Identities involving harmonic and hyperharmonic numbers

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R E S E A R C H

Open Access

Identities involving harmonic and

hyperharmonic numbers

Dae San Kim

1

_{and Taekyun Kim}

2*

*_{Correspondence: [email protected]} 2_{Department of Mathematics,}

Kwangwoon University, Seoul, 139-701, Republic of Korea Full list of author information is available at the end of the article

Abstract

In this paper, we give some new and interesting identities involving harmonic and hyperharmonic numbers which are derived from the transfer formula for the associated sequences.

1 Introduction

LetFbe the set of all formal power series in the variabletoverCwith

F=

f(t) =

∞

k=

ak

k!t

k_a k∈C

. ()

Suppose thatPis the algebra of polynomials in the variablexoverCand thatP∗is the vector space of all linear functionals onP. The action of the linear functionalLon a poly-nomialp(x) is denoted byL|p(x).

Letf(t)∈F. Then we consider a linear functional onPby setting

f(t)|xn=an (n≥) (see [, ]). ()

From () and (), we note that

tk|xn=n!δn,k (n,k≥) (see [, –]), ()

whereδn,kis the Kronecker symbol.

LetfL(t) =

_∞

k=L|x

n

k! tk. Then we see thatfL(t)|xn=L|xn. The mapL−→fL(t) is a

vector space isomorphism fromP∗ ontoF. Henceforth,F is thought of as both a for-mal power series and a linear functional. We callFthe umbral algebra. The umbral cal-culus is the study of umbral algebra. The orderO(f(t)) of the nonzero power seriesf(t) is the smallest integerk for which the coeﬃcient oftk _{does not vanish. If}_O(f_{(t)) = ,}

thenf(t) is called an invertible series. IfO(f(t)) = , thenf(t) is called a delta series. Let O(f(t)) =  andO(g(t)) = . Then there exists a unique sequencesn(x) of polynomials such

thatg(t)f(t)k_|_s

n(x)=n!δn,kforn,k≥. The sequencesn(x) is called the Sheﬀer sequence

for (g(t),f(t)) which is denoted bysn(x)∼(g(t),f(t)) (see [, , ]). Ifsn(x)∼(,f(t)), then

sn(x) is called the associated sequence forf(t). By (), we easily see thateyt|p(x)=p(y).

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Letf(t)∈Fandp(x)∈P. Then we have

f(t) =

∞

k=

f(t)|xk

k! t

k_, _{p(x) =}

∞

k=

tk_|_p(x)

k! x

k _{(see [, , ]).} _()

From (), we note that

p(k)() =tk|p(x), |p(k)(x)=p(k)(). ()

By (), we easily see that

tkp(x) =p(k)(x) =d

k_p(x)

dxk (k≥) (see [, , , ]). ()

Letφn(x) be exponential polynomials which are given by

∞

k= φk(x)

k! t

k₌_ex(et–) _{(see [, , ]).} _()

Thus, by (), we get

φn(x) = n

k=

S(n,k)xk∼ ,log( +t)

, ()

whereS(n,k) is the Stirling number of the second kind.

The Stirling number of the ﬁrst kind is deﬁned by

(x)n=x(x– )· · ·(x–n+ ) = n

k=

S(n,k)xk. ()

Thus, by (), we get

S(n,k) =

 k!

tk|(x)n

(see [, ]). ()

Let pn(x)∼(,f(t)), qn(x)∼(,g(t)). Then the transfer formula for the associated

se-quences is given by

qn(x) =x

f(t) g(t)

n

x–pn(x) (see [, ]). ()

Thenth harmonic number isHn=

n

i=i (n≥) andH= .

In general, the hyperharmonic numberHn(r)of orderris deﬁned by

H_n(r)=

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

 ifn≤ orr< ,



n ifr= ,n≥,

n i=H

(r–)

i ifr,n≥

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From (), we note thatHn()is the ordinary harmonic numberHn. It is known that

H_n(r)=

n+r–  r– 

(Hn+r––Hr–) (see [, ]). ()

The generating functions of the harmonic and hyperharmonic numbers are given by

∞

n=

Hntn= –

log( –t)

 –t ()

and

∞

n=

H_n(r)tn= –log( –t)

( –t)r , respectively. ()

The purpose of this paper is to give some new and interesting identities involving har-monic and hyperharhar-monic numbers which are derived from the transfer formula for the associated sequences.

