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Revista de la Unión Matemática Argentina

versão impressa ISSN 0041-6932

Rev. Unión Mat. Argent. vol.50 no.1 Bahía Blanca jun. 2009

 

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 ∑N ----1---- n=1 (n+x)(n+y) . 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  ∑n 1 γ = limn → ∞ i=1-i - ln(n) , see for example [7], [6]. Indeed the irrationality of γ would follow from criteria given in [3] (see also [5]).

The purpose of this note is to give integral representations for γ which seem to be new. As usual we write (x) = (x + n - 1)(x + n - 2) ... x n .

Theorem. If  n+x+ 1 f (x,n) := 32xn + 2 + -2n-12 ,  n+1 g(y,n) := n2n+y-- 2yn + 2n-21 then

 ∑∞ ∫ 1(- x) (x) i) γ = (- 1)n -----n---n f(x,n)dx. n=1 0 (x)2n+1

 ∑∞ ∫ 1 ii) γ = (- 1)n (--y)n(y)ng (y, n)dy 0 (2n )!y n=1

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

 ∫ 1 | (- x-)n(x)nf(x,n )dx| ≤--(6-)- 0 (x)2n+1 n2 2nn

Indeed this follows form the fact that for 0 ≤ x ≤ 1 one has |(-(xx))2nn(x+)1n| ≤ n212n-- (n) and f(x,n ) ≤ f (1,n) ≤ 6 if 1 ≤ n .

Proof. We use the following formula: if  (2n+x) 1 1 (x-y) f1(n,x, y) := (n+y) + (2n--1)(n + x + 2) +--2n-- and f (n, N, x,y) := --1--- + -N+x--- (x-y) 2 2(1-2n) (1- 2n) 2n , then

∑N 1 ---------------= n=1 (n + x)(n + y)
(1)

∑N (12 - (x - y)2)...((n - 1)2 - (x - y)2) (- 1)n-1--------------------------------------f1(n, x,y)+ n=1 (2n + x)(2n - 1 + x)...(x + 1)

 ∑N (12 - (x - y )2) ...((n - 1 )2 - (x - y)2) + (- 1)n-1------------------------------------------------f2(n,N, x,y) =: n=1 (N + n + x)(N + n - 1 + x )...(N - n + x + 1))

AN (x,y ) + BN (x, y),

where we set (12 - (x - y)2)...((n - 1)2 - (x - y)2) = 1 if n = 1 .

Recall the well-known representation

 ′ ∑∞ Γ-(x +-1) = - γ + x ----1----. Γ (x + 1 ) n (n + x) n=1
(2)

Notice that in (1), BN (x,y ) → 0 as N → ∞ if x and y are bounded . We prove this in a moment.

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

Now we prove (1): set Ck := Ck (n, x,y) = (n+k+bx1).....b.(kn-k+x)(n1+y) ; C := C (n,x,y ) = ----1---- 0 0 (n+x)(n+y) where b := b (x,y) = (x - y)2 - k2 k k , and define b1...bi- 1 = 1 if i = 1 .

Add from n = 1 to N the trivial identity C0 - Cn-1 = (C0 - C1 ) + ⋅⋅⋅ + (Cn- 2 - Cn -1) to get

∑N 1 N∑ b ...b ---------------- -------------1----n-1----------- = n=1(n + x )(n + y ) n=1(n + y ).((2n - 1 + x)...(x + 1))

∑N n∑-1 ∑N n∑- 1 b1 ...bk- 1 (Ck-1- Ck ) = --------------------------(n + 2x - y) = n=1 k=1 n=1k=1 (n + k + x) ...(n - k + x)
(3)

 N n-1 N N ∑ ∑ ∑ ∑ = ϵn,k(x,y) - ϵn-1,k(x,y ) = ϵN,k(x,y) - ϵk,k(x,y), n=1 k=1 k=1 k=1

with  ( ) 2(1-12k)+(n1+-x2k)+(x-y)(- 12k) ϵn,k(x,y) := b1 ...bk-1.------------------------ (n+k+x )...(n- k+1+x ) .

