Lerch-harmonic numbers related to Lerch transcendent

ABSTRACT

Harmonic numbers and generalized harmonic numbers have been studied in connection with combinatorial problems, many expressions involving special functions in analytic number theory and analysis of algorithms. We introduce Lerch-harmonic numbers which generalize the harmonic numbers and the generalized harmonic numbers. The aim of this note is to derive some identities expressing certain finite sums as the infinite sums involving the Lerch-harmonic numbers. Then, by taking limits of such identities we obtain the corresponding infinite sums of the finite sums as the infinite sums involving the Lerch transcendents.

KEYWORDS:

• Lerch-harmonic numbers
• Lerch transcendent

1. Introduction

The harmonic numbers are defined by

(1) $H_0=0,H_n=1+\frac{1}{2}+\cdots +\frac{1}{n},\left(n\ge 1\right)$(1)

(see [1,2]) and more generally, for any $r\in \mathbb{N}$, the generalized harmonic numbers of order $r$ are given by

(2) $H_0^{\left(r\right)}=0,H_n^{\left(r\right)}=1+\frac{1}{2^r}+\frac{1}{3^r}+\cdots +\frac{1}{n^r},\left(n\ge 1\right)$(2)

(see [3,4]). For $n,r\in \mathbb{N}$, we recall that

$(-1{)}^{r}r!\sum _{k=0}^n\left(\begin{array}{c}n\\ k\end{array}\right)(-1{)}^k\frac{1}{{(2k+1)}^{r+1}}=\frac{2^{2n}}{(2n+1)\left(\begin{array}{c}2n\\ n\end{array}\right)}{f}_{n,r}(0),$

$\sum _{n=1}^{\mathrm{\infty }}\frac{2^{2n-1}}{n(2n+1)\left(\begin{array}{c}2n\\ n\end{array}\right)}{f}_{n,r}(0)=(-1{)}^r(r+1)!,$

where $f_{n,s}(x)$ are determined by the recurrence relation

$f_{n,s+1}(x)={\beta }_n(x)f_{n,s}(x)+\frac{d}{dx}{f}_{n,s}(x),(s\ge 1),f_{n,1}(x)={\beta }_n(x)=\sum _{k=0}^n\frac{1}{x+2k+1},$

so that $f_{n,s}$ is a polynomial in ${\beta }_n(x),{\beta }_n^{(1)}(x),\dots ,{\beta }_n^{(s-1)}(x)$, with

${\beta }_n^{(j)}(0)=(-1{)}^{j-1}j!\left(H_{2n+1}^{(j+1)}-\frac{1}{2^{j+1}}{H}_n^{(j+1)}\right),(j\ge 0)$

(See [4]). In [3], it was shown that, for every $r=2,3,4,5$,

$\sum _{k=0}^n\left(\begin{array}{c}n\\ k\end{array}\right)(-1{)}^k\frac{1}{{(k+s+1)}^r},\zeta (x+1,r),r!$

are all expressed in terms of the generalized harmonic numbers $H_n^{(r)}$ and the generalized harmonic functions $H_n(x,r)={\sum }_{k=0}^n\frac{1}{{(k+x+1)}^r},(x\ge 0)$(See Figure 1).

Figure 1. Contour plot of the generalized harmonic functions H_n(x,r) on the region (x,r)∈[0.1,1.0]² by varying n = 10,20,40.

The Lerch transcendent is given by

(3) $\mathrm{\Phi }(z,s,\alpha )=\sum _{n=0}^{\mathrm{\infty }}\frac{z^n}{{(n+\alpha )}^s},$(3)

where $\alpha \in \mathbb{R}$ with $\alpha >0$, and $z,s\in \mathbb{C}$ with $|z|<1$,(see [1–16]).

