A Symbolic Approach to Some Indentities for Bernoulli-Barnes Polynomials

A symbolic method is used to establish some properties of the Bernoulli-Barnes polynomials.Comment: 12 page

Identities for generalized Euler polynomials

For $N \in \mathbb{N}$, let $T_{N}$ be the Chebyshev polynomial of the first kind. Expressions for the sequence of numbers $p_{\ell}^{(N)}$, defined as the coefficients in the expansion of $1/T_{N}(1/z)$, are provided. These coefficients give formulas for the classical Euler polynomials in terms of the so-called generalized Euler polynomials. The proofs are based on a probabilistic interpretation of the generalized Euler polynomials recently given by Klebanov et al. Asymptotics of $p_{\ell}^{(N)}$ are also provided

Pochhammer Symbol with Negative Indices. A New Rule for the Method of Brackets

The method of brackets is a method of integration based upon a small number of heuristic rules. Some of these have been made rigorous. An example of an integral involving the Bessel function is used to motivate a new evaluation rule

Joint inversion of Rayleigh wave phase velocity and ellipticity using USArray: Constraining velocity and density structure in the upper crust

Rayleigh wave ellipticity, or H/V ratio, observed on the surface is particularly sensitive to shallow earth structure. In this study, we jointly invert measurements of Rayleigh wave H/V ratio and phase velocity between 24–100 and 8–100 sec period, respectively, for crust and upper mantle structure beneath more than 1000 USArray stations covering the western United States. Upper crustal structure, in particular, is better constrained by the joint inversion compared to inversions based on phase velocities alone. In addition to imaging Vs structure, we show that the joint inversion can be used to constrain Vp/Vs and density in the upper crust. New images of uppermost crustal structure (<3 km depth) are in excellent agreement with known surface features, with pronounced low Vs, low density, and high Vp/Vs anomalies imaged in the locations of several major sedimentary basins including the Williston, Powder River, Green River, Denver, and San Juan basins. These results demonstrate not only the consistency of broadband H/V ratios and phase velocity measurements, but also that their complementary sensitivities have the potential to resolve density and Vp/Vs variations

Topology optimization of freeform large-area metasurfaces

We demonstrate optimization of optical metasurfaces over $10^5$--$10^6$ degrees of freedom in two and three dimensions, 100--1000+ wavelengths ($\lambda$) in diameter, with 100+ parameters per $\lambda^2$. In particular, we show how topology optimization, with one degree of freedom per high-resolution "pixel," can be extended to large areas with the help of a locally periodic approximation that was previously only used for a few parameters per $\lambda^2$. In this way, we can computationally discover completely unexpected metasurface designs for challenging multi-frequency, multi-angle problems, including designs for fully coupled multi-layer structures with arbitrary per-layer patterns. Unlike typical metasurface designs based on subwavelength unit cells, our approach can discover both sub- and supra-wavelength patterns and can obtain both the near and far fields

The finite Fourier transform of classical polynomials

The finite Fourier transform of a family of orthogonal polynomials $A_{n}(x)$, is the usual transform of the polynomial extended by $0$ outside their natural domain. Explicit expressions are given for the Legendre, Jacobi, Gegenbauer and Chebyshev families

Recursion Rules for the Hypergeometric Zeta Functions

The hypergeometric zeta function is defined in terms of the zeros of the Kummer function M(a, a + b; z). It is established that this function is an entire function of order 1. The classical factorization theorem of Hadamard gives an expression as an infinite product. This provides linear and quadratic recurrences for the hypergeometric zeta function. A family of associated polynomials is characterized as Appell polynomials and the underlying distribution is given explicitly in terms of the zeros of the associated hypergeometric function. These properties are also given a probabilistic interpretation in the framework of Beta distributions

An Extension of the Method of Brackets. Part 1

The method of brackets is an efficient method for the evaluation of a large class of definite integrals on the half-line. It is based on a small collection of rules, some of which are heuristic. The extension discussed here is based on the concepts of null and divergent series. These are formal representations of functions, whose coefficients $a_{n}$ have meromorphic representations for $n \in \mathbb{C}$, but might vanish or blow up when $n \in \mathbb{N}$. These ideas are illustrated with the evaluation of a variety of entries from the classical table of integrals by Gradshteyn and Ryzhik