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

We now prove that if 0 ≤ x ≤ 1 , 0 ≤ y ≤ 1 then BN (x, y) → 0 as N → ∞ (the proof for x, y bounded is similar). Indeed in this range of x and y one has

 ∑N ∑ | B (x, y) |= O ( (---1)(-)-N-) = O ( (---1)-(-)-N-)+O (1∕N ) N N+n 2n n3 N+n 2n n3 n=1 2n n 1≤n≤N ∕4 2n n

But  ∑ ----1----N- ∑ 1-N- O ( 1≤n≤N ∕4 (N+2nn)(2nn) n3) = O( 1≤n ≤N∕4 N2n3) = O (1∕N ) , where we have used N (N + 1)∕2 ≤ (N+n ) 2n for 1 ≤ n ≤ N ∕4 , N ≥ 4 .

This finishes the proof of the theorem. ■

A corollary of formula (1) is the following

Corollary. Let h(n, M, y) := 1∕2+2Mn+-n1--- y2n + MM++2nn+y-- . Set

 ∫ M∑ 1- 1 ∑ n-1(12 --y2)-...((n---1)2 --y2) DM := ( n )- ln(M +1 )+ 0 y (- 1) (2n)!(2n+M ) h(n,M, y)dy n=1 1≤n≤M ∕2 2n

Then

γ - DM = O (1∕8M )

Proof. From (1), letting N → ∞ one gets

∑∞ 1 (n +-x)(n-+-y)-= n=1

 ∞ ∑ n-1 (12---(x---y)2)...((n---1)2---(x---y)2)- (- 1) (2n + x )(2n - 1 + x)...(x + 1) f1(n,x,y ). n=1

Now substitute y by M + y and x by M to get for 0 ≤ y ≤ 1

 ∑∞ ----1----= n(n + y) n=M+1

∑∞ (12 - y2)...((n - 1)2 - y2) (- 1)n-1-------------(2n+M-)--------f1(n,M, M + y) = n=1 (2n)! 2n

 ∑ ∑ ∑ M + = +O (1∕8 ) 1≤n≤M ∕2 M ∕2<n<∞ 1≤n≤M ∕2

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

 ∫ 1 ∑M 1 ∑∞ 1 ∑M 1 ∫ 1 ∑∞ 1 γ = y{ --------+ --------}dy = -- ln (M +1)+ y --------dy, 0 n=1 n(n + y) n=M+1 n(n + y) n=1 n 0 n=M+1 n(n + y)

observing that h(n, M, y) = f1(n,M, M + y). ■

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

D = 1 - ln 2,D = 283-- ln 4,D = 35-- ln5, 1 2 144 3 16

 169553 192809 D4 = -------- ln 7,D5 = -------- ln 8 67200 72576

Numerically we have checked that DM is always of the form r - ln n , with r a rational number and 2 ≤ n ≤ 2M , n ϵZ .

Notice that (1) or derivates of (1) give formulae for Hurwitz-Riemann's zeta function ∑ ∞ --1--- n=1(n+x)s for s = 2, 3,4,... .

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

 ∫ 1 γ = {y 3F2[1∕2,1- y,1+y; 3∕2,3∕2;- 1∕4]+ --y-- 3F2[1- y,1+y, 1+y; 3∕2,2+y;- 1∕4] 0 1 + y

 y2 1 - ---4F3[1,1,1- y,1 + y;3∕2,2,2;- 1∕4 ]---Sinh2 (y.ArcSinh (1∕2))}dy. 4 y

Here  F [a ,...,a ;b ,...,b ;z] = ∑ ∞ (a1)k...(ap)k zk p q 1 p 1 q k=0 (b1)k...(bq)k k! is the general hypergeometric function.

 

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. Sums 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. Criteria for irrationality of Euler's Constant. Proc. Amer. Math. Soc. 131, 2003, 3335-3344.        [ Links ]

[4]    J. Sondow. Double 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. Euler'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