158 research outputs found
Euler-related sums
The purpose of this paper is to develop a set of identities for Euler type sums of products of harmonic numbers and reciprocal binomial coefficients
Harmonic number sums in closed form
We extend some results of Euler related sums. Integral and closed form representation of sums with products of harmonic numbers and cubed binomial coefficients are developed in terms of Polygamma functions. The given representations are new
Some BBP-type series for polylog integrals
An investigation into a family of definite integrals containing log-polylog functions will be undertaken in this paper. It will be shown that Euler sums play an important part in the solution of these integrals and may be represented as a BBP-type formula. In a special case we prove that the corresponding log integral can be represented as a linear combination of the product of zeta functions and the Dirichlet beta function
Some Remarks on the Trapezoid Rule In Numerical Integration
In this paper, by the use of some classical results from the Theory of Inequalities, we point out quasi-trapezoid quadrature formulae for which the error of approximation is smaller than in the classical case. Examples are given to demonstrate that the bounds obtained within this paper may be tighter than the classical ones. Some applications for special means are also given
Evaluation of integrals with hypergeometric and logarithmic functions
We provide an explicit analytical representation for a number of logarithmic integrals in terms of the Lerch transcendent function and other special functions. The integrals in question will be associated with both alternating harmonic numbers and harmonic numbers with positive terms. A few examples of integrals will be given an identity in terms of some special functions including the Riemann zeta function. In general none of these integrals can be solved by any currently available mathematical package
Explicit evaluations of log–log integrals
By investigating a family of log-log type integrals on the unit domain and on the positive half line, we produce a substantial number of new identities, representing the value of the integral with the aid of Euler sums. A new family of Euler sum identities will also be given, thereby extending the current knowledge
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