. The derivation of this combinatorial identity is done in an elemental way.]]>

**Formulas for the Euler-Mascheroni constant**

**Pablo A. Panzone**

* Abstract.* We give several integral representations for the Euler-Mascheroni constant using a combinatorial identity for . The derivation of this combinatorial identity is done in an elemental way.

*2000 Mathematics Subject Classification.* 11Mxx

* Key words and phrases.* Euler-Mascheroni constant.

**Introduction.** There exist many formulas for Euler-Mascheroni constant , see for example [7], [6]. Indeed the irrationality of would follow from criteria given in [3] (see also [5]).

The purpose of this note is to give integral representations for which seem to be new. As usual we write .

**Theorem.** *If* *,* *then*

**Remark 1.** The formulae stated converge more rapidly than the usual definition. For example, notice that for

Indeed this follows form the fact that for one has and if .

*Proof.* We use the following formula: if and , then

| (1) |

where we set if .

Recall the well-known representation

| (2) |

Notice that in (1), as if and are bounded . We prove this in a moment.

Now i) follows from integrating (2) from 0 to 1 and using (1) with , letting . The first formula of ii) is proved in the same way putting in (1).

Now we prove (1): set ; where , and define if .

]]> Add from to the trivial identity to get | (3) |

with .

From the equality of the first expression in (3) and the last one, we obtain (1).

We now prove that if , then as (the proof for bounded is similar). Indeed in this range of and one has

But , where we have used for , .

This finishes the proof of the theorem.

A corollary of formula (1) is the following

**Corollary.** *Let* *. Set*

*Then*

*Proof.*From (1), letting one gets

Now substitute by and by to get for

Now the corollary follows from this last formula inserted in (recall formula (2))

observing that

**Remark 2.** The corollary stated seems to give clean approximation formulas. Indeed

Numerically we have checked that is always of the form , with a rational number and , .

Notice that (1) or derivates of (1) give formulae for Hurwitz-Riemann's zeta function for .

We mention without proof that formula ii) of theorem 1 is equivalent to

Here is the general hypergeometric function.

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**References**

[1] A. J. Van der Poorten, *A* proof that Euler missed... Math.Intelligencer, **1** **(Nr.4)**, 1979, 195-203. [ Links ]

[2] P. A. Panzone. *S*ums for Riemann's Hurwitz function II. Actas del Quinto Congreso A. Monteiro, Bahía Blanca, Universidad Nacional del Sur, 1999, 109-125. [ Links ]

[3] J. Sondow. *C*riteria for irrationality of Euler's Constant. Proc. Amer. Math. Soc. **131**, 2003, 3335-3344. [ Links ]

[4] J. Sondow. *D*ouble integrals for Euler's constant and ln(4/p) and an analog of Hadjicostas's formula. Amer. Math. Monthly, **112**, 2005, 61-65 [ Links ]

[5] J. Sondow and W. Zudilin. *E*uler's constant, q-logarithms, and formulas of Ramanujan and Gosper. Ramanujan J. **12**, 2006, 225-244. [ Links ]

[6] X. Gourdon and P. Sebah. *A* collection of formulae for Euler Constant. 2003, http://numbers.computation.free.fr/Constants/Gamma/gammaFormulas.pdf. [ Links ]

[7] Eric W. Weisstein. Euler-Mascheroni Constant. From MathWorld—A Wolfram Web Resource. http://mathworld.wolfram.com/Euler-MascheroniConstant.html [ Links ]

**Pablo A. Panzone**** ** Departamento e Instituto de Matemática, ]]>
Universidad Nacional del Sur,

Av. Alem 1253,

(8000) Bahía Blanca, Argentina.

ppanzone@uns.edu.ar

**Recibido**: 18 de octubre de 2008

**Aceptado**: 3 de Junio de 2009