Lin Jiu

Recursion rules for the hypergeometric zeta function

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.

Identities for generalized Euler polynomials

For N∈ℕ, let TN be the Chebyshev polynomial of the first kind. Expressions for the sequence of numbers , defined as the coefficients in the expansion of 1/TN(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 are also provided.

Pochhammer symbol with negative indices. A new rule for the method of brackets

Abstract 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.

A symbolic approach to some identities for Bernoulli–Barnes polynomials

The Bernoulli–Barnes polynomials are defined as a natural multidimensional extension of the classical Bernoulli polynomials. Many of the properties of the Bernoulli polynomials admit extensions to this new family. A specific expression involving the Bernoulli–Barnes polynomials has recently appeared in the context of self-dual sequences. The work presented here provides a proof of this self-duality using the symbolic calculus attached to Bernoulli numbers and polynomials. Several properties of the Bernoulli–Barnes polynomials are established by this procedure.

A symbolic approach to multiple zeta values at negative integers

Symbolic computation techniques are used to derive some closed form expressions for an analytic continuation of the Euler-Zagier zeta function evaluated at the negative integers as recently proposed by B. Sadaoui. This approach allows to compute explicitly some contiguity identities, recurrences on the depth of the zeta values and generating functions.

An Extension of the Method of Brackets: Part 1

Abstract The method of brackets is an efficient method for the evaluation of alarge 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 an have meromorphic representations for n ∈ ℂ, but might vanish or blow up when n ∈ ℕ. These ideas are illustrated with the evaluation of a variety of entries from the classical table of integrals by Gradshteyn and Ryzhik.

THE FINITE FOURIER TRANSFORM OF CLASSICAL POLYNOMIALS

The finite Fourier transform of a family of orthogonal polynomials

The integrals in Gradshteyn and Ryzhik Part 30: Trigonometric functions

A_{n}(x)

The unimodality of a polynomial coming from a rational integral. Back to the original proof

, is the usual transform of the polynomial extended by

Connection Coefficients for Higher-order Bernoulli and Euler Polynomials: A Random Walk Approach

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