From [7], we note that

(4) $\mathrm{\Phi }(z,s,a)=\frac{1}{\mathrm{\Gamma }(s)}{\int }_0^{\mathrm{\infty }}\frac{t^{s-1}}{1-z{e}^{-t}}{e}^{-at}dt.$(4)

Hurwitz zeta function is a special case of the Lerch transcendent as it is seen from

(5) $\zeta \left(s,\alpha \right)=\mathrm{\Phi }\left(1,s,\alpha \right)=\sum _{n=0}^{\mathrm{\infty }}\frac{1}{{\left(n+\alpha \right)}^s},(\mathrm{R}\mathrm{e}\left(s\right)>1)$(5)

(see [5–8,15,16]). In particular, for $\alpha =1$, we have

(6) $\zeta \left(s,1\right)=\sum _{n=0}^{\mathrm{\infty }}\frac{1}{{\left(n+1\right)}^s}=\sum _{n=1}^{\mathrm{\infty }}\frac{1}{n^s}=\zeta \left(s\right),(\mathrm{R}\mathrm{e}\left(s\right)>1)$(6)

(see [1–16]). By [7], we see that the series transformation of Lerch transcendent is given by

(7) $\mathrm{\Phi }(z,s,\alpha )=\frac{1}{1-z}\sum _{n=0}^{\mathrm{\infty }}(\frac{-z}{1-z}{)}^n\sum _{k=0}^n(-1{)}^k\left(\begin{array}{c}n\\ k\end{array}\right)\frac{1}{{(\alpha +k)}^s}.$(7)

The harmonic numbers and generalized harmonic numbers have appeared in such diverse areas as combinatorial problems, special functions in analytic number theory and analysis of algorithms. The aim of this note is to introduce the Lerch-harmonic numbers, which generalize the harmonic numbers and the generalized harmonic numbers, and to derive some identities expressing certain finite sums as infinite sums involving the Lerch-harmonic numbers by using elementary methods. Moreover, as corollaries we obtain from such identities representations of the corresponding infinite sums of the finite sums as infinite sums involving the Lerch transcendents.

In more detail, the outline of this note is as follows. As a generalization of the generalized harmonic numbers $H_n^{(r)}$, we introduce the Lerch-harmonic numbers $H_n(\beta ,a,\alpha )$, for any real numbers $\alpha ,\beta$ with $\alpha >0$, and $a\in \mathbb{C}$ (see (8)). These numbers $H_n(\beta ,a,\alpha )$ are just partial sums of the Lerch transcendent $\mathrm{\Phi }(\beta ,a,\alpha )$, for any $\beta$ with $|\beta |<1$. In Theorem 1, we find the first identity expressing a finite sum as an infinite sum involving the Lerch-harmonic numbers. By taking the limit as $m$ tends to $\mathrm{\infty }$, as a corollary to Theorem 1 we represent the corresponding infinite sum of the finite sum as the infinite sum involving the Lerch transcendent. Then, by integrating the identity in Theorem 1, we obtain the second identity expressing a finite sum as an infinite sum involving the Lerch-harmonic numbers in Theorem 3. Again, by taking the limit as $m$ tends to $\mathrm{\infty }$, as a corollary to Theorem 3 we represent the corresponding infinite sum of the finite sum as the infinite sum involving the Lerch transcendent. As a special case of the identity in Theorem 3, we have an expression of the generating function of the sequence $\frac{{(-1)}^{n-1}}{n}{H}_m(b,n,\alpha )$, $(n=1,2,3,\dots )$ in Theorem 5. From a special case of Corollary 4, we obtain a representation of the infinite sum ${\sum }_{n=1}^{\mathrm{\infty }}\frac{{(-1)}^{n-1}}{n}\mathrm{\Phi }(b,n+a,\alpha )$ as an infinite sum plus the derivative $\frac{d}{da}\mathrm{\Phi }(b,a,\alpha )$ in Theorem 6.

Finally, we would like to give some brief account on literatures that are related to this paper. In [17], some interesting identities were derived as to certain finite or infinite series involving harmonic numbers and generalized harmonic numbers. In [18], reciprocity laws were proven for some new classes of finite sums involving the numbers $y(m,\lambda )$, the generalized harmonic functions $H_m^{(s)}(w)={\sum }_{n=1}^m\frac{1}{{(n+w)}^s}$, some special numbers and polynomials, Dedekind sums, and other combinatorial sum. A lot of series and integrals involving the Riemann zeta function, binomial coefficients and the harmonic numbers were presented by using elementary methods in [19–22]. The reader refers to [23,24] for some related infinite series involving Riemann zeta function or Lerch transcendent. In [25], the author constructed Lerch-type zeta functions which interpolate higher-order Apostol-type numbers and Apostol-type polynomials at negative integers. In [26], interpolation functions for two new classes of special combinatorial numbers and polynomials were obtained by applications of the $p$-adic Volkenborn and fermionic integrals. The interpolation function, which interpolates the numbers $y_1(n,k;\lambda )$ at negative integers, was found in [27].

