Euler sums of generalized hyperharmonic numbers Ce Xu∗ School of Mathematical Sciences, Xiamen University 7 Xiamen 361005,P.R. China 1 0 2 n (m) Abstract Thegeneralizedhyperharmonicnumbersh (k)aredefinedbymeansofthemultiple a n (m) J harmonicnumbers. Weshowthatthehyperharmonicnumbersh (k)satisfycertain recurrence n 2 relation which allow us to write them in terms of classical harmonic numbers. Moreover, we prove that the Euler-type sums with hyperharmonic numbers: ] T N ∞ h(m)(k) n S(k,m;p) := (p ≥ m+1, k = 1,2,3) . np h n=1 t X a m can be expressed as a rational linear combination of products of Riemann zeta values and harmonic numbers. This is an extension of the results of Dil (2015) [11] and Mezo¨ (2010) [17]. [ Some interesting new consequences and illustrative examples are considered. 1 v Keywords Euler sums; generalized hyperharmonic numbers; harmonic numbers; Riemann zeta 1 function; Stirling numbers. 9 3 AMS Subject Classifications (2010): 11B73; 11B83; 11M06; 11M32; 11M99 0 0 . 1 1 Introduction 0 7 1 Let N := {1,2,3,...} be the set of natural numbers, and N0 := N∪{0}, N\{1} := {2,3,4,···}. : Hyperharmonic numbers and their generalizations are classically defined by v i X 1 h(m)(k) := , (1.1) r n n n ···n a 1≤nm+kX−1<···<nm m m+1 m+k−1 ≤nm−1≤···≤n1≤n 1 h(m)(1) ≡ h(m) := , (1.2) n n n m 1≤nm≤X···≤n1≤n where k,m,n ∈ N and for any n < k, we set h(m)(k) := 0. When k = 1 in (1.1), the number n (m) (m) h (1) ≡ h is called classical hyperharmonic number (see [8–11, 17]). In special, the hy- n n (1) perharmonic number h is simply called classical harmonic number, which is the sum of the n reciprocals of the first n natural numbers:: n 1 h(1) ≡ H := . n n k k=1 X ∗ Corresponding author. Email: [email protected] (C. Xu) 1 (m) Moreover, in [17], Mezo¨ and Dil shown that the h can be expressed by binomial coefficients n and classical harmonic numbers: n+m−1 h(m) = (H −H ). n m−1 n+m−1 m−1 (cid:18) (cid:19) (k) The n-th generalized harmonic numbers of order k, denoted by H , is defined by n n 1 H(k) := , n,k ∈ N, (1.3) n jk j=1 X (k) (1) where the empty sum H is conventionally understood to be zero, and H ≡ H . The limit 0 n n as n tends to infinity exists if k > 1. In the limit of n → ∞, the generalized harmonic number converges to the Riemann zeta value: lim H(k) =ζ(k), ℜ(k) >1, k ∈ N, n n→∞ where the Riemann zeta function is defined by ∞ 1 ζ(s):= ,ℜ(s) > 1. (1.4) ns n=1 X In general, for r ∈ N, s := (s ,s ,...,s ) ∈ (N)r, and a non-negative integer n, the multiple 1 2 r harmonic number is defined by 1 H(s1,s2,···,sr) := . (1.5) n ns1ns2···nsr 1≤nr<nrX−1<···<n1≤n 1 2 r (s) (∅) By convention, we put H = 0, if