Print version ISSN 0041-6932
Rev. Unión Mat. Argent. vol.50 no.1 Bahía Blanca June 2009
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 , . Indeed the irrationality of would follow from criteria given in  (see also ).
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
where we set if .
Recall the well-known representation
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
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
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))
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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Pablo A. Panzone
Departamento e Instituto de Matemática,
Universidad Nacional del Sur,
Av. Alem 1253,
(8000) Bahía Blanca, Argentina.
Recibido: 18 de octubre de 2008
Aceptado: 3 de Junio de 2009