Narrow Results

This paper deals with the study of the reliability of multicomponent stress–strength model assuming that the strength components are independently and identically distributed as power transformed half-logistic (PHL) distribution. The strength components which are subject to a common stress are assumed to be independent with either Weibull distribution or PHL distribution. The maximum likelihood estimates of the multicomponent stress–strength reliability and its asymptotic confidence interval under the above said conditions are obtained. The Bayes estimates of the multicomponent stress–strength reliability are also obtained under squared error loss function and using gamma priors for the parameters. To evaluate the performance of the procedure, a simulation study is considered. For illustration purpose of the proposed model, two real life examples are given.

We prove that certain functions involving the gamma and q-gamma function are monotone. We also prove that (x^mψ(x))^(m+1) is completely monotonic. We conjecture that -(x^mψ^(m)(x))^(m) is completely monotonic for m ≥ 2; and we prove it, with help from Maple, for 2 ≤ m ≤ 16. We give a very useful Maple procedure to verify this for higher values of m. A stronger result is also formulated where we conjecture that the power series coefficients of a certain function are all positive.

This paper is concerned with new results for the circular Eisenstein series ${\epsilon }_r(z)$ as well as with a novel approach to Hilbert–Eisenstein series ${𝔥}_r(z)$, introduced by Michael Hauss in 1995. The latter turns out to be the product of the hyperbolic sinh function with an explicit closed form linear combination of digamma functions. The results, which include differentiability properties and integral representations, are established by independent and different argumentations. Highlights are new results on the Butzer–Flocke–Hauss Omega function, one basis for the study of Hilbert–Eisenstein series, which have been the subject of several recent papers.

For a real number $\beta \ne 0$ and positive integers $m$ and $n$ with $m\ge 2$, we evaluate the sum of multiple zeta values

We offer two new Mellin transform evaluations for the Riemann zeta function in the region $0<\Re (s)<1$. Some discussion is offered in the way of evaluating some further Fourier integrals involving the Riemann xi function.