Computation and theory of Euler sums of generalized hyperharmonic numbers

TL;DR

This paper derives explicit formulas for sums of generalized hyperharmonic numbers, expressing them in terms of multiple zeta values, harmonic numbers, and Stirling numbers, extending previous results on harmonic number sums.

Contribution

It introduces new explicit formulas for sums involving generalized hyperharmonic numbers with complex index sequences, connecting them to multiple zeta values and Stirling numbers.

Findings

Expressed sums in terms of multiple zeta values and harmonic numbers

Extended previous formulas to more complex index sequences

Provided explicit formulas for generalized hyperharmonic sums

Abstract

Recently, Dil and Boyadzhiev \cite{AD2015} proved an explicit formula for the sum of multiple harmonic numbers whose indices are the sequence $\left( {{{\left\{ 0 \right\}}_r},1} \right)$. In this paper we show that the sums of multiple harmonic numbers whose indices are the sequence $\left( {{{\left\{ 0 \right\}}_r,1};{{\left\{ 1 \right\}}_{k-1}}} \right)$ can be expressed in terms of (multiple) zeta values, multiple harmonic numbers and Stirling numbers of the first kind, and give an explicit formula.

Equations

$${\zeta }_n\left(s_1,s_2,\cdots ,s_m\right):=\sum _{n\ge n_1>\cdots >n_m\ge 1}\frac{1}{n_1^{s_1}{n}_2^{s_2}\cdots n_m^{s_m}},$$

$${\zeta }_n^{\star }\left(s_1,s_2,\cdots ,s_m\right):=\sum _{n\ge n_1\ge \cdots \ge n_m\ge 1}\frac{1}{n_1^{s_1}{n}_2^{s_2}\cdots n_m^{s_m}},$$

$$\zeta \left(s_1,s_2,\cdots ,s_m\right)=\underset{n\to \infty }{\lim }\zeta \left(s_1,s_2,\cdots ,s_m\right),$$

$${\zeta }^{\star }\left(s_1,s_2,\cdots ,s_m\right)=\underset{n\to \infty }{\lim }{\zeta }^{\star }\left(s_1,s_2,\cdots ,s_m\right)$$

$${\zeta }_n\left(s_1,\cdots ,s_m;s_{m+1},\cdots ,s_{m+k}\right):=\sum _{\genfrac{}{}{0ex}{}{0<n_{m+k}<\cdots <n_{m+1}\text{hfill}}{<n_m\le n_{m-1}\cdots \le n_1\le n\text{hfill}}}\frac{1}{n_1^{s_1}\cdots n_{m+k}^{s_{m+k}}}.$$

$${\zeta }_n\left(\varnothing ;s_1,\cdots ,s_k\right)={\zeta }_n\left(s_1,\cdots ,s_k\right),$$

$${\zeta }_n\left(s_1,\cdots ,s_m;\varnothing \right)={\zeta }_n^{\star }\left(s_1,\cdots ,s_m\right).$$

$$S_{\mathbf{S},q}:=\sum _{n=1}^{\infty }\frac{H_n^{\left(s_1\right)}{H}_n^{\left(s_2\right)}\cdots H_n^{\left(s_r\right)}}{n^q},$$

$$H_n^{\left(p\right)}\equiv {\zeta }_n(p):=\sum _{j=1}^n\frac{1}{j^p},p,n\in \mathbb{N}.$$

$$\zeta (s):=\sum _{n=1}^{\infty }\frac{1}{n^s},\Re (s)>1.$$

$$h_n^{\left(m\right)}:=\sum _{1\le n_m\le \cdots \le n_1\le n}\frac{1}{n_m}={\zeta }_n^{\star }\left({\left\{0\right\}}_{m-1},1\right).$$

$$h_n^{\left(m\right)}\left(k\right):=\sum _{\genfrac{}{}{0ex}{}{1\le n_{m+k-1}<\cdots <n_{m+1}\text{hfill}}{<n_m\le n_{m-1}\le \cdots \le n_1\le n\text{hfill}}}\frac{1}{n_m{n}_{m+1}\cdots n_{m+k-1}}={\zeta }_n\left({\left\{0\right\}}_{m-1},1;{\left\{1\right\}}_{k-1}\right),$$

$$S\left(k,m;p\right):=\sum _{n=1}^{\infty }\frac{h_n^{\left(m\right)}\left(k\right)}{n^p}\left(p\ge m+1\right)$$

