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

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

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

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

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

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

Analytical studies on transient groundwater flow induced by land reclamation

In many coastal areas, land has been reclaimed by dumping fill materials into the sea. Land reclamation may have a significant effect on groundwater regimes, especially when the reclamation is at large scale. Analytical studies on the impact of land reclamation on steady-state ground water flow conditions were conducted previously, but transient analytical solutions are not yet available. Transient analytical solutions are derived to illustrate the temporal change of groundwater systems in response to land reclamation using two hypothetical models: a hillside aquifer and an oceanic elongated island. The analytical solutions show that when time is short, the water level in the reclaimed area increases significantly after reclamation while that in the original aquifer remains almost unchanged. When time is great, the change of water level in the reclaimed site becomes small but the increase of water level propagates into the original aquifer. For the specific parameters and aquifer geometry used in the examples, it takes at least over 100 years for the whole system to approach a new equilibrium. The island example demonstrates that land reclamation on one side of the island will eventually modify the groundwater regimes over the entire island, including the water level, water divide, and submarine groundwater discharge. The degree of the modification of the groundwater system and the time required for the system to approach a new equilibrium depend mainly on the hydraulic conductivity and storativity of the fill materials and the reclamation length. It is suggested that for a large reclamation project, the response of the groundwater regime to reclamation should be studied in detail to evaluate the long-term change of the flow system and the consequent environmental and engineering impacts. Copyright 2008 by the American Geophysical Union.published_or_final_versio

Vanishing viscosity limits for the degenerate lake equations with Navier boundary conditions

The paper is concerned with the vanishing viscosity limit of the two-dimensional degenerate viscous lake equations when the Navier slip conditions are prescribed on the impermeable boundary of a simply connected bounded regular domain. When the initial vorticity is in the Lebesgue space $L^q$ with $2<q\le\infty$, we show the degenerate viscous lake equations possess a unique global solution and the solution converges to a corresponding weak solution of the inviscid lake equations. In the special case when the vorticity is in $L^\infty$, an explicit convergence rate is obtained