Computation and theory of Euler sums of generalized hyperharmonic numbers

0.12 MB·

by  Ce Xu

Computation and theory of Euler sums of generalized
hyperharmonic numbers
7 Ce Xu∗
1
0 School of Mathematical Sciences, Xiamen University
2 Xiamen 361005,P.R. China

n
a
J
Abstract Recently, Dil and Boyadzhiev [10] proved an explicit formula for the sum of multiple
1
1 harmonicnumberswhoseindicesarethesequence({0} ,1). Inthispaperweshowthatthesums
r

of multiple harmonic numberswhose indices are the sequence {0} ,1;{1} can be expressed
] r k−1
T in terms of (multiple) zeta values, multiple harmonic numbers and Stirling numbers of the first
(cid:0) (cid:1)
N kind, and give an explicit formula.
.
h Keywords Euler sums; hyperharmonic numbers; harmonic numbers; multiple harmonic num-
t
a bers; Riemann zeta function; multiple zeta (star) values; Stirling numbers of the first kind.
m
AMS Subject Classifications (2010): 11B73; 11B83; 11M06; 11M32; 11M99
[

1
v 1 Introduction
3
2
Let s ,...,s be positive integers. The classical multiple harmonic numbers (MHNs) and mul-
7 1 m
3 tiple harmonic star numbers (MHSNs) are defined by the partial sums (see [17, 23])
0
. 1
1 ζ (s ,s ,··· ,s ) := , (1.1)
0 n 1 2 m ns1ns2···nsm
7 n≥n1>X···>nk≥1 1 2 m
1
1
v: ζn⋆(s1,s2,··· ,sm) := ns1ns2···nsm, (1.2)
Xi n≥n1≥X···≥nk≥1 1 2 m
r when n <m, then ζn(s1,s2,··· ,sm) = 0, and ζn(∅) =ζn⋆(∅) = 1. The limit cases of MHNs and
a
MHSNs give rise to multiple zeta values (MZVs) and multiple zeta star values (MZSVs) (see
[12, 16, 17, 23, 25]):
ζ(s ,s ,··· ,s )= lim ζ(s ,s ,··· ,s ),
1 2 m 1 2 m
n→∞
ζ⋆(s ,s ,··· ,s ) = lim ζ⋆(s ,s ,··· ,s )
1 2 m 1 2 m
n→∞
defined for s ,...,s ≥ 1 and s ≥ 2 to ensure convergence of the series. For non-negative
2 m 1
integers s ,...,s , we define the following a generalized multiple harmonic numbers
1 m+k
1
ζ (s ,··· ,s ;s ,··· ,s ) := . (1.3)
n 1 m m+1 m+k ns1···nsm+k
0<nm+k<X···<nm+1 1 m+k
<nm≤nm−1···≤n1≤n
∗
Corresponding author. Email: [email protected] (C. Xu)
1
Obviously, if m = 0 or k = 0 in (1.3) and s ∈ N := {1,2,···}, then
j
ζ (∅;s ,··· ,s )= ζ (s ,··· ,s ),
n 1 k n 1 k
ζ (s ,··· ,s ;∅)= ζ⋆(s ,··· ,s ).
n 1 m n 1 m
There are a lot of recent contributions on MZVs and MZSVs (for example, see [12, 16, 17, 23,
25]). The earliest results on MZVs or MZSVs are due to Euler who elaborated a method to
reduce double sums ζ(s ,s ) (also called linear Euler sums [11, 21]) of small weight to certain
1 2
rational linear combinations of products of zeta values. In [11], Flajolet and Salvy introduced
the following generalized series
∞ H(s1)H(s2)···H(sr)
n n n
SS,q := nq , (1.4)
n=1
X
which is called the generalized (nonlinear) Euler sums. Here S := (s ,s ,...,s ) (r,s ∈ N,i =
1 2 r i
1,2,...,r) with s ≤ s ≤ ... ≤ s and q ≥ 2. The quantity w := s +···+s +q is called the
1 2 r 1 r
(p)
weight andthequantity r is called thedegree. Thenotation H denotes theordinaryharmonic
n
numbers defined by
n
1
H(p) ≡ ζ (p) := , p,n ∈ N.
n n jp
j=1
X
It has been discovered in the course of the years that many nonlinear Euler sums admit expres-
sions involving finitely ”zeta values”, that is say values of the Riemann zeta function,
∞
1
ζ(s):= ,ℜ(s) > 1.
ns
n=1
X
with positive integer arguments, and linear Euler sums. The relationship between the values of
the Riemann zeta values and Euler sums has been studied by many authors. For details and
historical introductions, please see [1, 3–6, 8, 11, 15, 18–22] and references therein.