2. Lerch-harmonic numbers related to Lerch transcendent

For $\alpha ,\beta \in \mathbb{R}$ with $\alpha >0$, and $a\in \mathbb{C}$, we consider the Lerch-harmonic numbers which are given by

(8) $H_n(\beta ,a,\alpha )=\sum _{k=0}^n\frac{{\beta }^k}{{(k+\alpha )}^a},(n\ge 0).$(8)

Note that $H_n(1,1,1)=H_{n+1},H_n(1,r,1)=H_{n+1}^{(r)},(r\in \mathbb{N})$ (See Figure 2).

Figure 2. Contour plot of Lerch-harmonic numbers H₂₀(β,a,α) on the region (α,β)∈[0.2,1.0]² by varying a = 0.5,1.0,1.5.

Before proceeding further we would like to provide some tables containing some values of the Lerch-harmonic numbers in a few special cases. For positive integers $m,l$, we note that

(9) $H_n\left(\frac{1}{m},1,\frac{1}{l}\right)=l\sum _{k=0}^n\frac{1}{m^k(lk+1)}.$(9)

Taking $m=2$ or $m=3$, and $l=1,2,3,4$ in [12], we obtain the following tables for Lerch-harmonic numbers (See Table 1).

Table 1. H_n(β,a,α).

$n\to$	1	2	3	4	5	6	7	8	9	10
$H_n(\frac{1}{2},1,1)$	$\frac{5}{4}$	$\frac{4}{3}$	$\frac{131}{96}$	$\frac{661}{480}$	$\frac{1327}{960}$	$\frac{1163}{840}$	$\frac{148969}{107520}$	$\frac{447047}{322560}$	$\frac{44711}{32256}$	$\frac{983705}{709632}$
$H_n(\frac{1}{2},1,\frac{1}{2})$	$\frac{7}{3}$	$\frac{73}{30}$	$\frac{1037}{420}$	$\frac{6257}{2520}$	$\frac{137969}{55440}$	$\frac{3590659}{1441440}$	$\frac{7184321}{2882880}$	$\frac{244311959}{98017920}$	$\frac{9284620207}{3724680960}$	$\frac{2652847607}{1064194560}$
$H_n(\frac{1}{2},1,\frac{1}{3})$	$\frac{27}{8}$	$\frac{195}{56}$	$\frac{1971}{560}$	$\frac{1608}{455}$	$\frac{824661}{232960}$	$\frac{15679479}{4426240}$	$\frac{172526139}{48688640}$	$\frac{862744809}{243443200}$	$\frac{3451183011}{973772800}$	$\frac{106989526191}{30186956800}$
$H_n(\frac{1}{2},1,\frac{1}{4})$	$\frac{22}{5}$	$\frac{203}{45}$	$\frac{5323}{1170}$	$\frac{181567}{39780}$	$\frac{2545253}{556920}$	$\frac{25466453}{5569200}$	$\frac{1477402349}{323013600}$	$\frac{32506216403}{7106299200}$	$\frac{2405571049747}{525866140800}$	$\frac{197260934408479}{43121023545600}$
$H_n(\frac{1}{3},1,1)$	$\frac{7}{6}$	$\frac{65}{54}$	$\frac{131}{108}$	$\frac{1969}{1620}$	$\frac{17731}{14580}$	$\frac{13793}{11340}$	$\frac{744857}{612360}$	$\frac{20111419}{16533720}$	$\frac{20111503}{16533720}$	$\frac{24580757}{20207880}$
$H_n(\frac{1}{3},1,\frac{1}{2})$	$\frac{20}{9}$	$\frac{34}{15}$	$\frac{2152}{945}$	$\frac{58174}{25515}$	$\frac{640124}{280665}$	$\frac{8322382}{3648645}$	$\frac{14980688}{6567561}$	$\frac{254673698}{111648537}$	$\frac{14516434820}{6363966609}$	$\frac{130648005758}{57275699481}$
$H_n(\frac{1}{3},1,\frac{1}{3})$	$\frac{13}{4}$	$\frac{277}{84}$	$\frac{4169}{1260}$	$\frac{162731}{49140}$	$\frac{1953227}{589680}$	$\frac{111341219}{33611760}$	$\frac{3674329387}{1109188080}$	$\frac{55115245109}{16637821200}$	$\frac{165346007027}{49913463600}$	$\frac{15377186261111}{4641952114800}$
$H_n(\frac{1}{3},1,\frac{1}{4})$	$\frac{64}{15}$	$\frac{1748}{405}$	$\frac{22784}{5265}$	$\frac{129196}{29835}$	$\frac{24422464}{5638815}$	$\frac{122118508}{28194075}$	$\frac{10624464896}{2452884525}$	$\frac{1051826511004}{242835567975}$	$\frac{12972543418816}{2994972005025}$	$\frac{4786870347467204}{1105144669854225}$

