adde

Tight bounds on the mutual coherence of sensing matrices for Wigner D-functions on regular grids

Many practical sampling patterns for function approximation on the rotation group utilizes regular samples on the parameter axes. In this paper, we relate the mutual coherence analysis for sensing matrices that correspond to a class of regular patterns to angular momentum analysis in quantum mechanics and provide simple lower bounds for it. The products of Wigner d-functions, which appear in coherence analysis, arise in angular momentum analysis in quantum mechanics. We first represent the product as a linear combination of a single Wigner d-function and angular momentum coefficients, otherwise known as the Wigner 3j symbols. Using combinatorial identities, we show that under certain conditions on the bandwidth and number of samples, the inner product of the columns of the sensing matrix at zero orders, which is equal to the inner product of two Legendre polynomials, dominates the mutual coherence term and fixes a lower bound for it. In other words, for a class of regular sampling patterns, we provide a lower bound for the inner product of the columns of the sensing matrix that can be analytically computed. We verify numerically our theoretical results and show that the lower bound for the mutual coherence is larger than Welch bound. Besides, we provide algorithms that can achieve the lower bound for spherical harmonics

The importance of the Selberg integral

It has been remarked that a fair measure of the impact of Atle Selberg's work is the number of mathematical terms which bear his name. One of these is the Selberg integral, an n-dimensional generalization of the Euler beta integral. We trace its sudden rise to prominence, initiated by a question to Selberg from Enrico Bombieri, more than thirty years after publication. In quick succession the Selberg integral was used to prove an outstanding conjecture in random matrix theory, and cases of the Macdonald conjectures. It further initiated the study of q-analogues, which in turn enriched the Macdonald conjectures. We review these developments and proceed to exhibit the sustained prominence of the Selberg integral, evidenced by its central role in random matrix theory, Calogero-Sutherland quantum many body systems, Knizhnik-Zamolodchikov equations, and multivariable orthogonal polynomial theory.Comment: 43 page

Current Trends in Symmetric Polynomials with their Applications

This Special Issue presents research papers on various topics within many different branches of mathematics, applied mathematics, and mathematical physics. Each paper presents mathematical theories, methods, and their application based on current and recently developed symmetric polynomials. Also, each one aims to provide the full understanding of current research problems, theories, and applications on the chosen topics and includes the most recent advances made in the area of symmetric functions and polynomials

A simple algorithm for expanding a power series as a continued fraction

I present and discuss an extremely simple algorithm for expanding a formal power series as a continued fraction. This algorithm, which goes back to Euler (1746) and Viscovatov (1805), deserves to be better known. I also discuss the connection of this algorithm with the work of Gauss (1812), Stieltjes (1889), Rogers (1907) and Ramanujan, and a combinatorial interpretation based on the work of Flajolet (1980).Comment: LaTeX2e, 48 pages. Version 2 contains a few additional historical remarks, and adds a new Remark 4 at the end of Section 10. To be published in Expositiones Mathematica

Symbolic Evaluations Inspired by Ramanujan's Series for 1/pi

We introduce classes of Ramanujan-like series for $\frac{1}{\pi}$, by devising methods for evaluating harmonic sums involving squared central binomial coefficients, as in the family of Ramanujan-type series indicated below, letting $H_{n} = 1 + \frac{1}{2} + \cdots + \frac{1}{n}$ denote the $n^{\text{th}}$ harmonic number: \begin{align*} & \sum _{n=1}^{\infty } \frac{\binom{2 n}{n}^2 H_n}{16^n(2 n - 1)} = \frac{ 8 \ln (2) - 4 }{\pi}, \\ & \sum _{n=1}^{\infty } \frac{\binom{2 n}{n}^2 H_n}{16^n(2 n-3)} = \frac{120 \ln (2)-68 }{27 \pi}, \\ & \sum _{n = 1}^{\infty } \frac{ \binom{2 n}{n}^2 H_n}{ 16^{ n} (2 n - 5)} = \frac{10680 \ln (2) -6508}{3375 \pi }, \\ & \cdots \end{align*} In this direction, our main technique is based on the evaluation of a parameter derivative of a beta-type integral, but we also show how new integration results involving complete elliptic integrals may be used to evaluate Ramanujan-like series for $\frac{1}{\pi}$ containing harmonic numbers. We present a generalization of the recently discovered harmonic summation formula $$\sum_{n=1}^{\infty} \binom{2n}{n}^{2} \frac{H_{n}}{32^{n}} = \frac{\Gamma^{2} \left( \frac{1}{4} \right)}{4 \sqrt{\pi}} \left( 1 - \frac{4 \ln(2)}{\pi} \right)$$ through creative applications of an integration method that we had previously introduced. We provide explicit closed-form expressions for natural variants of the above series. At the time of our research being conducted, up-to-date versions of Computer Algebra Systems such as Mathematica and Maple could not evaluate our introduced series, such as $$\sum _{n=1}^{\infty } \frac{ \binom{2 n}{n}^2 H_n}{32^n (n + 1)} = 8-\frac{2 \Gamma^2 \left(\frac{1} {4}\right)}{\pi ^{3/2}}-\frac{4 \pi ^{3/2}+16 \sqrt{\pi } \ln (2)}{\Gamma^2 \left(\frac{1}{4}\right)}.$$ We also introduce a class of harmonic summations for Catalan's constant $G$ and $\frac{1}{\pi}$ such as the series $$\sum _{n=1}^{\infty } \frac{ \binom{2 n}{n}^2 H_n}{ 16^{n} (n+1)^2} = 16+\frac{32 G-64 \ln (2)}{\pi }-16 \ln (2),$$ which we prove through a variation of our previous integration method for constructing $\frac{1}{\pi}$ series. We also present a new integration method for evaluating infinite series involving alternating harmonic numbers, and we apply a Fourier--Legendre-based technique recently introduced by Campbell et al., to prove new rational double hypergeometric series formulas for expressions involving $\frac{1}{\pi^2}$, especially the constant $\frac{\zeta(3)}{\pi^2}$, which is of number-theoretic interest