$$S\left({k,r+1;p}\right)=\displaystyle\frac{1}{{r!}}\displaystyle\sum\limits_{l=1}^{r+1}{\left[{\begin{array}[]{*{20}{c}}{r+1}\\ l\\ \end{array}}\right]}\displaystyle\sum\limits_{\scriptstyle i+j=k\hfill\atop\scriptstyle i,j\geq 0\hfill}{{{\left({-1}\right)}^{i}}{\zeta_{r}^{\star}}\left({{{\left\{1\right\}}_{i}}}\right){U_{j,r}}\left({p+1-l}\right)},$$

$$n!x\left({1+x}\right)\left({1+\displaystyle\frac{x}{2}}\right)\cdots\left({1+\displaystyle\frac{x}{n}}\right)=\displaystyle\sum\limits_{k=0}^{n}{\left[{\begin{array}[]{*{20}{c}}{n+1}\\ {k+1}\\ \end{array}}\right]{x^{k+1}}},$$

$${\log^{k}}\left({1-x}\right)={\left({-1}\right)^{k}}k!\displaystyle\sum\limits_{n=1}^{\infty}{\left[{\begin{array}[]{*{20}{c}}n\\ k\\ \end{array}}\right]\displaystyle\frac{{{x^{n}}}}{{n!}}},\;x\in\left[{-1,1}\right).$$

$$U_{j,r}\left(p\right):=\sum _{n=1}^{\infty }\frac{{\zeta }_{n+r}\left({\left\{1\right\}}_j\right)}{n^p}.$$

$$U_{m,r}\left(p\right)=\zeta \left(p,{\left\{1\right\}}_m\right)+\zeta \left(p+1,{\left\{1\right\}}_{m-1}\right)$$

$$+{\left(-1\right)}^{p-1}\sum _{1\le i_m<\cdots <i_1\le r}\sum _{l=1}^m\prod _{\genfrac{}{}{0ex}{}{a=1\text{hfill}}{a\ne l\text{hfill}}}^m\frac{H_{i_l}}{\left(i_a-i_l\right)i_l^p}$$

$$+\sum _{1\le i_m<\cdots <i_1\le r}\sum _{b=1}^{p-1}\sum _{l=1}^m\prod _{\genfrac{}{}{0ex}{}{a=1\text{hfill}}{a\ne l\text{hfill}}}^m\frac{\zeta \left(p+1-b\right)}{\left(i_a-i_l\right)i_l^b}{\left(-1\right)}^{b-1}$$

$$+{\left(-1\right)}^{p-1}\sum _{j=1}^{m-1}\sum _{1\le i_j<\cdots <i_1\le r}\sum _{l=1}^j\prod _{\genfrac{}{}{0ex}{}{a=1\text{hfill}}{a\ne l\text{hfill}}}^j\frac{\zeta \left(m-j+1\right)+{\zeta }_{i_l}^{\star }\left({\left\{1\right\}}_{m-j+1}\right)-\frac{{\zeta }_{i_l}^{\star }\left({\left\{1\right\}}_{m-j}\right)}{i_l}}{\left(i_a-i_l\right)i_l^p}$$

$$+\sum _{j=1}^{m-1}\sum _{1\le i_j<\cdots <i_1\le r}\sum _{b=1}^{p-1}\sum _{l=1}^j\prod _{\genfrac{}{}{0ex}{}{a=1\text{hfill}}{a\ne l\text{hfill}}}^j\frac{\zeta \left(p+1-b,{\left\{1\right\}}_{m-j}\right)+\zeta \left(p+2-b,{\left\{1\right\}}_{m-j-1}\right)}{\left(i_a-i_l\right)i_l^b}{\left(-1\right)}^{b-1},$$

$$\left[{\begin{array}[]{*{20}{c}}n\\ k\\ \end{array}}\right]=\left({n-1}\right)!{\zeta_{n-1}}\left({{{\left\{1\right\}}_{k-1}}}\right).$$

$$h_n^{\left(m\right)}\left(k\right)=\frac{{\left(-1\right)}^{k-1}}{k}\sum _{i=0}^{k-1}{\left(-1\right)}^i{h}_n^{\left(m\right)}\left(i\right)\left\{H_{m+n-1}^{\left(k-i\right)}-H_{m-1}^{\left(k-i\right)}\right\},$$

$$h_{n}^{\left(m\right)}\left(0\right)=\left({\begin{array}[]{*{20}{c}}{m+n-1}\\ {m-1}\\ \end{array}}\right).$$