From [7, 9, 10, 14], we know that the classical hyperharmonic numbers are defined by
1
h(m) := = ζ⋆ {0} ,1 . (1.5)
n n n m−1
m
1≤nm≤X···≤n1≤n (cid:0) (cid:1)
(m)
In [24], we define the generalized hyperharmonic numbers h (k) by
n
1
h(m)(k) := = ζ {0} ,1;{1} , (1.6)
n n n ···n n m−1 k−1
m m+1 m+k−1
1<≤nnmm≤+nkm−X−1<1≤····<··≤nnm1+≤1n (cid:0) (cid:1)
where m,k ∈ N (The notation {} means that the sequence in the bracket is repeated p times).
p
In this paper, we prove the result: for positive integers m and k, the Euler-type sums with
hyperharmonic numbers
∞ (m)
h (k)
n
S(k,m;p) := (p ≥ m+1)
np
n=1
X
are related to the multiple zeta values, multiple harmonic numbers and Stirling numbers of the
first kind. For k = 1,2,3, the above results have been proved in Dil et al.[10] and our paper
[24]. The purpose of the present paper is to prove the following theorems.
2
Theorem 1.1 For integers k ∈ N and r ∈ N with r+2 ≤ p ∈ N, then the following identity
0
holds:
r+1
1 r+1
S(k,r+1;p) = (−1)iζ⋆({1} )U (p+1−l), (1.7)
r! l r i j,r
l=1(cid:20) (cid:21) i+j=k
X X
i,j≥0
n
where denotes the (unsigned) Stirling number of the first kind, which is defined by [7, 13]
k
(cid:20) (cid:21)
n
x x n+1
n!x(1+x) 1+ ··· 1+ = xk+1, (1.8)
2 n k+1
(cid:16) (cid:17) (cid:16) (cid:17) Xk=0(cid:20) (cid:21)
n n 0 0
with = 0, if n < k and = = 0, = 1, or equivalently, by the generating
k 0 k 0
(cid:20) (cid:21) (cid:20) (cid:21) (cid:20) (cid:21) (cid:20) (cid:21)
function:
∞ n xn
logk(1−x) = (−1)kk! , x ∈ [−1,1). (1.9)
k n!
n=1(cid:20) (cid:21)
X
and
∞ ζ {1}
n+r j
U (p):= . (1.10)
j,r n(cid:16)p (cid:17)
n=1
X
Theorem 1.2 For integers m > 0,p > 1 and r ∈ N := N∪{0}, we have
0
U (p)= ζ(p,{1} )+ζ p+1,{1}
m,r m m−1
m m
+(−1)p−1 (cid:0) Hil (cid:1)
p
(i −i )i
1≤im<X···<i1≤rXl=1 aaY=6=l1 a l l
p−1 m m
ζ(p+1−b)
+ (−1)b−1
(i −i )ib
1≤im<X···<i1≤rXb=1Xl=1 aaY=6=l1 a l l
ζ⋆ {1}
+(−1)p−1m−1 j j ζ(m−j +1)+ζi⋆l {1}m−j+1 − il(cid:16) ilm−j(cid:17)
(cid:16) p (cid:17)
(i −i )i
Xj=1 1≤ij<X···<i1≤rXl=1 aaY=6=l1 a l l
m−1 p−1 j j ζ p+1−b,{1} +ζ p+2−b,{1}
+ m−j m−j−1 (−1)b−1.
(cid:16) (i(cid:17) −i(cid:16))ib (cid:17)
Xj=1 1≤ij<X···<i1≤rXb=1Xl=1 aaY=6=l1 a l l
(1.11)
2 Some Lemmas and Theorems
To prove the Theorem 1.1 and Theorem 1.2, we need the following lemmas.
