Abstract: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 2 C, but might vanish or blow up when n 2 N. These ideas are illustrated with the evaluation of a variety of entries from the classical table of integrals by Gradshteyn and Ryzhik.

Tewodros Amdeberhan obtained his Ph.D. in mathematics from Temple University under the supervision of Doron Zeiberger. He currently teaches at Tulane University and holds a permanent membership at DIMACS, Rutgers University. Xiao Guan was born in Beijing, China. He is currently a graduate student working under the guidance of V. Moll at Tulane University. Inspired by his advisor, he is currently concentrating on problems related to the evaluation of definite integrals. He is also interested in random matrices. Lin Jiu is a doctoral student working under the supervision of V. Moll at Tulane University. He obtained both bachelor and master degrees in Mathematics at Beijing Institute of Technology. His research interests involve Symbolic Computations, Special Functions and Number Theory.

The method of brackets, developed in the context of evaluation of integrals coming from Feynman diagrams, is a procedure to evaluate definite integrals over the half-line. This method consists of a small number of operational rules devoted to convert the integral into a bracket series. A second small set of rules evaluates this bracket series and produces the result as a regular series. The work presented here combines this method with the classical Mellin transform to extend the class of integrands where the method of brackets can be applied. A selected number of examples are used to illustrate this procedure.

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We evaluate the Hankel determinants of various sequences related to Bernoulli and Euler numbers and special values of the corresponding polynomials. Some of these results arise as special cases of Hankel determinants of certain sums and differences of Bernoulli and Euler polynomials, while others are consequences of a method that uses the derivatives of Bernoulli and Euler polynomials. We also obtain Hankel determinants for sequences of sums and differences of powers and for generalized Bernoulli polynomials belonging to certain Dirichlet characters with small conductors. Finally, we collect and organize Hankel determinant identities for numerous sequences, both new and known, containing Bernoulli and Euler numbers and polynomials.

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.

Abstract. For N ∈ N, let TN be the Chebyshev polynomial of the first kind. Expressions for the sequence of numbers p

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