Throughout this paper, we assume that

(10) $\alpha ,\beta \in \mathbb{R}\mathrm{w}\mathrm{i}\mathrm{t}\mathrm{h}\alpha ,\beta >0\mathrm{a}\mathrm{n}\mathrm{d}|x|<{\alpha }^{\beta }.$(10)

Theorem 1.

For $\alpha \in \mathbb{R}$ with $\alpha ,\beta >0$, and $|x|<{\alpha }^{\beta }$, we have

(11) $\sum _{k=0}^m\frac{b^k}{{(k+\alpha )}^a(x+{(k+\alpha )}^{\beta })}=\sum _{n=1}^{\mathrm{\infty }}(-1{)}^{n-1}{x}^{n-1}{H}_m(b,a+n\beta ,\alpha ).$(11)

Proof.

Observing that $|\frac{x}{{(k+\alpha )}^{\beta }}|<1$, for any nonnegative integer $k$, we have

(12) $\sum _{k=0}^m\frac{b^k}{{(k+\alpha )}^a(x+{(k+\alpha )}^{\beta })}=\sum _{k=0}^m\frac{b^k}{{(k+\alpha )}^{a+\beta }}\left(\frac{1}{1+\frac{x}{{(k+\alpha )}^{\beta }}}\right)$(12)

$=\sum _{k=0}^m\frac{b^k}{{(k+\alpha )}^{a+\beta }}\sum _{l=0}^{\mathrm{\infty }}\frac{{(-1)}^l{x}^l}{{(k+\alpha )}^{l\beta }}$

$=\sum _{n=1}^{\mathrm{\infty }}(-1{)}^{n-1}{x}^{n-1}{H}_m(b,a+n\beta ,\alpha ).$

Therefore, by (12), we obtain the result.

Corollary 2.

For $\alpha ,\beta ,b\in \mathbb{R}$ with $\alpha ,\beta >0,|b|<1$, and for $|x|<{\alpha }^{\beta }$, we have

$\sum _{k=0}^{\mathrm{\infty }}\frac{b^k}{{(k+\alpha )}^a(x+{(k+\alpha )}^{\beta })}=\sum _{n=1}^{\mathrm{\infty }}(-1{)}^{n-1}{x}^{n-1}\mathrm{\Phi }(b,a+n\beta ,\alpha ).$

Proof.

From (3) and (11), for $|b|<1$, we note that

(13) $\sum _{k=0}^{\mathrm{\infty }}\frac{b^k}{{(k+\alpha )}^a(x+{(k+\alpha )}^{\beta })}=\sum _{n=1}^{\mathrm{\infty }}(-1{)}^{n-1}{x}^{n-1}\underset{m\to \mathrm{\infty }}{lim}{H}_m(b,a+n\beta ,\alpha )$(13)

$=\sum _{n=1}^{\mathrm{\infty }}(-1{)}^{n-1}{x}^{n-1}\mathrm{\Phi }(b,a+n\beta ,\alpha ).$

Thus, by (13), we get what we wanted.

Theorem 3.

For $\alpha ,\beta \in \mathbb{R}$ with $\alpha ,\beta >0$, and for $|x|<{\alpha }^{\beta }$, we have

(14) $\sum _{n=1}^{\mathrm{\infty }}{H}_m(b,a+n\beta ,\alpha )(-1{)}^{n-1}\frac{x^n}{n}=\sum _{k=0}^m\frac{b^k}{{(k+\alpha )}^a}log\left(1+\frac{x}{{(k+\alpha )}^{\beta }}\right).$(14)

Proof.