$${\bar{B}}_m\left(n\right)=\frac{{\left(-1\right)}^{m-1}}{m}\sum _{i=0}^{m-1}{\left(-1\right)}^i{\bar{B}}_i\left(n\right)X_n\left(m-i\right),$$

$$X_n\left(m\right):=\sum _{i=1}^n{x}_i^m,x_i\in \mathbb{C},$$

$${\bar{B}}_m(n):=\underset{k_1=1}{\sum ^n}{x}_{k_1}\underset{k_2=1}{\sum ^{k_1-1}}{x}_{k_2}\cdots \underset{k_m=1}{\sum ^{k_{m-1}-1}}{x}_{k_m},{\bar{B}}_0(n)=1.$$

$$\left({p-1}\right)!\displaystyle\sum\limits_{n=1}^{\infty}{\displaystyle\frac{{\left[{\begin{array}[]{*{20}{c}}{n+1}\\ p\\ \end{array}}\right]}}{{n!n\left({n+k}\right)}}}=\displaystyle\frac{1}{k}\left\{{\left({p-1}\right)!\zeta(p)+\displaystyle\frac{{{Y_{p}}\left(k\right)}}{p}-\displaystyle\frac{{{Y_{p-1}}\left(k\right)}}{k}}\right\},$$

$$\exp \left(\sum _{m\ge 1}{x}_m\frac{t^m}{m!}\right)=1+\sum _{k\ge 1}{Y}_k\left(x_1,x_2,\cdots \right)\frac{t^k}{k!}.$$

$${\zeta }_n^{\star }\left({\{1\}}_m\right)=\frac{1}{m!}{Y}_m\left(n\right),n,m\in {\mathbb{N}}_0.$$

$$\sum _{\genfrac{}{}{0ex}{}{i+j=m\text{hfill}}{i,j\ge 0\text{hfill}}}{\left(-1\right)}^i{\bar{A}}_i\left(n\right)A_j\left(n\right)=0,$$

$${\bar{A}}_m\left(n\right):=\sum _{1\le k_m<\cdots <k_1\le n}{a}_{k_1}\cdots a_{k_m},a_k\in \mathbb{C},$$

Full text

1. 1 Introduction
2. 2 Some Lemmas and Theorems
3. 3 Proof of Theorem 1.1
4. 4 Proof of Theorem 1.2
5. 5 Conclusion

1 Introduction

Let $s_{1},\ldots,s_{m}$ be positive integers. The classical multiple harmonic numbers (MHNs) and multiple harmonic star numbers (MHSNs) are defined by the partial sums (see [18, 24])

[TABLE]

[TABLE]

when $n<m$, then ${\zeta_{n}}\left({{s_{1}},{s_{2}},\cdots,{s_{m}}}\right)=0$, and ${\zeta_{n}}\left(\emptyset\right)={\zeta^{\star}_{n}}\left(\emptyset\right)=1$. The limit cases of MHNs and MHSNs give rise to multiple zeta values (MZVs) and multiple zeta star values (MZSVs) (see [13, 17, 18, 24, 26]):

[TABLE]

defined for $s_{2},\ldots,s_{m}\geq 1$ and $s_{1}\geq 2$ to ensure convergence of the series. For non-negative integers $s_{1},\ldots,s_{m+k}$, we define the following a generalized multiple harmonic numbers

[TABLE]

Obviously, if $m=0$ or $k=0$ in (1.3) and $s_{j}\in\mathbb{N}:=\{1,2,\cdots\}$, then

[TABLE]

There are a lot of recent contributions on MZVs and MZSVs (for example, see [13, 17, 18, 24, 26]). The earliest results on MZVs or MZSVs are due to Euler who elaborated a method to reduce double sums $\zeta\left({{s_{1}},{s_{2}}}\right)$ (also called linear Euler sums [11, 22]) of small weight to certain rational linear combinations of products of zeta values. In [11], Flajolet and Salvy introduced the following generalized series

[TABLE]

which is called the generalized (nonlinear) Euler sums. Here ${\bf S}:=(s_{1},s_{2},\ldots,s_{r})\ (r,s_{i}\in\mathbb{N},i=1,2,\ldots,r)$ with $s_{1}\leq s_{2}\leq\ldots\leq s_{r}$ and $q\geq 2$. The quantity $w:={s_{1}}+\cdots+{s_{r}}+q$ is called the weight and the quantity $r$ is called the degree. The notation $H_{n}^{\left(p\right)}$ denotes the ordinary harmonic numbers defined by

[TABLE]

It has been discovered in the course of the years that many nonlinear Euler sums admit expressions involving finitely ”zeta values”, that is say values of the Riemann zeta function,

[TABLE]

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, 4, 5, 6, 8, 11, 16, 19, 20, 21, 23, 22] and references therein.

