ON TWO FUNDAMENTAL IDENTITIES FOR EULER SUMS

Contents 1. Introduction 2. Telescoping and Partial Fractions 3. Finite Series Transformations 4. Geometric Series 4.1. Convolution of Geometric Series 4.2. A Sum Formula 4.3. A q -Analog 5. Integral Representations 5.1. Single Integrals I 5.2. Single Integrals II 5.3. Double Integrals I 5.4. Double Integrals II 5.5. Integration by Parts 5.6. Triple Integrals I 5.7. Triple Integrals II 5.8. Complex Line Integrals I 2 4 5 6 6 7 8 9 9 10 11 11 12 12 14 14
Date : February 1, 2005. 1991 Mathematics Subject Classification. Primary: 11M41; Secondary: 11M06, 40B05. Key words and phrases. Multiple harmonic series, multiple zeta values, Euler sums, Euler-Zagier sums. Research of the first author supported by NSERC.
m
ζ (s 1 , . . . , s m ) =
k1 >···>km >0 j =1
kj j .
−s
(1.1)
The identity ζ (2, 1) = ζ (3) (1.2)
EULER SUMS
3
goes back to Euler [34] [35, p. 228] and has since been repeatedly rediscovered [24, 25, 36, 45]. The more general formula
?
n ∈ Z+ ,
(1.6)
4
J. M. BORWEIN AND D. M. BRADLEY
2. Telescoping and Partial Fractions For a quick proof of (1.2), consider 1 1 = S := nk (n + k ) n,k>0 n2 n,k>0 On the other hand, S=
15 17 17 19 19 21 23 24 24 25 26 26 27 29 30 31 32
1. Introduction For positive integers s1 , . . . , sm and signs σj = ±1, consider [9] the m-fold Euler sum ζ (s1 , . . . , sm ; σ1 , . . . , σm ) :=
k1 >···>km >0
σj j sj . k j j =1
m
k
As is now customary, we combine strings of exponents and signs by replacing sj by sj in the argument list if and only if σj = −1, and denote n repetitions of a substring S by {S }n . Thus, for example, ζ (1) = − log 2, ζ ({2}3) = ζ (2, 2, 2) = π 6 /7! and
1
2
J. M. BORWEIN AND D. M. BRADLEY
5.9. Complex Line Integrals II 5.10. Contours Integrals and Residues 6. A Stirling Number Generating Function 7. Polylogarithm Identities 7.1. Dilogarithm and Trilogarithm 7.2. Convolution of Polylogarithms 8. Fourier Series 9. Further Generating Functions 9.1. Hypergeometric Functions 9.2. A Generating Function for Sums 9.3. An Alternating Generating Function 9.4. The Digamma Function 9.5. The Beta Function 10. A Decomposition Formula of Euler 11. Equating Shuffles and Stuffles 12. Conclusion References
n,k>0
1 1 − k n+k
∞
=
n=1
1 n2Biblioteka nk =11 = ζ (3) + ζ (2, 1). k
1 1 + n k
1 1 1 = + = 2ζ (2, 1), 2 2 (n + k ) n(n + k ) k (n + k )2 n,k>0 n,k>0
by symmetry. The above argument goes back at least to Steinberg [45]. See also [46]. For (1.5), first consider
m−2
2ζ (m, 1) = mζ (m + 1) −
j =1
ζ (j + 1)ζ (m − j ),
2≤m∈Z
(1.3)
is also due to Euler [34] [35, p. 266]. Nielsen [50, p. 229] [51, p. 198] [52, pp. 47–49] developed a method for obtaining (1.3) and related results based on partial fractions. Formula (1.3) has also been rediscovered a number of times [63, 57, 55, 38, 17, 60]. Crandall and Buhler [32] deduced (1.3) from their general infinite series formula which expresses ζ (s, t) for real s > 1 and t ≥ 1 in terms of Riemann zeta values. Study of the the multiple zeta function (1.1) led to the discovery of a new generalization of (1.2), involving nested sums of arbitrary depth: ζ ({ 2 , 1 } n ) = ζ ({ 3 } n ), n ∈ Z+ .
(1.4)
Although numerous proofs of (1.2) and (1.3) are known (we give several in the sequel), the only proof of (1.4) of which we are aware involves making a simple change of variable in a multiple iterated integral (see [9, 10, 14] and (5.11) below). An alternating version of (1.2) is 8ζ (2, 1) = ζ (3), (1.5)
arXiv:math.NT/0502034 v1 1 Feb 2005
ON TWO FUNDAMENTAL IDENTITIES FOR EULER SUMS
JONATHAN M. BORWEIN AND DAVID M. BRADLEY
Abstract. We give diverse proofs of the fundamental identities ζ (2, 1) = ζ (3) = 8ζ (2, 1). We also discuss various generalizations for multiple harmonic (Euler) sums and some connections, thereby illustrating the wide variety of techniques fruitfully used to study such sums.
which has also resurfaced from time to time [52, p. 50] [58, (2.12)] [28, p. 267] and hints at the generalization originally conjectured in [9], and which still remains open—despite abundant evidence [7]. The purpose of this paper is to illustrate some of the techniques used to study Euler sums by focusing on the identities (1.2), (1.5) and various generalizations. For some of the broader issues relating to Euler sums, we refer the reader to the survey articles [14, 29, 61, 62, 64]. Computational issues are discussed in [31] and to an extent in [10]. Notation and Terminology. For positive integer N , denote the N th partial sum of the ′ harmonic series by HN := N n=1 1/n. We also use ψ = Γ /Γ to denote the logarithmic derivative of the Euler gamma function (also referred to as the digamma function), and recall the identity ψ (N + 1) + γ = HN , where γ = 0.5772156649 . . . is Euler’s constant. As usual, the Kronecker δm,n is 1 if m = n and 0 otherwise. We organize the proofs by technique, although clearly this is somewhat arbitrary as many proofs fit well within more than one category. In some of the later sections the proofs become more schematic. We invite readers to add selections to our collection. 8n ζ ({2, 1}n ) = ζ ({3}n),