Observe that, under the assumption of (10), $x+(k+\alpha {)}^{\beta }>0$, for any nonnegative integer $k$. Taking indefinite integrals on the both sides of (11), we have

(15) $\sum _{k=0}^m\frac{b^k}{{(k+\alpha )}^a}log(x+(k+\alpha {)}^{\beta })=\sum _{k=0}^m\frac{b^k}{{(k+\alpha )}^a}\int \frac{1}{x+{(k+\alpha )}^{\beta }}dx$(15)

$=\sum _{n=1}^{\mathrm{\infty }}(-1{)}^{n-1}{H}_m(b,a+n\beta ,\alpha )\int x^{n-1}dx=\sum _{n=1}^{\mathrm{\infty }}(-1{)}^{n-1}{H}_m(b,a+n\beta ,\alpha )\frac{x^n}{n}+C.$

Let $x=0$ in (15). Then we have

(16) $C=\sum _{k=0}^m\frac{b^k}{{(k+\alpha )}^a}log(k+\alpha {)}^{\beta }.$(16)

From (15) and (16), we have

(17) $\sum _{k=0}^m\frac{b^k}{{(k+\alpha )}^a}log(x+(k+\alpha {)}^{\beta })$(17)

$=\sum _{n=1}^{\mathrm{\infty }}\frac{{(-1)}^{n-1}}{n}{H}_m(b,a+n\beta ,\alpha )x^n+\sum _{k=0}^m\frac{b^k}{{(k+\alpha )}^a}log(k+\alpha {)}^{\beta }.$

Thus, by (17), we obtain the result.

Corollary 4.

For $\alpha ,\beta ,b\in \mathbb{R}$ with $\alpha ,\beta >0,|b|<1$, and for $|x|<{\alpha }^{\beta }$, we have

(18) $\sum _{k=0}^{\mathrm{\infty }}\frac{b^k}{{(k+\alpha )}^a}log(1+\frac{x}{{(k+\alpha )}^{\beta }})=\sum _{n=1}^{\mathrm{\infty }}(-1{)}^{n-1}\mathrm{\Phi }(b,a+n\beta ,\alpha )\frac{x^n}{n}.$(18)

Proof.

From (14), for $|b|<1$ we note that

(19) $\sum _{k=0}^{\mathrm{\infty }}\frac{b^k}{{(k+\alpha )}^a}log\left(1+\frac{x}{{(k+\alpha )}^{\beta }}\right)=\sum _{n=1}^{\mathrm{\infty }}\underset{m\to \mathrm{\infty }}{lim}{H}_m(b,a+n\beta ,\alpha )\frac{{(-1)}^{n-1}}{n}{x}^n$(19)

$=\sum _{n=1}^{\mathrm{\infty }}\mathrm{\Phi }(b,a+n\beta ,\alpha )(-1{)}^{n-1}\frac{x^n}{n}.$

Therefore, by (19), we get what we wanted.

Theorem 5.

For $\alpha \in \mathbb{R}$ with $\alpha >0$, and for $|x|<\alpha$, we have

$\sum _{n=1}^{\mathrm{\infty }}(-1{)}^{n-1}{H}_m(b,n,\alpha )\frac{x^n}{n}=log\left(\frac{\mathrm{\Gamma }(\alpha )\mathrm{\Gamma }(\alpha +m+x+1)}{\mathrm{\Gamma }(\alpha +m+1)\mathrm{\Gamma }(\alpha +x)}\right)+log\prod _{k=0}^m{\left(\frac{k+\alpha +x}{k+\alpha }\right)}^{b^k-1}.$

In particular, for $b=\alpha =1$, it reduces to the interesting identity:

$\sum _{n=1}^{\mathrm{\infty }}(-1{)}^{n-1}{H}_{m+1}^{(n)}\frac{x^n}{n}=log\left(\frac{\mathrm{\Gamma }(m+x+2)}{(m+1)!\mathrm{\Gamma }(1+x)}\right),(|x|<1).$

Proof.