From [7, 9, 10, 15, 12], we know that the classical hyperharmonic numbers are defined by

[TABLE]

In [25], we define the generalized hyperharmonic numbers $h_{n}^{\left(m\right)}(k)$ by

[TABLE]

where $m,k\in\mathbb{N}$  (The notation $\{\}_{p}$ means that the sequence in the bracket is repeated $p$ times). In this paper, we prove the result: for positive integers $m$ and $k$, the Euler-type sums with hyperharmonic numbers

[TABLE]

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 [25]. The purpose of the present paper is to prove the following theorems.

Theorem 1.1

For integers $k\in\mathbb{N}$ and $r\in\mathbb{N}_{0}$ with $r+2\leq p\in\mathbb{N}$, then the following identity holds:

[TABLE]

where \left[{\begin{array}[]{*{20}{c}}n\\ k\\ \end{array}}\right] denotes the (unsigned) Stirling number of the first kind, which is defined by [7, 14]

[TABLE]

with \left[{\begin{array}[]{*{20}{c}}n\\ k\\ \end{array}}\right]=0, if $n<k$ and \left[{\begin{array}[]{*{20}{c}}n\\ 0\\ \end{array}}\right]=\left[{\begin{array}[]{*{20}{c}}0\\ k\\ \end{array}}\right]=0,\ \left[{\begin{array}[]{*{20}{c}}0\\ 0\\ \end{array}}\right]=1, or equivalently, by the generating function:

[TABLE]

and

[TABLE]

Theorem 1.2

For integers $m>0,p>1$ and $r\in\mathbb{N}_{0}:=\mathbb{N}\cup\{0\}$, we have

[TABLE]

where $H_{i_{l}}$ is harmonic number.

It is clear that the Theorem 1.2 implies that the sums ${U_{m,r}}\left(p\right)$ can be expressed in terms of series of Riemann zeta function and harmonic numbers.

2 Some Lemmas and Theorems

To prove the Theorem 1.1 and Theorem 1.2, we need the following lemmas.

Lemma 2.1

([25]) For positive integers $n$ and $k$, then the following identity holds:

[TABLE]

Lemma 2.2

([25]) For positive integers $m,n$ and $k$, we have the recurrence relation

[TABLE]

where

[TABLE]

Lemma 2.3

([24]) For positive integers $m$ and $n$, then the recurrence relation holds:

[TABLE]

where

[TABLE]

[TABLE]

Lemma 2.4

([23, 25]) For integers $k\in\mathbb{N}$ and $p\in\mathbb{N}\setminus\{1\}:=\{2,3,\ldots\}$, then the following identity holds:

[TABLE]

where ${Y_{k}}\left(n\right)={Y_{k}}\left({{H_{n}},1!{H^{(2)}_{n}},2!{H^{(3)}_{n}},\cdots,\left({r-1}\right)!{H^{(r)}_{n}},\cdots}\right)$, ${Y_{k}}\left({{x_{1}},{x_{2}},\cdots}\right)$ stands for the complete exponential Bell polynomial defined by (see [14])

[TABLE]

Noting that, in [22], we find the relation

[TABLE]

Lemma 2.5

For positive integers $m$ and $n$, then

[TABLE]

where

[TABLE]

[TABLE]

For convenience, we set ${{\bar{A}}_{0}}\left(n\right)={A_{0}}\left(n\right)=1$. If $n<m$, we let ${{\bar{A}}_{m}}\left(n\right)=0$.

Proof. By a direct calculation, the following identities are easily derived

[TABLE]

Hence, by using Cauchy product of power series, we have

[TABLE]

Thus, comparing the coefficients of $t^{m}$ in above equation, we obtain the formula (2.7). The proof of Theorem 2.3 is finished. $\square$

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 $r\geq 0$ and $m,n>1$, then

[TABLE]

where

[TABLE]

Proof. The proof is by induction on $m$. For $m=1$ we have $\displaystyle\sum\limits_{1\leq{k_{1}}\leq n}{\displaystyle\frac{1}{{{k_{1}}+r}}}={\zeta_{n+r}}\left(1\right)-\zeta_{r}^{\star}\left(1\right)$, and the formula is true. For $m>1$ we proceed as follow. Let

[TABLE]

[TABLE]

Then by the definition (2.9) and the induction hypothesis, we have that

[TABLE]

On the other hand, from Lemma 2.5, setting ${a_{k}}=\displaystyle\frac{1}{k}$ and $n=r$ we get

[TABLE]

Hence, combining (2.10) and (2.11) we can prove that the formula (2.8) holds. $\square$

Similarly, by a similar argument as in the proof of Theorem 2.6 with the help of formula (5.2) in the reference [24], we obtain the more general theorem.

