Moments Of Polynomials Research Articles

Hankel determinants of linear combinations of moments of orthogonal polynomials, II

We present a formula that expresses the Hankel determinants of a linear combination of length d+1 of moments of orthogonal polynomials in terms of a dtimes d determinant of the orthogonal polynomials. This formula exists somehow hidden in the folklore of the theory of orthogonal polynomials but deserves to be better known, and be presented correctly and with full proof. We present four fundamentally different proofs, one that uses classical formulae from the theory of orthogonal polynomials, one that uses a vanishing argument and is due to Elouafi (J Math Anal Appl 431:1253–1274, 2015) (but given in an incomplete form there), one that is inspired by random matrix theory and is due to Brézin and Hikami (Commun Math Phys 214:111–135, 2000), and one that uses (Dodgson) condensation. We give two applications of the formula. In the first application, we explain how to compute such Hankel determinants in a singular case. The second application concerns the linear recurrence of such Hankel determinants for a certain class of moments that covers numerous classical combinatorial sequences, including Catalan numbers, Motzkin numbers, central binomial coefficients, central trinomial coefficients, central Delannoy numbers, Schröder numbers, Riordan numbers, and Fine numbers.

Chebyshev’s Problem of the Moments of Nonnegative Polynomials

Chebyshev’s Problem of the Moments of Nonnegative Polynomials

Explicit bounds for even moments of Bernstein’s polynomials

We prove explicit and optimal lower and upper bounds for even-order moments of Bernstein polynomials.

On a distinguished family of random variables and Painlevé equations

A family of random variables $\mathbf{X}(s)$, depending on a real parameter $s>-\frac{1}{2}$, appears in the asymptotics of the joint moments of characteristic polynomials of random unitary matrices and their derivatives, in the ergodic decomposition of the Hua-Pickrell measures and conjecturally in the asymptotics of the joint moments of Hardy's function and its derivative. Our first main result establishes a connection between the characteristic function of $\mathbf{X}(s)$ and the $\sigma$-Painlev\'e III' equation in the full range of parameter values $s>-\frac{1}{2}$. Our second main result gives the first explicit expression for the density and all the complex moments of the absolute value of $\mathbf{X}(s)$ for integer values of $s$. Finally, we establish an analogous connection to another special case of the $\sigma$-Painlev\'e III' equation for the Laplace transform of the sum of the inverse points of the Bessel point process.

Symmetric function theory and unitary invariant ensembles

Representation theory and the theory of symmetric functions have played a central role in random matrix theory in the computation of quantities such as joint moments of traces and joint moments of characteristic polynomials of matrices drawn from the circular unitary ensemble and other circular ensembles related to the classical compact groups. The reason is that they enable the derivation of exact formulas, which then provide a route to calculating the large-matrix asymptotics of these quantities. We develop a parallel theory for the Gaussian Unitary Ensemble (GUE) of random matrices and other related unitary invariant matrix ensembles. This allows us to write down exact formulas in these cases for the joint moments of the traces and the joint moments of the characteristic polynomials in terms of appropriately defined symmetric functions. As an example of an application, for the joint moments of the traces, we derive explicit asymptotic formulas for the rate of convergence of the moments of polynomial functions of GUE matrices to those of a standard normal distribution when the matrix size tends to infinity.

Computation of Al-Salam Carlitz and Askey-Wilson Moments using Motzkin Paths

In this paper we study the moments of polynomials from the Askey scheme, and we focus on Askey-Wilson polynomials. More precisely, we give a combinatorial proof for the case where $d=0$. Their values have already been computed by Kim and Stanton in 2015, however, the proof is not completely combinatorial, which means that an explicit bijection has not been exhibited yet. In this work, we use a new combinatorial approach for the simpler case of Al-Salam-Carlitz, using a sign reversing involution that directly operates on Motzkin path. We then generalize this method to Askey-Wilson polynomials with $d=0$ only, providing the first fully combinatorial proof for that case.

Conjectures for Moments Associated With Cubic Twists of Elliptic Curves

We extend the heuristic introduced by Conrey, Farmer, Keating, Rubinstein, and Snaith in order to formulate conjectures for the -moments of L-functions of elliptic curves twisted by cubic characters. We also apply the work of Keating and Snaith on the -moments of characteristic polynomials of unitary matrices to extend our conjecture to such that and . Our conjectures are then numerically tested for two families. A novelty of our conjectures is that cubic twists naturally lead us to consider the possibility .

Integer moments of complex Wishart matrices and Hurwitz numbers

We give formulae for the cumulants of complex Wishart (LUE) and inverse Wishart matrices (inverse LUE). Their large- N expansions are generating functions of double (strictly and weakly) monotone Hurwitz numbers which count constrained factorisations in the symmetric group. The two expansions can be compared and combined with a duality relation proved in [F. D. Cunden, F. Mezzadri, N. O’Connell, and N. J. Simm, Moments of random matrices and hypergeometric orthogonal polynomials, Comm. Math. Phys. 369 (2019), no. 3, 1091–1145] to obtain: i) a combinatorial proof of the reflection formula between moments of LUE and inverse LUE at genus zero and, ii) a new functional relation between the generating functions of monotone and strictly monotone Hurwitz numbers. The main result resolves the integrality conjecture formulated in [F. D. Cunden, F. Mezzadri, N. J. Simm, and P. Vivo, Correlators for the Wigner–Smith time-delay matrix of chaotic cavities, J. Phys. A 49 (2016), no. 18, 18LT01, 20 pp] on the time-delay cumulants in quantum chaotic transport. The precise combinatorial description of the cumulants given here may cast new light on the concordance between random matrix and semiclassical theories.