From (14), with $a=0$ and $\beta =1$, we have

(20) $\sum _{n=1}^{\mathrm{\infty }}(-1{)}^{n-1}{H}_m(b,n,\alpha )\frac{x^n}{n}=\sum _{k=0}^m{b}^{k}log\left(1+\frac{x}{k+\alpha }\right)$(20)

$=\sum _{k=0}^m{b}^{k}log\left(\frac{k+\alpha +x}{k+\alpha }\right)=log\prod _{k=0}^m{\left(\frac{k+\alpha +x}{k+\alpha }\right)}^{b^k}$

$=log\left(\frac{(\alpha +x)(\alpha +x+1)\cdots (\alpha +x+m)}{\alpha (\alpha +1)\cdots (\alpha +m)}\right)+log\prod _{k=0}^m{\left(\frac{k+\alpha +x}{k+\alpha }\right)}^{b^k-1}$

$=log\left(\frac{\mathrm{\Gamma }(\alpha )\mathrm{\Gamma }(\alpha +m+x+1)}{\mathrm{\Gamma }(\alpha +m+1)\mathrm{\Gamma }(\alpha +x)}\right)+log\prod _{k=0}^m{\left(\frac{k+\alpha +x}{k+\alpha }\right)}^{b^k-1}.$

Therefore, by (20), we obtain the result.

Theorem 6.

For $\alpha ,b\in \mathbb{R}$ with $\alpha >1,|b|<1$, we have

$\sum _{n=1}^{\mathrm{\infty }}\frac{{(-1)}^{n-1}}{n}\mathrm{\Phi }(b,n+a,\alpha )=\sum _{k=0}^{\mathrm{\infty }}\frac{b^{k}log(k+\alpha +1)}{{(k+\alpha )}^a}+\frac{d}{da}\mathrm{\Phi }(b,a,\alpha ),(\alpha >1,|b|<1).$

Proof.

Let $x=\beta =1$ in (18). Then we have

(21) $\sum _{n=1}^{\mathrm{\infty }}\mathrm{\Phi }(b,n+a,\alpha )\frac{{(-1)}^{n-1}}{n}=\sum _{k=0}^{\mathrm{\infty }}\frac{b^k}{{(k+\alpha )}^a}log\left(1+\frac{1}{k+\alpha }\right)$(21)

$=\sum _{k=0}^{\mathrm{\infty }}\frac{b^{k}log(k+\alpha +1)}{{(k+\alpha )}^a}-\sum _{k=0}^{\mathrm{\infty }}\frac{b^{k}log(k+\alpha )}{{(k+\alpha )}^a}$

$=\sum _{k=0}^{\mathrm{\infty }}\frac{b^{k}log(k+\alpha +1)}{{(k+\alpha )}^a}+\frac{d}{da}\mathrm{\Phi }(b,a,\alpha ).$

Thus, by (21), we get what we wanted.

3. Conclusion

In this note, we introduced the Lerch-harmonic numbers as generalizations of the harmonic numbers and the generalized harmonic numbers. Then by using elementary methods we expressed certain finite sums as infinite sums involving Lerch-harmonic numbers. By taking limits of such identities we represented the corresponding infinite sums of the finite sums as infinite sums involving Lerch transcendents. For example, we showed the following:

$\sum _{n=1}^{\mathrm{\infty }}(-1{)}^{n-1}{H}_m(b,n,\alpha )\frac{x^n}{n}=\sum _{k=0}^m{b}^{k}log\left(1+\frac{x}{k+\alpha }\right)$

$=log\left(\frac{\mathrm{\Gamma }(\alpha )\mathrm{\Gamma }(\alpha +m+x+1)}{\mathrm{\Gamma }(\alpha +m+1)\mathrm{\Gamma }(\alpha +x)}\right)+log\prod _{k=0}^m{\left(\frac{k+\alpha +x}{k+\alpha }\right)}^{b^k-1},(\alpha >0,|x|<\alpha ),$

$\sum _{k=0}^{\mathrm{\infty }}{b}^{k}log\left(1+\frac{x}{k+\alpha }\right)=\sum _{n=1}^{\mathrm{\infty }}(-1{)}^{n-1}\mathrm{\Phi }(b,n,\alpha )\frac{x^n}{n},(\alpha >0,|b|<1,|x|<\alpha ).$

Acknowledgments

We would like to thank the reviewers for their comments and suggestions that helped improve the original manuscript to its present form.

Disclosure statement

No potential conflict of interest was reported by the author(s).