Theorem 2.7

For integers $r\geq 0,m,n>1$ and real $p>0$, then

[TABLE]

Remark 2.1

In fact, in the same way as above, the results of Theorem 2.6 and 2.7 can be extended to the following generalized conclusion.

[TABLE]

where ${{A}_{m}}\left(n\right)$ and ${{\bar{A}}_{m}}\left(n\right)$ are defined in Lemma 2.5. It is clear that Lemma 2.5 and Theorem 2.7 are immediate corollaries of Remark 2.1.

Theorem 2.8

For integers $r\geq 0$ and $m,n\geq 1$, then the following identity holds:

[TABLE]

Proof. In Lemma 2.3, taking ${x_{j}}=\displaystyle\frac{1}{{j+r}}$, then we have

[TABLE]

From Lemma 2.1 and formula (2.15) we deduce that

[TABLE]

Substituting (2.8) into (2.16) we may easily obtain the desired result. This completes the proof of Theorem 2.7. $\square$

3 Proof of Theorem 1.1

By replacing $x$ by $n$ and $n$ by $r$ in (1.9), we deduce that

[TABLE]

Therefore, from (2.12) and (3.1) we obtain

[TABLE]

Thus, by the definition of $S\left({k,m;p}\right)$ and (3.2) we can prove (1.7). $\square$

4 Proof of Theorem 1.2

By the definition of multiple harmonic number (1.1), we can find that

[TABLE]

On the other hand, we consider the expansion

[TABLE]

where

[TABLE]

Therefore, the equation (4.1) can be written as

[TABLE]

For $r>0$, we have the partial fraction decomposition

[TABLE]

Moreover, from identities (2.1), (2.4) and (2.6), we deduce the following result

[TABLE]

Hence, combining (4.4), (4.5) and (4.6), by a simple calculation, we obtain the the desired result. This completes the proof of Theorem 1.2. $\square$

Similarly, applying the same arguments as in the proof of formula (4.1), we also deduce a similar result

[TABLE]

In fact, we hope to obtain a similar result of Theorem 1.2, but, so far, we have been unable to continue any progress with this sums.

5 Conclusion

From [13, 17], we know that the Aomoto-Drinfel d-Zagier formula reads

[TABLE]

which implies that for any $m,\ n\in\mathbb{N}$, the multiple zeta value ${\zeta\left({m+1,{{\left\{1\right\}}_{n-1}}}\right)}$ can be represented as a polynomial of zeta values with rational coefficients, and we have the duality formula

[TABLE]

In particular, one can find explicit formulas for small weights.

[TABLE]

Hence, from formulas (1.11) and (5.1), we see that the sums ${U_{m,r}}\left(p\right)$ can be expressed in terms of series of Riemann zeta function and harmonic numbers. Thus, we show that the Euler-type sums with hyperharmonic numbers $S\left({k,r+1;p}\right)$ can be expressed in terms of zeta values and Stirling numbers of the first kind, for integers $k\in\mathbb{N}$ and $r\in\mathbb{N}_{0}$ with $r+2\leq p\in\mathbb{N}$.

To conclude, note that it would be useful to be able to extend the approach described above to include other similar and related sums. In particular, it would be very interesting to consider sums of the form

[TABLE]

where $H_{n}^{\left(m\right)}\left(k\right)$ is called the generalized hyperharmonic star numbers, defined by

[TABLE]

Obviously, by the definitions of $H_{n}^{\left(m\right)}\left(k\right)$ and $h_{n}^{\left(m\right)}(k)$, we have

[TABLE]

Hence $S\left({1,m;p}\right)={S^{\star}}\left({1,m;p}\right)$. However, we have been unable, so far, to make any progress with this sums. Unfortunately, it appears that, even in the case $k=2$, the method used in this work gives rise to several complex and intractable summations.

Acknowledgments. The authors would like to thank the anonymous referee for his/her helpful comments, which improve the presentation of the paper.