Some special finite sums related to the three term polynomial relations and their applications

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R E S E A R C H

Open Access

Some special ﬁnite sums related to the

three-term polynomial relations and their

applications

Elif Cetin

1*

_{, Yilmaz Simsek}

2

_{and Ismail Naci Cangul}

1

*_{Correspondence:}

[email protected] 1_{Department of Mathematics,}

Uludag University, Gorukle, Bursa 16059, Turkey

Full list of author information is available at the end of the article

Abstract

We define some finite sums which are associated with the Dedekind type sums and Hardy-Berndt type sums. The aim of this paper is to prove a reciprocity law for one of these sums. Therefore, we define a new function which is related to partial derivatives of the three-term polynomial relations. We give a partial differential equation (PDE) for this function. For some special values, this PDE reduces the three-term relations for Hardy-Berndt sums (cf.Apostol and Vu in Pac. J. Math. 98:17-23, 1982; Berndt and Dieter in J. Reine Angew. Math. 337:208-220, 1982; Simsek in Ukr. Math. J. 56(10): 1434-1440, 2004; Simsek in Turk. J. Math. 22:153-162, 1998; Simsek in Bull. Calcutta Math. Soc. 85:567-572, 1993; Pettet and Sitaramachandraro in J. Number Theory 25:328-339, 1989), to the generalized Carlitz polynomials, which are defined by Beck (Diophantine Analysis and Related Fields, pp. 11-18, 2006), to the Gauss law of quadratic reciprocity (cf.Beck in Diophantine Analysis and Related Fields, pp. 11-18, 2006; Berndt and Dieter in J. Reine Angew. Math. 337:208-220, 1982; Simsek in Turk. J. Math. 22:153-162, 1998), and also to the well-known identity on the greatest integer function which was proved by Berndt and Dieter (J. Reine Angew. Math. 337:208-220, 1982), p.212, Corollary 3.5. Finally, we prove the reciprocity law for ann-variable new sum which is related to the Dedekind type and Hardy-Berndt type sums. We also raise some open questions on the reciprocity laws of our new finite sums.

MSC: Primary 11F20; secondary 11C08

Keywords: Hardy-Berndt sums; Dedekind sums; three-term polynomial relations;

greatest integer function;Y(h,k) sums

1 Introduction

The Dedekind sums are very useful in analytic number theory, in combinatorial theory and also in other branches of mathematics. That is, these sums arise in many areas of mathe-matics and also mathematical physics. Recently, there are many papers on the Dedekind sums which are related to elliptic modular functions, geometry (lattice point enumeration in polytopes, topology (signature defects of manifolds), algorithmic complexity (pseudo random number generators), character theory, the family of zeta functions, the Bernoulli functions, and other special functions. In , Dedekind gave, under the modular trans-formation, an elegant functional equation for the Dedekind eta function, which contains the Dedekind sums.

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On the other hand, Berndt [], Goldberg [] and also Simsek [] gave, under the modular transformation, other elegant functional equations for the theta functions, which contain six diﬀerent arithmetic sums (Hardy-Berndt sums). These sums are also related to the Dedekind sums and other special functions which have been mentioned before. Motivated largely by a number of recent investigations of the Dedekind sums and the Hardy-Berndt sums, we introduce and investigate various properties of a certain new family of ﬁnite arithmetic sums. We are ready to summarize our results in detail as follows.

In this section, some elementary properties and definitions on the Dedekind sums, the Hardy-Berndt sums, and the Simsek sum are given. In Section , we define some new finite arithmetic sums which are associated with the Dedekind sums, the Hardy-Berndt sums, and Simsek’s sum. We gave reciprocity laws for one of these sums. We also raise two open questions for the reciprocity laws. In the last section, we give a PDE for three-term polynomial relations. We give many applications for this PDE, which are related to the Dedekind-Rademacher sums, the Hardy-Berndt sums, and other finite arithmetic sums. Finally, by using this equation we give a proof of the reciprocity law of our new sums.

In the customary notation, we have

(x)=

x– [x] –_, ifxis not an integer,

, otherwise,

where [x] denotes the largest integer≤x(cf. [–], and the references cited in each of

these earlier works). Letnbe a positive natural number andαbe a real number, then

(α)n=

n

k=(α+k– ), ifn≥,

, ifn= ,

and

[α]n=

n

k=([α] +k– ), ifn≥,

, ifn= .

