New representations for the Lugo and Euler-Mascheroni constants. (English) Zbl 1216.33005
Summary: Lugo’s constant \(L\) given by
\[ L=-\tfrac12- \gamma+\ln 2 \]
is defined as the limit of the sequence \((L_n)_{n\in\mathbb N}\) defined by
\[ L_n:= \sum_{i=1}^n \sum_{j=1}^n \frac{1}{i+j}- (2\ln 2)n+\ln n \quad(n\in\mathbb N) \]
as \(n\to\infty\), \(\mathbb N\) being the set of positive integers. In this paper, we establish new analytical representations for the Euler-Mascheroni constant \(\gamma\) in terms of the psi function. We also give bounds of \(L-L_n\) and present a new sequence which converges to Lugo’s constant \(L\).
\[ L=-\tfrac12- \gamma+\ln 2 \]
is defined as the limit of the sequence \((L_n)_{n\in\mathbb N}\) defined by
\[ L_n:= \sum_{i=1}^n \sum_{j=1}^n \frac{1}{i+j}- (2\ln 2)n+\ln n \quad(n\in\mathbb N) \]
as \(n\to\infty\), \(\mathbb N\) being the set of positive integers. In this paper, we establish new analytical representations for the Euler-Mascheroni constant \(\gamma\) in terms of the psi function. We also give bounds of \(L-L_n\) and present a new sequence which converges to Lugo’s constant \(L\).
MSC:
| 33B15 | Gamma, beta and polygamma functions |
| 33C45 | Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.) |
Keywords:
Euler-Mascheroni constant; gamma function; Weierstrass formula; Lugo’s constant; psi (or digamma) function; arithmetical functions; speed of convergence and approximationReferences:
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