n < r, and H = 1. The limit cases of multiple harmonic n n numbers give rise to multiple zeta values: ζ(s ,s ,··· ,s ) = lim H(s1,s2,···,sr) 1 2 r n n→∞ defined for s ,s ,...,s ≥ 1 and s ≥ 2 to ensure convergence of the series. Here s +···+s is 2 3 r 1 1 r calledtheweightandristhemultiplicity. Tosimplifythereadingofsuchformulas,whenastring of arguments is repeated an exponent is used. In other words, we treat string multiplication as concatenation. For example, 1,··· ,1 2,··· ,2,3,··· ,3     H r  = H({1}r), H p r  = H({2}p,{3}r). n n n n | {z } | {z }| {z } (m) With this notation, then the definition of hyperharmonic number h (k) of formula (1.1) can n be rewritten as ({1}k−1) H h(m)(k):= nm−1 . (1.6) n n m 1≤nm≤nmX−1≤···≤n1≤n The subject of this paper is Euler-type sums S(k,m;p), which is the infinite sum whose general term is a productof hyperharmonicnumbersand a power of n−1. Here, p > m is both necessary 2 and sufficient for the sum S(k,m;p) to converge. The classical linear Euler sum is defined by ([12]) ∞ (p) H S := n , p ∈ N,q ∈ N\{1}. (1.7) p,q nq n=1 X The number w = p+q is defined as the weight of S . The evaluation of S in terms of values p,q p,q of Riemann zeta function at positive integers is known when p = 1, p = q, (p,q) = (2,4),(4,2) or p+q is odd (see [1, 3, 6, 12]). For example, Euler discovered the following formula ∞ k−2 H 1 n S = = (k+2)ζ(k+1)− ζ(k−i)ζ(i+1) . (1.8) 1,k nk 2 ( ) n=1 i=1 X X RelatedserieswerestudiedbyMezo¨in[18],Sofo[20]andXu.etal[21,24],forinstance. Similarly, it has been discovered in the course of the years that many Euler type sums S(k,m;p) admit expressions involving finitely the zeta values, that is to say values of the Riemann zeta function at the positive integer arguments, for more details, see for instance [11, 17]. For example, Dil and Boyadzhiev [11] gave explicit reductions to zeta values and (unsigned) Stirling numbers of the first kind for all sums S(k,m;p) with k = 1. Here the (unsigned) Stirling number of the n first kind is defined by [8, 15] k (cid:20) (cid:21) n x x n+1 n!x(1+x) 1+ ··· 1+ = xk+1, (1.9) 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). (1.10) k n! n=1(cid:20) (cid:21) X n Moreover, the (unsigned) Stirling numbers of the first kind satisfy a recurrence relation in k (cid:20) (cid:21) the form n n−1 n−1 = +(n−1) . (1.11) k k−1 k (cid:20) (cid:21) (cid:20) (cid:21) (cid:20) (cid:21) n By the definition of , we see that we may rewrite (1.7) as k (cid:20) (cid:21) n n n+1 x xk = n!exp ln 1+ k+1  j  Xk=0(cid:20) (cid:21) Xj=1 (cid:18) (cid:19)  n ∞ xk = n!exp (−1)k−1  kjk Xj=1Xk=1  ∞ H(k)xk = n!exp (−1)k−1 n . k ( ) k=1 X 3 n Therefore, we know that is a rational linear combination of products of harmonic numbers. k (cid:20) (cid:21) Moreover, we deduce the following identities n n n (n−1)! = (n−1)!, = (n−1)!H , = H2 −H(2) , 1 2 n−1 3 2 n−1 n−1 (cid:20) (cid:21) (cid:20) (cid:21) (cid:20) (cid:21) h i n (n−1)! = H3 −3H H(2) +2H(3) , 4 6 n−1 n−1 n−1 n−1 (cid:20) (cid:21) h i n (n−1)! = H4 −6H(4) −6H2 H(2) +3(H(2) )2+8H H(3) . 