### A computer proof of Turán's inequality.

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2000 Math. Subject Classification: 33E12, 65D20, 33F05, 30E15The paper deals with analysis of several techniques and methods for the numerical evaluation of the Wright function. Even if the focus is mainly on the real arguments’ values, the methods introduced here can be used in the complex plane, too. The approaches presented in the paper include integral representations of the Wright function, its asymptotic expansions and summation of series. Because the Wright function depends on two parameters ...

Zeta-generalized-Euler-constant functions, $$\gamma \left(s\right):=\sum _{k=1}^{\infty}\left(\frac{1}{{k}^{s}}-{\int}_{k}^{k+1}\frac{dx}{{x}^{s}}\right)$$ and $$\tilde{\gamma}\left(s\right):=\sum _{k=1}^{\infty}{\left(-1\right)}^{k+1}\left(\frac{1}{{k}^{s}}-{\int}_{k}^{k+1}\frac{dx}{{x}^{s}}\right)$$ defined on the closed interval [0, ∞), where γ(1) is the Euler-Mascheroni constant and $$\tilde{\gamma}$$ (1) = ln $$\frac{4}{\pi}$$ , are studied and estimated with high accuracy.

In this work, a symbolic encoding of generalized Di-richlet generating series is found thanks to combinatorial techniques of noncommutative rational power series. This enables to explicit periodic generalized Dirichlet generating series – particularly the coloured polyzêtas – as linear combinations of Hurwitz polyzêtas. Moreover, the noncommutative version of the convolution theorem gives easily rise to an integral representation of Hurwitz polyzêtas. This representation enables us to build the...

Accurate estimates of real Pochhammer products, lower (falling) and upper (rising), are presented. Double inequalities comparing the Pochhammer products with powers are given. Several examples showing how to use the established approximations are stated.