Lemma 2.1 ([24]) For positive integers n and k, then the following identity holds:
n
= (n−1)!ζ {1} . (2.1)
k n−1 k−1
(cid:20) (cid:21)
(cid:0) (cid:1)
3
Lemma 2.2 ([24]) For positive integers m,n and k, we have the recurrence relation
(−1)k−1 k−1
h(m)(k) = (−1)ih(m)(i) H(k−i) −H(k−i) , (2.2)
n k n m+n−1 m−1
Xi=0 n o
where
m+n−1
h(m)(0) = .
n m−1
(cid:18) (cid:19)
Lemma 2.3 ([23]) For positive integers m and n, then the recurrence relation holds:
(−1)m−1 m−1
B¯ (n)= (−1)iB¯ (n)X (m−i), (2.3)
m i n
m
i=0
X
where
n
X (m) := xm, x ∈ C,
n i i
i=1
X
n k1−1 km−1−1
B¯ (n):= x x ··· x ,B¯ (n) = 1.
m k1 k2 km 0
kX1=1 kX2=1 kXm=1
Lemma 2.4 ([22, 24]) For integers k ∈ N and p ∈ N \{1} := {2,3,...}, then the following
identity holds:
n+1
∞
p 1 Y (k) Y (k)
(p−1)! (cid:20) (cid:21) = (p−1)!ζ(p)+ p − p−1 , (2.4)
n!n(n+k) k p k
n=1 (cid:26) (cid:27)
X
(2) (3) (r)
where Y (n)= Y H ,1!H ,2!H ,··· ,(r−1)!H ,··· , Y (x ,x ,···) stands for the com-
k k n n n n k 1 2
plete exponential B(cid:16)ell polynomial defined by (see [13]) (cid:17)
tm tk
exp x = 1+ Y (x ,x ,···) . (2.5)
m k 1 2
 m! k!
m≥1 k≥1
X X
 
Noting that, in [21], we find the relation
1
ζ⋆({1} ) = Y (n),n,m ∈ N . (2.6)
n m m! m 0
Lemma 2.5 For positive integers m and n, then
(−1)iA¯ (n)A (n)= 0, (2.7)
i j
i+j=m
X
i,j≥0
where
A¯ (n):= a ···a , a ∈ C,
m k1 km k
1≤km<X···<k1≤n
A (n):= a ···a , a ∈ C.
m k1 km k
1≤km≤X···≤k1≤n
For convenience, we set A¯ (n)= A (n)= 1. If n < m, we let A¯ (n)= 0.
0 0 m
4
Proof. By a direct calculation, the following identities are easily derived
n ∞
(1−a t)−1 = A (n)tm,
i m
i=1 m=0
Y X
n ∞
(1+a t)= A¯ (n)tm.
i m
i=1 m=0
Y X
Hence, by using Cauchy product of power series, we have
n n
1= (1−a t)−1 (1−a t)
i i
i=1 i=1
Y Y
∞ ∞
= A (n)tm (−1)mA¯ (n)tm
m m
! !
m=0 m=0
X X
∞
=  (−1)iA¯i(n)Aj(n)tm.
 
mX=0i+Xj=m 
i,j≥0
 
Thus,comparingthecoefficients oftm inabove equation, weobtain theformula(2.7). Theproof
of Theorem 2.3 is finished. (cid:3)
The above lemmas will be useful in the development of the main theorems. Next, we give some
important theorems and it’s proofs by using these lemmas.
Theorem 2.6 For integers r ≥ 0 and m,n > 1, then
1
= (−1)iζ⋆({1} )ζ {1} , (2.8)
(k +r)···(k +r) r i n+r j
1 m
1≤km<X···<k1≤n i+Xj=m (cid:16) (cid:17)
i,j≥0
where
ζ⋆(∅) = ζ (∅) = 1, ζ⋆({1} ) = ζ {1} = 0, (i,j ≥ 1).