The Dedekind sums(h,k), arising in the theory of the Dedekind eta function, is deﬁned by

s(h,k) = k–

j=

j k

hj k

,

wherehis an integer andkis a positive integer (cf. [–], and the references cited in each of these earlier works). The most important property of Dedekind sums is the following reciprocity theorem: Ifhandkare coprime positive integers, then

s(h,k) +s(k,h) = –

+

 

h k +

k h+



hk

. ()

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Ifhandkare integers withk> , the Hardy-Berndt sums are deﬁned by

S(h,k) = k–

j=

(–)j++[jhk]_,

s(h,k) =

k–

j=

(–)[jhk]

j k

,

s(h,k) =

k–

j=

(–)j

j k

jh k

,

s(h,k) =

k–

j=

(–)j

jh k

,

s(h,k) =

k–

j=

(–)[jhk],

s(h,k) =

k–

j=

(–)j+[hjk]

j k

.

()

Fors(h,k), the equality below also holds true:

s(h,k) =



k

k–

j=

j(–)j+[hjk] _()

when handk are odd []. Besides, the following equations will be very useful for the

remaining sections []:

b–

j=

(–)j+[cjb]

j b

=s(c,b) –

 S(c,b),

c–

j=

(–)j+[bjc]

j c

=s(b,c) –

 S(b,c).

()

The reciprocity law for thes(h,k) is given by the following theorem.

Theorem  Let h and k be coprime positive integers.If h and k are odd,then

s(h,k) +s(k,h) =



–



hk ()

(cf. [, , , , , ]and the references cited in each of these earlier works).In the follow-ing theorem,Sitaramachandraro[]showed that the Hardy-Berndt sum s(h,k)can be

expressed explicitly in terms of Dedekind sums.

Theorem  Let h and k be coprime positive integers.If h+k is even,then

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and if h+k is odd,then

s(h,k) = . ()

The next theorem will be useful for the further sections.

Theorem  If both h and k are odd and(h,k) = ,then

S(h,k) =S(k,h) = .

A proof of this theorem was given by Apostol in []. In [], Simsek deﬁned a new sum related to the sumss(h,k) as follows:

Lethandkbe integers with (h,k) = 

Y(h,k) = k

k–

j=

(–)j+[hjk]

j k

.

The reciprocity law for the Simsek sumY(h,k) is given by (cf. [, p., Theorem ])

hY(h,k) +kY(k,h) = hk– .

1.1 Three-term polynomial relations for the Hardy sums

Here, two and three-term polynomial relations, which were studied thoroughly in [] and [], are recalled. In [, , ], and [] some new theorems on three-term relations for the Hardy sums were found by applying derivative operator to the three-term polynomial relation. Throughout this section, we assume thata,b, andcare pairwise coprime positive integers anda,b, andcsatisfy

aa≡ (modb), bb≡ (modc), and cc≡ (moda).

The following corollary was given by Pettet and Sitaramachandrarao [].

Corollary (Three and two-term polynomial relations) If a,b,and c are pairwise coprime positive integers,then

(u– ) a–

x=

ux–v[bxa]_w[cxa]_{+ (}_v_{– )}

b–

y=

vy–w[cyb]_u[ ay

b]_{+ (}_w_{– )}

c–

z=

wz–u[azc]_v[bzc]

=ua–vb–wc–– , ()

(u– ) a–

x=

ux–v[bxa]_{+ (}_v_{– )}

b–

y=

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The identity () is originally due to Berndt and Dieter []. The next corollary, which is equivalent to (), was ﬁrst established by Carlitz [].

Corollary [] If a and b are coprime positive integers,then

(u– )

We need the following relations, which were proved by Pettet and Sitaramachandrarao []:

In this section, we define some new finite sums which are related to not only the functions [x] and ((x)), but also the Dedekind sums, the Hardy-Berndt sums, the Simsek sumY(h,k), and the other finite sums. We also investigate the reciprocity laws of these sums. We also ask two open questions for these reciprocity laws.

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fol-C(a,a, . . . ,an–;an;k) = an–

j=

jk(–)j+[aanj]+···+[an– j an ]_,

wheren≥ is a positive integer.

Substitutingn=  andk=  into (), we have

Y(a;a) =

a– j=

(j– )(–)j+[

aj a] aj

a

, ()

B(a;a) =

a–

j=

(–)j+[

aj a_] aj

a

, ()

and

C(a;a; ) =

a–

j=

j(–)j+[

aj a_]_.