Hankel determinants of linear combinations of moments of orthogonal polynomials

We prove evaluations of Hankel determinants of linear combinations of moments of orthogonal polynomials (or, equivalently, of generating functions for Motzkin paths), thus generalizing known results for Catalan numbers.

Moments of characteristic polynomials in certain random neural networks

We consider large neural networks in cognitive neuropsychology whose synaptic connectivity matrices are randomly chosen from correlated Gaussian random matrices. We focus on the moments of characteristic polynomials and prove that the limiting even and odd moments at the edge are given by the largest eigenvalue distribution in the Gaussian Symplectic Ensemble (GSE) and in the induced GSE ensemble, respectively. Our results show that there exists a duality relation between the real Ginibre ensemble and the GSE ensemble via the moment of characteristic polynomials and the largest eigenvalue.

On the moments of the moments of ζ(1/2 + it)

Taking t at random, uniformly from [0,T], we consider the kth moment, with respect to t, of the random variable corresponding to the 2βth moment of ζ(1/2+ix) over the interval x∈(t,t+1], where ζ(s) is the Riemann zeta function. We call these the ‘moments of moments’ of the Riemann zeta function, and present a conjecture for their asymptotics, when T→∞, for integer k,β. This is motivated by comparisons with results for the moments of moments of the characteristic polynomials of random unitary matrices and is shown to follow from a conjecture for the shifted moments of ζ(s) due to Conrey, Farmer, Keating, Rubinstein, and Snaith [18]. Specifically, we prove that a function which, the shifted-moment conjecture of [18] implies, is a close approximation to the moments of moments of the zeta function does satisfy the asymptotic formula that we conjecture. We motivate as well similar conjectures for the moments of moments for other families of primitive L-functions.

Moments of Moments and Branching Random Walks

We calculate, for a branching random walk X_n(l) to a leaf l at depth n on a binary tree, the positive integer moments of the random variable frac{1}{2^{n}}sum _{l=1}^{2^n}e^{2beta X_n(l)}, for beta in {mathbb {R}}. We obtain explicit formulae for the first few moments for finite n. In the limit nrightarrow infty , our expression coincides with recent conjectures and results concerning the moments of moments of characteristic polynomials of random unitary matrices, supporting the idea that these two problems, which both fall into the class of logarithmically correlated Gaussian random fields, are related to each other.

Chebyshev's Problem of the Moments of Nonnegative Polynomials

Изучается задача П. Л. Чебышева об экстремальных значениях моментов неотрицательных многочленов на отрезке $[-1,1]$ c весом при фиксированном нулевом моменте, поставленная им в 1883 г.в более общей форме. В случае первого момента задача была решена П. Л. Чебышевым (1883 г.) для единичного веса и Г. Сегe (1927 г.) для произвольного веса. Нами ранее было получено решение задачи Чебышева для моментов нечетного порядка, в значительной степени основанное на монотонности функции $x^{2k+1}$, $k\in\mathbb{N}$. Функция $x^{2k}$ не является монотонной на отрезке $[-1,1]$ и задача для моментов четного порядка становится сложнее. В работе получено решение задачи Чебышева о наибольших значениях моментов четного порядка для многочленов четной степени. Решение задачи о наименьшем значении второго момента для многочленов четной степени получено при дополнительном условии на вес. Библиография: 8 названий.

Remarks on results by Müger and Tuset on the moments of polynomials

Let f(x) be a non-zero polynomial with complex coefficients, and Mp=∫01f(x)pdx for p a positive integer. In a recent paper, Müger and Tuset showed that lim supp→∞|Mp|1∕p>0, and conjectured that this limit is equal to the maximum amongst the critical values of f together with the values |f(0)| and |f(1)|. We give an example that shows that this conjecture is false. It also may be natural to guess that lim supp→∞|Mp|1∕p is equal to the maximum of |f(x)| on [0,1]. However, we give a counterexample to this as well. We also provide a few more guesses as to the behavior of the quantity lim supp→∞|Mp|1∕p.

Extremal Values of Moments of Nonnegative Polynomials

Extremal Values of Moments of Nonnegative Polynomials

Ramanujan–Bernoulli numbers as moments of Racah polynomials

The classical sequence of Bernoulli numbers is known to the the sequence of moments of a family of orthogonal polynomials. Some similar statements are obtained for another sequence of rational numbers, which is similar in many ways to the Bernoulli numbers.

Lecture hall tableaux

We introduce lecture hall tableaux, which are fillings of a skew Young diagram satisfying certain conditions. Lecture hall tableaux generalize both lecture hall partitions and anti-lecture hall compositions, and also contain reverse semistandard Young tableaux as a limit case. We show that the coefficients in the Schur expansion of multivariate little q-Jacobi polynomials are generating functions for lecture hall tableaux. Using a Selberg-type integral we show that moments of multivariate little q-Jacobi polynomials, which are equal to generating functions for lecture hall tableaux of a Young diagram, have a product formula. We also explore various combinatorial properties of lecture hall tableaux.

A note on the moments of sequences of complex numbers

We give a short proof that the limsup of the pth root of the modulus of the pth moment of a sequence of complex numbers is equal to the modulus of the maximum of the sequence. This strengthens known results, and provides an analog to a recent result concerning moments of complex polynomials.

Moments of moments of characteristic polynomials of random unitary matrices and lattice point counts

In this note, we give a combinatorial and noncomputational proof of the asymptotics of the integer moments of the moments of the characteristic polynomials of Haar distributed unitary matrices as the size of the matrix goes to infinity. This is achieved by relating these quantities to a lattice point count problem. Our main result is a new explicit expression for the leading order coefficient in the asymptotic as a volume of a certain region involving continuous Gelfand–Tsetlin patterns with constraints.

Extremal Values of Moments of Nonnegative Polynomials

Extremal Values of Moments of Nonnegative Polynomials