0 0 0 i 0 j
(cid:16) (cid:17)
1
Proof. The proof is by induction on m. For m = 1 we have = ζ (1)−ζ⋆(1),
k +r n+r r
1
1≤Xk1≤n
and the formula is true. For m > 1 we proceed as follow. Let
1
ζ (s ,s ,··· ,s |r+1) := , (2.9)
n 1 2 m (k +r)s1···(k +r)sm
1 m
1≤km<X···<k1≤n
ζ (∅|r+1)= 1.
n
Then by the definition (2.9) and the induction hypothesis, we have that
n
ζ ({1} |r+1)
ζ {1} |r+1 = k−1 m
n m+1 k+r
k=1
(cid:0) (cid:1) X
n
1
= (−1)iζ⋆({1} )ζ {1}
k+r r i k+r−1 j
Xk=1 i+Xj=m (cid:16) (cid:17)
i,j≥0
5
n ζ {1}
k+r−1 j
= (−1)iζ⋆({1} )
r i k+(cid:16)r (cid:17)
i+j=m k=1
X X
i,j≥0
= (−1)iζ⋆({1} ) ζ {1} −ζ {1}
r i n+r j+1 n+r j+1
i+Xj=m n (cid:16) (cid:17) (cid:16) (cid:17)o
i,j≥0
= (−1)iζ⋆({1} )ζ {1} − (−1)iζ⋆({1} )ζ {1} .
r i n+r j r i r j
i+jX=m+1 (cid:16) (cid:17) i+jX=m+1 (cid:16) (cid:17)
i,j≥0 i,j≥0
(2.10)
1
On the other hand, from Lemma 2.5, setting a = and n = r we get
k
k
(−1)iζ⋆({1} )ζ {1} = 0, m ≥ 1. (2.11)
r i r j
i+Xj=m (cid:16) (cid:17)
i,j≥0
Hence, combining (2.10) and (2.11) we can prove that the formula (2.8) holds. (cid:3)
In fact, by a similar argument as in the proof of Theorem 2.6 with the help of formula (5.2) in
the reference [23], we obtain the more general theorem.
Theorem 2.7 For integers r ≥ 0,m,n > 1 and real p > 0, then
1
= (−1)iζ⋆({p} )ζ {p} , (2.12)
(k +r)p···(k +r)p r i n+r j
1 m
1≤km<X···<k1≤n i+Xj=m (cid:16) (cid:17)
i,j≥0
1
= (−1)iζ ({p} )ζ⋆ {p} . (2.13)
(k +r)p···(k +r)p r i n+r j
1 m
1≤km≤X···≤k1≤n i+Xj=m (cid:16) (cid:17)
i,j≥0
Theorem 2.8 For integers r ≥ 0 and m,n ≥ 1, then the following identity holds:
n+r
h(r+1)(m)= (−1)iζ⋆({1} )ζ {1} . (2.14)
n r r i n+r j
(cid:18) (cid:19) i+Xj=m (cid:16) (cid:17)
i,j≥0
1
Proof. In Lemma 2.3, taking x = , then we have
j
j +r
n m
1
X (m)= = H(m) −H(m),
n j+r n+r r
j=1(cid:18) (cid:19)
X
(−1)m−1 m−1
ζ ({1} |r+1)= (−1)iζ ({1} |r+1) H(m−i)−H(m−i) . (2.15)
n m m n i n+r r
Xi=0 (cid:16) (cid:17)
From Lemma 2.1 and formula (2.15) we deduce that
n+r
h(r+1)(m) = ζ ({1} |r+1). (2.16)
n r n m
(cid:18) (cid:19)
Substituting (2.8) into (2.16) we may easily obtain the desired result. This completes the proof
of Theorem 2.7. (cid:3)
6
3 Proof of Theorem 1.1
By replacing x by n and n by r in (1.9), we deduce that
r+1
n+r 1 r+1
= nl−1. (3.1)
r r! l
(cid:18) (cid:19) l=1(cid:20) (cid:21)
X
Therefore, from (2.12) and (3.1) we obtain
r+1
1 r+1
h(r+1)(k) = nl−1 (−1)iζ⋆({1} )ζ {1} . (3.2)
n r! l r i n+r j
Xl=1(cid:20) (cid:21) i+Xj=k (cid:16) (cid:17)
i,j≥0
Thus, by the definition of S(k,m;p) and (3.2) we can prove (1.7). (cid:3)
4 Proof of Theorem 1.2