Combining the function [x] =x– ((x)) –_ with () and (), we get

Y(a,a) = –(a+a)s(a,a) +C(a,a) – C(a,a)

+a

a

C(a;a; ) –



S(a,a) and

B(a,a) =as(a,a) –C(a,a) +



S(a,a), where

C(a,a) =

a– j=

(–)j+[

aj a]

aj

a

and

C(a,a) =

a– j=

j(–)j+[

aj a_]

aj

a

.

Thus, our new deﬁnitions are related to the Hardy-Berndt sums and also the Simsek sum

Y(a,a). The reciprocity laws for the special ﬁnite sums, that is, the Dedekind type sums,

the Hardy-Berndt type sums, and the Simsek sum, are very important. Therefore, we are ready to give the reciprocity law of the sumsYn–(a,a, . . . ,an–;an) by the following the-orem.

Theorem  Let n be a natural number with n≥and a,a, . . . ,anbe positive integers,

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Yn–(a,a, . . . ,an;a) +Yn–(a,a, . . . ,an;a) +· · ·+Yn–(a,a, . . . ,an–;an) =

n

m=

(–)am–₍_a

m– ).

Leta,a, . . . ,anbe positive integers, relatively prime in pairs. Then we deﬁne the fol-lowing sums:

YSk,n(a, . . . ,an–;an) = an–

j=

jk

ak n

aj

an

aj

an

· · ·

an–j

an

. ()

Substitutingk=  andn=  into (), we arrive at the Dedekind sums

YS,(a;a) =s(a;a) =



a

a–

j=

j

aj

a

,

where (a;a) = .

Substitutingn=  into (), we obtain

YSk,(a;a) =

a–

j=

jk

ak_

aj

a

.

Open questions

() Forn≥, ﬁnd the reciprocity laws of the sumsYSk,n(a,a, . . . ,an–;an)and

Bn–(a,a, . . . ,an;a). That is, ﬁnd

YSk,n(a,a, . . . ,an;a) +YSk,n(a,a, . . . ,an;a) +· · ·+YSk,n(a,a, . . . ,an–;an) and

Bn–(a,a, . . . ,an;a) +Bn–(a,a, . . . ,an;a) +· · ·+Bn–(a, . . . ,an–;an). () Fork> , ﬁnd the reciprocity law ofC(a,a, . . . ,an–;an;k). That is, evaluate

C(a,a, . . . ,an;a;k) +C(a,a, . . . ,an;a;k) +· · ·+C(a, . . . ,an–;an;k).

3 PDE for the Carlitz polynomials and their applications

In this section, we study on the Carlitz polynomials and their properties (cf.[, , , , ], and the references cited in each of these earlier works). We find a PDE for this polynomial. We give many applications for this PDE, which are related to the Dedekind-Rademacher sums, the Hardy-Berndt sums, and the other finite sums. In [], Beck defined generalized the Carlitz polynomials as follows.

Deﬁnition (The Carlitz polynomial) c(u,u, . . . ,un;a,a, . . . ,an), where u,u, . . . ,un are indeterminants anda,a, . . . ,anare positive integers, is deﬁned as the polynomial

c(u,u, . . . ,un;a,a, . . . ,an) = a–

k=

uk_–u[ ka_

a_]

 u [kaa_]

 · · ·u [kana_]

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Theorem (Berndt-Dieter) If a,a, . . . ,anare pairwise relatively prime positive integers,

Proof of Theorem  was given by Beck in []. Now we will give a new deﬁnition:

Deﬁnition  Letu,v,wbe indeterminants. Leta,b,cbe positive integers, relatively prime in pairs, and letk,k,kbe positive natural numbers, which are the orders of the

deriva-By using (), we derive the following theorem, which is very important and valuable to

obtain some new and old identities related to the function [·], the Dedekind sums, the

Hardy-Berndt sums, and the Simsek sumY(h,k).

Theorem  The following identity holds true:

F(u,v,w;a,b,c;k,k,k) = (a– )k(b– )k(c– )ku

a–_vb–_wc–_. _()

Proof By using (), we have

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First we take the partial derivative ofP(u,v,w) with respect tou, then we have

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That is,

G(u,v,w) =k

a–

x=

(x– )k

bx a

k

ux–k_v[bxa]–k_w[cxa]

+ (u– ) a–

x=

(x– )k+

bx a

k

ux–(k+)_v[bxa]–k_w[cxa]

+k

b–

y=

ay b

k

(y– )kv

y–k_w[cy_b]_u[ay_b]–k

+ (v– ) b–

y=

ay b

k

(y– )k+vy–(k+)w[

cy b]u[

ay b]–k

+ (w– ) c–

z=

az c

k

bz c

k

wz–u[azc]–k_v[bzc]–k

= (a– )k(b– )ku

a–(k+)_vb–(k+)_wc–_.

