On a sequence transformation with integral coefficients for Euler’s constant

Abstract:

Let $\gamma$ denote Euler’s constant, and let \[ {s_n} = \left ( {1 + \frac {1}{2} + \cdots + \frac {1}{{n - 1}}} \right ) - \log n\quad (n \geq 2).\] We prove by Ser’s formula for the remainder $\gamma - {s_n}$ that for all integers $n \geq 1$ and $\tau \geq 2$ there are integers ${\mu _{n,0,}}{\mu _{n,1}}, \ldots ,{\mu _{n,n}}$ such that \[ {\mu _{n,0}}{s_\tau } + {\mu _{n,1}}{s_{\tau + 1}} + \cdots + {\mu _{n,n}}{s_{\tau + n}} = \gamma + {O_\tau }({(n(n + 1)(n + 2) \bullet \cdots \bullet (n + \tau ))^{ - 1}}),\] where the constant in ${O_\tau }$ depends only on $\tau$. The coefficients ${\mu _{n,k}}$ are explicitly given and are bounded by ${2^{3n + \tau - 1}}$.