By the definition of multiple harmonic number (1.1), we can find that
∞ ∞ n+r
U (p) = ζn+r({1}m) = 1 ζk−1 {1}m−1
m,r np np k
n=1 n=1 k=1 (cid:0) (cid:1)
X X X
∞ r ∞
= ζn({1}m) + ζn+k−1 {1}m−1
np np(n+k)
n=1 k=1n=1 (cid:0) (cid:1)
X XX
r ∞ ζ {1} r i1−1 ∞ ζ {1}
= ζ(p,{1} )+ζ p+1,{1} + n m−1 + n+i2−1 m−2
m m−1 np(n+i ) np(n+i )(n+i )
(cid:0) (cid:1) iX1=1nX=1 (cid:0) 1 (cid:1) iX1=1iX2=1nX=1 (cid:0)1 2(cid:1)
= ···
m ∞ ζ {1}
n m−j
= ζ(p,{1} )+ζ p+1,{1} + . (4.1)
m m−1 np(n+(cid:16)i )···(n(cid:17)+i )
1 j
(cid:0) (cid:1) Xj=11≤ij<X···<i1≤rnX=1
On the other hand, we consider the expansion
k
1 A
= j (k ∈ N ;a ∈C\Z−) (4.2)
0 i
k n+a
j
(n+a ) j=1
i X
i=1
Q
where
k
n+a
A = lim j = (a −a )−1. (4.3)
j i j
n→−aj k
(n+a ) i=1,i6=j
i Y
i=1
Q
Therefore, the equation (4.1) can be written as
U (p)=ζ(p,{1} )+ζ p+1,{1}
m,r m m−1
m j (cid:0)j (cid:1) ∞ ζ {1}
+ (i −i )−1 n m−j . (4.4)
 a l  n(cid:16)p(n+i )(cid:17)
l
Xj=1 Xl=1a=Y1,a6=l 1≤ij<X···<i1≤rnX=1
 
7
For r > 0, we have the partial fraction decomposition
1 p−1 (−1)b−1 1 (−1)p−1 1
= · + · , (4.5)
np(n+r) rb np+1−b rp−1 n(n+r)
b=1
X
Moreover, from identities (2.1), (2.4) and (2.6), we deduce the following result
∞ ζ {1} ζ⋆ {1}
n p−1 1 k p−1
= ζ(p)+ζ⋆ {1} − , k ∈ N,p ∈ N/{1}. (4.6)
n(cid:16)(n+k)(cid:17) k  k p (cid:16) k (cid:17)
nX=1  (cid:16) (cid:17) 
Hence, combining(4.4), (4.5)and(4.6), byasimplecalculation,weobtain thethedesiredresult.
This completes the proof of Theorem 1.2. (cid:3)
5 Conclusion
From [12, 16], we know that the Aomoto-Drinfeld-Zagier formula reads
∞ ∞ xn+yn−(x+y)n
ζ m+1,{1} xmyn = 1−exp ζ(n) , (5.1)
n−1 n
!
n,m=1 n=2
X (cid:0) (cid:1) X
which implies that for any m, n ∈ N, the multiple zeta value ζ m+1,{1}n−1 can be repre-
sented as a polynomial of zeta values with rational coefficients, an(cid:16)d we have the(cid:17)duality formula
ζ n+1,{1} = ζ m+1,{1} .
m−1 n−1
In particular, one can find exp(cid:0)licit formulas f(cid:1)or sm(cid:0)all weights. (cid:1)
ζ(2,{1} ) = ζ(m+2),
m
m
m+2 1
ζ(3,{1} ) = ζ(m+3)− ζ(k+1)ζ(m+2−k).
m 2 2
k=1
X
Hence, from formulas (1.11) and (5.1), we see that the sums U (p) can be expressed in terms
m,r
of series of Riemann zeta function and harmonic numbers. Thus, we show that the Euler-type
sums with hyperharmonic numbers S(k,r+1;p) can be expressed in terms of zeta values and
Stirling numbers of the first kind, for integers k ∈ N and r ∈N with r+2 ≤ p ∈ N.
0
Acknowledgments. Theauthorswouldliketothanktheanonymousrefereeforhis/herhelpful
comments, which improve the presentation of the paper.