2 Identities involving harmonic and hyperharmonic numbers

From () and (), we note that

φn(x) = n

j=

S(n,j)xj∼ ,log( +t)

()

and

(–)nφn(–x)∼ , –log( –t)

. ()

Let us assume that

qn(x)∼ ,t( –t)r

. ()

From (), () andxn_∼_(,_{t), we note that}

qn(x) =x

t t( –t)r

n

x–xn=x( –t)–rnxn–

=x

n–

k=

–rn k

(–t)kxn–=x

n–

k=

rn+k–  k

tkxn–

=x

n–

k=

rn+k–  k

(n– )kxn––k= n–

k=

rn+k–  k

(n– )kxn–k

=

n

k=

rn+n–k–  n–k

(n– )n–kxk. ()

Now, we use the following fact:

∞

n=

H_n(r)tn= –log( –t)

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Forn≥, by (), () and (), we get

qn(x) =x

–log( –t) t( –t)r

n

x–(–)nφn(–x)

=x

_∞

l=

H_l(_+r)tl

n

x–(–)n

n

j=

S(n,j)(–x)j

= (–)n

n

j=

S(n,j)(–)jx _∞

l=

H_l(_+r)tl

n

xj–

= (–)n

n

j=

S(n,j)(–)jx _j_–

l=

l+···+ln=l

H_l(_+r) · · ·H_l(_nr_+)

tl

xj–

= (–)n

n

j=

j–

l=

l+···+ln=l

S(n,j)(–)jHl(+r) · · ·H (r)

ln+(j– )lx

j–l

= (–)n

n

j=

j

k=

l+···+ln=j–k

S(n,j)(–)jHl(+r) · · ·H (r)

ln+(j– )j–kx

k

= (–)n

n

k= _n

j=k

l+···+ln=j–k

(–)jS(n,j)Hl(+r) · · ·H (r)

ln+(j– )j–k

xk. ()

Therefore, by comparing coeﬃcients on both sides of () and (), we obtain the following theorem.

Theorem  For n≥,r≥, ≤k≤n,we have

(n– )n–k= (–)n n

j=k

l+···+ln=j–k

S(n,j)(–)jHl(+r) · · ·H (r)

ln+(j– )j–k.

We recall the following equation:

log( +t) t

n

=

∞

l=

n!

(l+n)!S(l+n,n)t

l_. _()

Forn≥, from (), () and (), we have

qn(x) =x

–log( –t) t( –t)r

n

x–(–)nφn(–x)

=x

log( –t) –t

n

( –t)–rnx–(–)nφn(–x)

= (–)n

n

j=

S(n,j)(–)jx

log( –t) –t

n

( –t)–rnxj–

= (–)n

n

j=

S(n,j)(–)jx

log( –t) –t

n j–

l=

rn+l–  l

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= (–)n

n

j=

S(n,j)(–)j

j–

l=

rn+l–  l

(j– )lx j––l

m=

n! (m+n)!

×S(m+n,n)(–t)mxj––l

= (–)n

n

j=

j–

l=

j––l m=

(–)j+m

rn+l–  l

n! (m+n)!

(j– )! (j–  –l–m)!

×S(m+n,n)S(n,j)xj–l–m

= (–)n

n

k= _n

j=k j–k

l=

(–)k+l

rn+l–  l

n! (j–l–k+n)!

(j– )! (k– )!

×S(j–l–k+n,n)S(n,j)

xk. ()

Therefore, by () and (), we obtain the following theorem.

Theorem  For r,n≥, ≤k≤n,we have

(n– )n–k

= (–)n

n

j=k j–k

l=

(–)k+l

rn+l–  l

n! (j–l–k+n)!

(j– )! (k– )!

×S(j–l–k+n,n)S(n,j).