5 24 n−1 n−1 n−1 n−1 n−1 n−1 n−1 (cid:20) (cid:21) h i In this paper we are interested in Euler-type sums with hyperharmonic numbers S(k,m;p). Such series could be of interest in analytic number theory. We will prove that the generalized (m) hyperharmonic number h (k) can be expressed as a rational linear combination of products n of harmonic numbers. Furthermore, we also provide an explicit evaluation of S(k,m;p) with k = 2,3 in a closed form in terms of zeta values and Stirling numbers of the first kind. The results which we present here can be seen as an extension of Mezo¨’s and Dil’s work. 2 Main Theorems and Proof (m) In this section, we will show that the hyperharmonic number h (k) is expressible in terms of n harmonic numbers and give recurrence formula. We need the following lemma. Lemma 2.1 For positive integers n and k, then the following identity holds: n ({1}k−1) = (n−1)!H . (2.1) k n−1 (cid:20) (cid:21) Proof. By considering the generating function (1.10), we know that we need to prove the following identity logk(1−x)= (−1)kk! ∞ H({1}k−1)xn. (2.2) n−1 n n=1 X To prove the identity we proceed by induction on k. Obviously, it is valid for k = 1. For k > 1 we use the integral identity x logk(1−t) logk+1(1−x)=−(k+1) dt 1−t Z 0 and apply the induction hypothesis, by using Cauchy product of power series, we arrive at x logk(1−t) logk+1(1−x)=−(k+1) dt 1−t Z 0 ∞ n ({1}k−1) 1 H = (−1)k+1(k+1)! i−1 xn+1 n+1 i n=1 i=1 X X 4 ∞ ({1}k) H = (−1)k+1(k+1)! n xn+1. n+1 n=1 X ({1}k) Nothing that H = 0 when n < k. Hence, we can deduce (2.2) holds. Thus, comparing the n coefficients of xn in (1.10) and (2.2), we obtain formula (2.1). The proof of lemma 2.1 is thus completed. (cid:3) (m) By using(1.6) and(2.2), we findthat thegenerating functionof hyperharmonicnumberh (k) n is given as ∞ (−1)k logk(1−z) h(m)(k)zn = , z ∈ [−1,1). (2.3) n k! (1−z)m n=1 X On the other hand, we note that the function on the right hand side of (2.3) is equal to (−1)k logk(1−z) 1 ∂k 1 = lim (k,m ∈ N ). (2.4) k! (1−z)m k! x→m∂xk (1−z)x 0 (cid:18) (cid:19) Therefore, the relations (2.3) and (2.4) yield the following result: ∞ 1 ∂k 1 h(m)(k)zn = lim . (2.5) n k! x→m∂xk (1−z)x n=1 (cid:18) (cid:19) X Moreover, we know that the generating function of (1−z)−x