References
[1] David H. Bailey, Jonathan M. Borwein and Roland Girgensohn. Experimental evaluation of Euler
sums. Experimental Mathematics., 1994, 3(1): 17-30.
[2] A.T. Benjamin, D. Gaebler, R. Gaebler. A combinatorial approach to hyperharmonic numbers.
Integers (Elec. J. Combi. Number Theory)., 2003, 3: 1-9.
[3] David Borwein, Jonathan M. Borwein and Roland Girgensohn. Explicit evaluation of Euler sums.
Proc. Edinburgh Math., 1995, 38: 277-294.
8
[4] J.Borwein,P. Borwein, R.Girgensohn, S.Parnes. Making sense of experimental mathematics. Math-
ematical Intelligencer., 1996, 18(4): 12-18.
[5] J.M.Borwein,I.J.Zucker,J.Boersma.TheevaluationofcharacterEulerdoublesums.Ramanujan
J., 2008, 15 (3): 377-405.
[6] J. M. Borwein, R. Girgensohn. Evaluation of triple Euler sums. Electron. J. Combin., 1996: 2-7.
[7] J. H. Conway, R.K. Guy. The book of numbers. Springer-Verlag,New York., 1996: 258-259.
[8] A.Dil,V.Kurt.Polynomialsrelatedtoharmonicnumbersandevaluationofharmonicnumberseries
II. Applicable Analysis and Discrete Mathematics., 2009, 5(2): 212-229.
[9] A. Dil, I. Mez¨o. A symmetric algorithm for hyperharmonic and Fibonacci numbers. Appl. Math.
Comput., 2008, 206: 942-951.
[10] A.Dil,K.N.Boyadzhiev.Eulersumsofhyperharmonicnumbers.JournalofNumber Theory.,2015,
147: 490-498.
[11] PhilippeFlajolet,BrunoSalvy.Eulersumsandcontourintegralrepresentations.ExperimentalMath-
ematics., 1998, 7(1): 15–35.
[12] Masanobu Kaneko, Yasuo Ohno. On a kind of duality of multiple zeta-star values. International
Journal of Number Theory., 2010, 8(8): 1927-1932.
[13] Comtet L. Advanced combinatorics.D Reidel Publishing Company, Boston., 1974.
[14] I. Mez¨o, A. Dil. Hyperharmonic series involving Hurwitz zeta function. J. Number Theory., 2010,
130: 360-369.
[15] I. Mez¨o. Nonlinear Euler sums. Pacific J. Math., 2014, 272: 201-226.
[16] YasuoOhnoandDonZagier.Multiplezetavaluesoffixedweight,depth,andheight.Indag.Mathem.,
N.S. 2001, 12(4): 483-487.
[17] Kh. Hessami Pilehrood, T. Hessami Pilehrood, R. Tauraso. New properties of multiple harmonic
sums modulo p and p-analogues of Leshchiner’s series. Trans. Amer. Math. Soc., 2013, 366(6):
3131-3159.
[18] A. Sofo. Quadratic alternating harmonic number sums. J. Number Theory., 2015, 154: 144-159.
[19] H.M. Srivastava, J. Choi. Zeta and q-Zeta functions and associated series and integrals. Elsevier
Science Publishers., Amsterdam, London and New York, 2012.
[20] Ce Xu, Jinfa Cheng. Some results on Euler sums. Functions et Approximatio., 2016, 54(1): 25-37.
[21] Ce Xu, Yuhuan Yan, Zhijuan Shi. Euler sums and integrals of polylogarithm functions. J. Number
Theory., 2016, 165: 84-108.
[22] CeXu,MingyuZhang,WeixiaZhu.Someevaluationofharmonicnumbersums.IntegralTransforms
and Special Functions., 2016, 27(12): 937-955.
[23] Ce Xu. Multiple zeta values and Euler sums. arXiv:1609.05863.
[24] Ce Xu. Euler sums of generalized hyperharmonic numbers. arXiv:1701.00391.
[25] D. Zagier.Evaluationof the multiple zeta values ζ(2,...,2,3,2,...,2).Annals ofMathematics., 2012,
2(2): 977-1000.
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