The proof of Muqattash-Yahdi conjecture (original) (raw)

Sharp inequalities for the psi function and harmonic numbers

2009

In this paper, two sharp inequalities for bounding the psi function ψ and the harmonic numbers Hn are established respectively, some results in [I. Muqattash and M. Yahdi, Infinite family of approximations of the Digamma function, Math. Comput. Modelling 43 (2006), 1329-1336.] are improved, and some remarks are given. 2000 Mathematics Subject Classification. Primary 33B15, Secondary 26D15.

Some results on the digamma function

Applied Mathematics & Information Sciences, 2013

The digamma function is defined for x > 0 as a locally summable function on the real line by ψ(x) = −γ

Rational Approximations to Values of the Digamma Function and a Conjecture on Denominators

—We explicitly construct rational approximations to the numbers ln(b) − ψ(a + 1), where ψ is the logarithmic derivative of the Euler gamma function. We prove formulas expressing the numerators and the denominators of the approximations in terms of hypergeometric sums. This generalizes the Aptekarev construction of rational approximations for the Euler constant γ. As a consequence, we obtain rational approximations for the numbers π/2 ± γ. The proposed construction is compared with rational Rivoal approximations for the numbers γ + ln(b). We verify assumptions put forward by Rivoal on the denominators of rational approximations to the numbers γ + ln(b) and on the general denominators of simultaneous approximations to the numbers γ and ζ(2) − γ 2 .

Some Inequalities for the k-Digamma Function

2014

Some inequalities involving the k-digamma function are presented. These results are the k-analogues of some recent results. Mathematics Subject Classification: 33B15, 26A48.

Some Inequalities for the p-Digamma Function

2014

Some inequalities involving the p-digamma function are presented. These results are the p-analogues of some recent results. Mathematics Subject Classification: 33B15, 26A48.