is given as ∞ 1 (x) = nzn, z ∈(−1,1), (2.6) x (1−z) n! n=0 X where (x) represents the Pochhammer symbol (or the shifted factorial) given by n (x) := x(x+1)···(x+n−1), (2.7) n with (x) := 1. Hence, upon differentiating both members of (2.6) k times with respect to x 0 then setting x = m, and combining (2.5), we readily arrive at the following relationship: 1 ∂k(x) h(m)(k) = lim n, k ∈ N. (2.8) n k!n!x→m ∂xk By convention, from (2.8), we define that 1 m+n−1 h(m)(0) := (x) = . (2.9) n n! m m−1 (cid:18) (cid:19) ∂k(x) By simple calculation, the n satisfy a recurrence relation in the form ∂xk ∂k(x) k−1 k−1 ∂i(x) n = n ψ(m−k−1)(x+n)−ψ(m−k−1)(x) , k ∈ N. (2.10) ∂xk i ∂xi Xi=0(cid:18) (cid:19) h i Here ψ(m)(x) stands for the polygamma function of order m defined as the (m+1)th derivative of the logarithm of the gamma function: dm dm ψ(m)(x) := ψ(x)= logΓ(x), dxm dxm 5 Thus Γ′(x) ψ(0)(x) = ψ(x) = Γ(x) holds where ψ(x) is the digamma function and Γ(x) is the gamma function. ψ(m)(x) satisfy the following relations ∞ 1 1 ψ(z) = −γ+ − , z ∈/ N− := {0,−1,−2...}, n+1 n+z 0 n=0(cid:18) (cid:19) X ∞ ψ(n)(z) = (−1)n+1n! 1/(z+k)n+1,n ∈ N, k=0 X 1 1 1 ψ(x+n) = + +···+ +ψ(x), n ∈ N. x x+1 x+n−1 Hence, combining (2.8), (2.9) and (2.10), we obtain the recurrence relation (−1)k−1 k−1 h(m)(k) = (−1)ih(m)(i) H(k−i) −H(k−i) . (2.11) n k n m+n−1 m−1 Xi=0 n o (m) By (2.11), we give the following description of hyperharmonic number h (k). n (m) Theorem 2.2 For positive integers n and k, then the hyperharmonic number h (k) can be n expressed in terms of ordinary harmonic numbers. For example, setting m = 1,2,3,4 in the above equation (2.11) we obtain n+m−1 h(m)(1) = (H −H ), (2.12) n m−1 n+m−1 m−1 (cid:18) (cid:19) 1 n+m−1 h(m)(2) = (H −H )2− H(2) −H(2) , (2.13) n 2 m−1 n+m−1 m−1 n+m−1 m−1 (cid:18) (cid:19) n (cid:16) (cid:17)o h(m)(3) = 1 n+m−1 (Hn+m−1−Hm−1)3+2 Hn(3+)m−1−Hm(3−)1 , (2.14) n 3! m−1 −3(H −H ) H(cid:16)(2) −H(2) (cid:17) (cid:18) (cid:19) n+m−1 m−1 n+m−1 m−1  (H −H )4 (cid:16) (cid:17)  n+m−1 m−1  2 (2) (2) (4) (4) 1 n+m−1 +3 Hn+m−1−Hm−1 −6 Hn+m−1−Hm−1  h(m)(4) =  . (2.15) n 4! (cid:18) m−1 (cid:19)−6(cid:16)(Hn+m−1−Hm−1)(cid:17)2 Hn(2+(cid:16))m−1−Hm(2−)1 (cid:17) +8(H −H ) (cid:16)H(3) −H(3) (cid:17) n+m−1 m−1 n+m−1 m−1      (cid:16) (cid:17)    By replacing x by n and n by rin (1.9), we deduce that  r+1 n+r 1 r+1 = nk−1. (2.16) r r! k (cid:18) (cid:19) k=1(cid:20) (cid:21) X Therefore, the relations (2.13), (2.14) and (2.16) yield the following results 1 n+r h(r+1)(2) = (H −H )2− H(2) −H(2) n 2 r n+r r n+r r (cid:18) (cid:19) n (cid:16) (cid:17)o 6 r+1 1 r+1 = nk−1 (H −H )2− H(2) −H(2) , (2.17) 2!r! k n+r r n+r r