Here we invoke the following identity:

∞

n= _n

m=

mH_m(r)

tn=t( –rlog( –t))

( –t)r+ . ()

Let us consider the following associated sequence:

qn(x)∼ ,t( –t)r+

. ()

Forn≥, by () and (), we get

qn(x) = n

k=

(r+ )n–k–  n–k

(n– )n–kxk. ()

Let us assume that

pn(x)∼ ,t  –rlog( –t)

. ()

Forn≥, by (), () andxn_∼_(,_{t), we get}

pn(x) = x

t t( –rlog( –t))

n

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=x

Therefore, by () and (), we obtain the following theorem.

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Here we use the following identity:

qn(x)∼ ,t( –t)r+

Let us assume that

pn(x)∼ ,t  –rlog( –t)

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Theorem  For n,r≥, ≤k≤n,we have

(r+ )n–k–  n–k

(n– )n–k

=

n

a=k n–a

l=

j+···+jn=a–k

(j+ )· · ·(jn+ )Hj(+r) · · ·H (r)

jn+

l!rl

×

n+l–  l

n–  a– 

S(n–a,l)(a– )a–k.

Now, we utilize the following identity:

∞

n=

(n+ )Hntn=

t–log( –t)

( –t) . ()

qn(x)∼ ,t( –t)

. ()

Forn≥, from () and (), we have

qn(x) = n

k=

n–k–  n–k

(n– )n–kxk. ()

Let us assume that

pn(x)∼ ,t–log( –t)

. ()

We observe that

t–log( –t) =t+

∞

n=

tn

n = t+

∞

n=

tn

n. ()

From (), (), () andxn_∼_(,_{t), we can derive the following equation:}

pn(x) =x

t (t+∞_n_=t_nn)

n

x–xn

= –nx

 +

∞

n=

tn–

n

–n

xn–

= –nx

∞

l=

–n l

∞

n=

tn–

n

l

xn–

= –nx

n–

l=

(–)l

n+l–  l

×

n––l m=

m+···+ml=m

 l_(m

+ )· · ·(ml+ )

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= –n

Therefore, by () and (), we obtain the following theorem.

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Now, we recall the following identity:

∞

n=

nHntn=

t{ + t– ( +t)log( –t)}

( –t) . ()

qn(x)∼ ,t( –t)

. ()

Forn≥, from () and (), we can derive the following equation:

qn(x) = n

k=

n–k–  n–k

(n– )n–kxk. ()

Let us assume that

pn(x)∼ ,t

 + t– ( +t)log( –t). ()

We observe that

 + t– ( +t)log( –t) =  + t+ ( +t)

∞

j=

tj j

=  + t+t+

∞

j=

tj

j +

∞

j=

tj+ j

=  + t+

∞

j=

tj+

j+ +

∞

j=

tj+

j+ 

=  + t+

∞

j=

j+  (j+ )(j+ )t

j+_. _()

Forn≥, by (), (), () andxn∼(,t), we get

pn(x) =x

t

t{ + t– ( +t)log( –t)}

n

x–xn

=x

 + t+

∞

j=

j+  (j+ )(j+ )t

j+ –n

xn–

=x

n–

l=

(–)l

n+l–  l

 +

∞

j=

j+  (j+ )(j+ )t

j+ l

tlxn–

=

n–

l=

n––l a=

n–a–l k=

j+···+ja=n–a–k–l

(–)l

n+l–  l

l a

l–a(n– )n–k

×

a

i=(ji+ )

a

i=(ji+ )(ji+ )

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=

Therefore, by () and (), we obtain the following theorem.

Theorem  For n≥, ≤k≤n,we have

Here we invoke the following identity:

∞

qn(x)∼ ,t( –t)

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From () and (), we note that

Let us assume that

pn(x)∼ ,t  + t– ( +t)log( –t)

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Theorem  For n≥, ≤k≤n,we have

n–k–  n–k

(n– )n–k

=

n

m=k n–m

l=

n–m–l a=

j+···+ja=n–a–m–l

b+···+bn=m–k (–)l

n+l–  l

l a

l–a

×(n– )n–m(m– )m–k a

i=(ji+ )

a

i=(ji+ )(ji+ )

b+

c=

· · ·

bn+

cn=

n

i=

c_iHci.