Xk=1(cid:20) (cid:21) n (cid:16) (cid:17)o h(r+1)(3) =1 n+r (Hn+r −Hr)3+2 Hn(3+)r −Hr(3) n 3! r −3(H −H ) H(cid:16)(2) −H(2) (cid:17) (cid:18) (cid:19) n+r r n+r r  = 1 r+1 r+1 nk−1 (Hn+r −(cid:16)Hr)3+2 H(cid:17)n(3+)r−Hr(3) , (2.18) 3!r! k −3(H −H ) H(cid:16)(2) −H(2) (cid:17) Xk=1(cid:20) (cid:21)  n+r r n+r r  (cid:16) 2 (cid:17) (Hn+r −Hr)4+3 Hn(2+)r −Hr(2)  h(r+1)(4) =1 n+r −6 Hn(4+)r −Hr(4)(cid:16) (cid:17)  n 4! (cid:18) r (cid:19)−6((cid:16)Hn+r −Hr)2 (cid:17)Hn(2+)r −Hr(2)  +8(H −H ) (cid:16)H(3) −H(3) (cid:17)  n+r r n+r r       (H −(cid:16)H )4+3 H(cid:17)(2) −H(2) 2 n+r r n+r r = 1 r+1 r+1 nk−1−6 Hn(4+)r −Hr(4)(cid:16) (cid:17) . (2.19) 4!r! Xk=1(cid:20) k (cid:21) −6((cid:16)Hn+r −Hr)2 (cid:17)Hn(2+)r −Hr(2)  +8(H −H ) (cid:16)H(3) −H(3) (cid:17)  n+r r n+r r       (cid:16) (cid:17)    Furthermore, using(2.17) and (2.18), by adirectcalculation, wecan give thefollowing corollary. Corollary 2.3 For integers r ∈N and n∈ N, we have 0 r+1 1 r+1 h(r+1)(2) = nk−1 H2 −H(2) −2H H +H2+H(2) , (2.20) n 2!r! k n+r n+r r n+r r r Xk=1(cid:20) (cid:21) n(cid:16) (cid:17) o h(r+1)(3) = 1 r+1 r+1 nk−1 Hn3+r −3Hn+rHn(2+)r +2Hn(3+)r −3Hr Hn2+r −Hn(2+)r . n 3!r! k +(cid:16)3 H2+H(2) H − H3+(cid:17)3H H(2(cid:16))+2H(3) (cid:17) Xk=1(cid:20) (cid:21)  r r n+r r r r r  (cid:16) (cid:17) (cid:16) (cid:17) (2.21)   (k) Moreover, from the definition of harmonic numbers H , we get n r 1 H(k) = H(k)+ , k,n ∈ N. (2.22) n+r n k (n+j) j=1 X By simple calculation, the following identities are easily derived 2 r r 1 1 H2 −H(2) = H + − H(2)+ n+r n+r  n n+j  n (n+j)2 j=1 j=1 X X     r 1 1 =H2−H(2)+2H +2 , (2.23) n n n n+j (n+i)(n+j) j=1 1≤i<j≤r X X   7 H3 −3H H(2) +2H(3) n+r n+r n+r n+r 3 r r 1 1 = H + +2 H(3)+  n n+j  n (n+j)3 j=1 j=1 X X     r r 1 1 −3 H + H(2) +  n n+j n (n+j)2 j=1 j=1 X X    r 1 = H3−3H H(2)+2H(3)+3 H2−H(2) n n n n n n  n+j (cid:16) (cid:17) Xj=1   1 1 +6H +6 . (2.24) n  (n+i)(n+j) (n+i)(n+j)(n+k) 1≤i<j≤r 1≤i<j<k≤r X X   Now, we use the notation W (m) to stands for the sum k,r n+r+1 ∞ k W (m):= (k−1)! (cid:20) (cid:21). (2.25) k,r (n+r)!nm n=1 X where k ∈ N, r ∈ N and m ∈ N\{1}. By using the above notation, we obtain 0 W (m) = ζ(m), 1,r ∞ H n+r W (m) = , 2,r nm n=1 X ∞ H2 −H(2) W (m) = n+r n+r, 3,r nm n=1 X ∞ H3 −3H H(2) +2H(3) W (m) = n+r n+r n+r n+r. 4,r nm n=1 X Next, we give a lemma. The following lemma will be useful in the development of the main theorems. Noting that when r = 0 and k > 1 in (2.25), then using (1.11), which can be rewritten as n+1 ∞ k W (m) := (k−1)! (cid:20) (cid:21) k,0 n!nm n=1 X n n ∞ ∞ k−1 k = (k−1)! (cid:20) (cid:21) + (cid:20) (cid:21)  n!nm n!nm−1 n=1 n=1 X X      = (k−1)! ζ m+1,{1}k−2 +ζ m,{1}k−1 . (2.26) (cid:16) (cid:16) (cid:17) (cid:16) (cid:17)(cid:17) On the other hand, the Aomoto-Drinfeld-Zagier formula reads ∞ ∞ xn+yn−(x+y)n ζ m+1,{1}n−1 xmyn = 1−exp ζ(n) , (2.27) n ! nX,m=1 (cid:16) (cid:17) nX=2 8 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}m−1 = ζ m+1,{1}n−1 . (cid:16) (cid:17) (cid:16) (cid:17) In particular, one can find explicit formulas for small weights. m ζ(2,{1} ) = ζ(m+2), m m+2 1 m ζ(3,{1} ) = ζ(m+3)− ζ(k+1)ζ(m+2−k). 2 2 k=1 X Hence, we know that for m, k ∈ N ,the sums W (m) can be expressed as a rational linear k,0 combination of zeta values. For example, from [12, 23] m−2 1 W (m)= (m+2)ζ(m+1)− ζ(m−i)ζ(i+1) , 2,0 2 ( ) i=1 X m(m+1) W (m)= mW (m+1)− ζ(m+2)+ζ(2)ζ(m). 3,0 2,0 6 Lemma 2.4 ([24]) For integers k ∈ N and p ∈ N\{1}, 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 , (2.28) 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 [15]) (cid:17) tm tk exp x = 1+ Y (x ,x ,···) . (2.29) m k 1 2  m! k! m≥1 k≥1 X X   From the definition of the complete exponential Bell polynomial, we have Y (n)= H ,Y (n) =H2+H(2),Y (n)= H3+3H H(2) +2H(3), 1 n 2 n n 3 n n n n Y (n) =H4+8H H(3) +6H2H(2)+3(H(2))2+6H(4), 4 n n n n n n n Y (n)= H5+10H3H(2) +20H2H(3) +15H (H(2))2+30H H(4)+20H(2)H(3) +24H(5), 5 n n n n n n n n n n n n In fact, Y (n) is a rational linear combination of products of harmonic numbers. Putting k p = 2,3,4 in (2.28), we obtain the corollary. Corollary 2.5 For integer k > 0, we have ∞ H 1 1 1 H n = H2+ H(2) +ζ(2)− k . (2.30) n(n+k) k 2 k 2 k k n=1 (cid:18) (cid:19) X 9 ∞ H2−H(2) 1 H3+3H H(2)+2H(3) H2+H(2) n n = 2ζ(3)+ k k k k − k k , (2.31) n(n+k) k 3 k ( ) n=1 X H4+8H H(3)+6H2H(2)+3(H(2))2+6H(4) ∞ H3−3H H(2) +2H(3) 1 k k k k k k k n n n n =  4 . (2.32) nX=1 n(n+k) k −Hk3+3HkHk(2) +2Hk(3) +6ζ(4)  k     Hence, combining (2.22)-(2.25), (2.30) and (2.31), we deduce the following identities ∞ r ∞ H 1 n W (m)= + 2,r nm nm(n+j) n=1 j=1n=1 X XX m−1 r H = W (m)+ (−1)l−1ζ(m+1−l)H(l)+(−1)m−1 j, (2.33) 2,0 r jm l=1 j=1 X X ∞ H2−H(2) r ∞ H ∞ 1 W (m)= n n +2 n +2 3,r nm nm(n+j) nm(n+i)(n+j) n=1 j=1n=1 1≤i<j≤rn=1 X XX X X m−1 = W (m)+2 (−1)i−1H(i)W (m+1−i) 3,0 r 2,0 i=1 X m−1 1 +2 (−1)l−1ζ(m+1−l) il(j −i) l=1 1≤i<j≤r X X m−1 1 −2 (−1)l−1ζ(m+1−l) jl(j−i) l=1 1≤i<j≤r X X r H2+H(2) r H j j +2ζ(2)H(m)−2 j +(−1)m−1j=1 jm r j=1 jm+1 , (2.34) X X  +2 Hi −2 Hj  im(j −i) jm(j −i) 1≤i<j≤r 1≤i<j≤r  X X   r ∞ H2−H(2) ∞ H W (m)= W (m)+3 n n +6 n 4,r 4,0 nm(n+j) nm(n+i)(n+j) j=1n=1 1≤i<j≤rn=1 XX X X ∞ 1 +6 , nm(n+i)(n+j)(n+k) 1≤i<j<k≤rn=1 X X m−1 = W (m)+3 (−1)l−1H(l)W (m+1−l) 4,0 r 3,0 l=1 X m−1 1 1 1 +6 (−1)l−1W (m+1−l) − 2,0 j −i il jl l=1 1≤i<j≤r (cid:18) (cid:19) X X 10