Here we use the following identity:

∞

n=

n(n+ )Hntn=

t{( +t) – (t+ )log( –t)}

( –t) . ()

qn(x)∼ ,t( –t)

. ()

By () and (), we get

qn(x) = n

k=

n–k–  n–k

(n– )n–kxk (n≥). ()

Let us assume that

pn(x)∼ ,t

( +t) – (t+ )log( –t). ()

We see that

( +t) – (t+ )log( –t) =  + t+

∞

n=

n+  n(n+ )t

n+_. _()

Forn≥, from (), (), () andxn_∼_(,_{t), we have}

pn(x) =x

t

t{( +t) – (t+ )log( –t)}

n

x–xn =x ( +t) – (t+ )log( –t)–nxn–

=x

 + t+

∞

j=

j+  j(j+ )t

j+ –n

xn–. ()

From (), by the same method of (), we get

pn(x) = –n n

k= _n_–_k

l=

n–k–l a=

j+···+ja=n–a–l–k (–)l

n+l–  l

l a

l–a

×(n– )n–k

_a

i=

(ji+ )

(ji+ )(ji+ )

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Forn≥, by (), (), (), () and (), we get

By the same method, we can derive the following identity from ():

qn(x) = –n

By comparing coeﬃcients on both sides of () and (), we get

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Remark Recently, several authors have studied theq-extension of harmonic and hyper-harmonic numbers (see [–]).

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors contributed equally to the manuscript and typed, read and approved the ﬁnal manuscript.

Author details

1_{Department of Mathematics, Sogang University, Seoul, 121-742, Republic of Korea.}2_{Department of Mathematics,}

Kwangwoon University, Seoul, 139-701, Republic of Korea.

Acknowledgements

The authors express their sincere gratitude to the referees for their valuable suggestions and comments. This work was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MOE) (No. 2012R1A1A2003786).

Received: 26 April 2013 Accepted: 22 July 2013 Published: 7 August 2013

References

1. Roman, S: More on the umbral calculus, with emphasis on theq-umbral calculus. J. Math. Anal. Appl.107, 222-254 (1985)

2. Roman, S: The Umbral Calculus. Dover, New York (2005)

3. Araci, S, Acikgoz, M, Kilicman, A: Extendedp-adicq-invariant integrals onZpassociated with applications of umbral

calculus. Adv. Diﬀer. Equ.2013, 96 (2013)

4. Kim, DS, Kim, T, Lee, S-H, Rim, S-H: Some identities of Bernoulli, Euler and Abel polynomials arising from umbral calculus. Adv. Diﬀer. Equ.2013, 15 (2013)

5. Kim, DS, Kim, T, Dolgy, DV, Rim, S-H: Some new identities of Bernoulli, Euler and Hermite polynomials arising from umbral calculus. Adv. Diﬀer. Equ.2013, 73 (2013)

6. Kim, DS, Kim, T, Lee, S-H, Rim, S-H: A note on the higher-order Frobenius-Euler polynomials and Sheﬀer sequences. Adv. Diﬀer. Equ.2013, 41 (2013)

7. Dere, R, Simsek, Y: Applications of umbral algebra to some special polynomials. Adv. Stud. Contemp. Math.22(3), 433-438 (2012)

8. Kim, DS, Kim, T: Some identities of Frobenius-Euler polynomials arising from umbral calculus. Adv. Diﬀer. Equ.2012, 196 (2012)

9. Dil, A, Kurt, V: Polynomials related to harmonic numbers and evaluation of harmonic number series. Integers12, 38 (2012)

10. Dil, A, Kurt, V: Polynomials related to harmonic numbers and evaluation of harmonic number series II. Appl. Anal. Discrete Math.5, 212-229 (2011)

11. Mansour, T, Shattuck, M, Song, C:q-Analogs of identities involving harmonic numbers and binomial coeﬃcients. Appl. Appl. Math.7(1), 22-36 (2012)

12. Mansour, T: Identities on harmonic andq-harmonic number sums. Afr. Math.23, 135-143 (2012) 13. Mansour, T, Shattuck, M: Aq-analog of the hyperharmonic numbers. Afr. Math. (2012).

doi:10.1007/s13370-012-0106-6

doi:10.1186/1687-1847-2013-235