Finally, if we also takektimes partial derivative ofG(u,v,w), with respect tow, then we

obtain the desired result.

Substitutingu=u=u=  into Deﬁnition , we get

F(, , ;a,b,c;k,k,k)

=k

a–

x=

(x– )k–

bx a

k

cx a

k +k

b–

y=

(y– )k–

ay b

k

cy b

k

+k

c–

z=

(z– )k–

az c

k

bz c

k

and by () we arrive at the following corollary.

Corollary 

F(, , ;a,b,c;k,k,k) := (a– )k(b– )k(c– )k. ()

Corollary  If we substitute k= ,k= ,and k= into(),we get

F(, , ;a,b,c; , , ) =a– ,

if we substitute k= ,k= ,and k= into(),we get

F(, , ;a,b,c; , , ) =b– ,

and ﬁnally if we substitute k= ,k= ,and k= into(),we get

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Remark  If we substitutek=k= ,k=  in (), then we have the following

well-known reciprocity law of the function [x], which is very important to prove the Gauss law of quadratic reciprocity:

a–

x=

bx a

+

b–

y=

ay b

= (a– )(b– ) ()

(cf. [, , ], and the references cited in each of these earlier works).

Remark  If we substituteu=w= –,v=  andk=k= ,k=  into (), we get

F(–, , –;a,b,c; , , ) = – a–

x=

bx a

(–)x+[cxa]–

+ b–

y=

(–)[cyb]+[ ay

b]– 

c–

z=

bz c

(–)z+[azc]–_. _()

We also know from () that

F(–, , –;a,b,c; , , ) = (b– )(–)a+c–. ()

By combining () and (), we get

– a–

x=

bx a

(–)x+[cxa]–₊

b–

y=

(–)[cyb]+[ ay

b]_{– }

c–

z=

bz c

(–)z+[azc]–

= (b– )(–)a+c–,

which gives us Theorem . in [], so we have

F(–, , –;a,b,c; , , ) =s

ac,b– s

cb,a– s

ab,c

or equivalently

F(–, , –;a,b,c; , , ) =b–ac

ac .

Remark  If we substituteu=v= ,w= – andk=k= ,k=  into (), we get

F(, , –;a,b,c; , , ) = a–

x=

(–)[cxa] bx a

+

b–

y=

(–)[cy_b] ay

b

–  c–

z=

(–)z– az

c bz

c

;

we also know from () that

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so if we use these two equations together, then we get

a–

x=

(–)[cxa] bx a

+

b–

y=

(–)[cyb] ay b

– 

c–

z=

(–)z– az

c bz

c

= (a– )(b– )(–)c–,

which gives us Theorem . in [], so we have

F(, , –;a,b,c; , , ) = s

ab,c–s

cb,a–s

ca,b

or equivalently

F(, , –;a,b,c; , , ) = –

+

 c

a c+

b c

.

Remark  If we substituteu= –,v=w=  andk= ,k=k=  into (), we get

F(–, , ;a,b,c; , , ) = – a–

x=

(–)x– bx

a

+

b–

y=

(–)[ayb].

We also know from () that

F(–, , ;a,b,c; , , ) = (b– )(–)a–

– a–

x=

(–)x– bx

a

+

b–

y=

(–)[ayb]_{= (}_b_{– )(–)}a–_,

which gives us Theorem . in [], so we have

F(–, , ;a,b,c; , , ) = s(b,a) –s(a,b)

or equivalently

F(–, , ;a,b,c; , , ) =  –b

a.

Remark  If we substituteu=v= –,w=  andk=k= ,k=  into (), we get

F(–, –, ;a,b,c; , , )

= – a–

x=

(–)x+[bxa]– cx a

– 

b–

y=

(–)y+[ayb]– cy b

+

c–

z=

(–)[azc]+[bzc]_;

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F(–, , ;a,b,c; , , ) = (b– )(c– )(–)a–,

–

We can also have some results from [] by using the same method as follows.

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F(, –, –;a,b,c; , , ) = (a– )(–)b+c,

a–

x=

(–)[bxa]+[cxa]_{– }

b–

y=

(–)y+[cyb] ay b

– 

c–

z=

(–)z+[bzc]– az c

= (a– )(–)b+c,

which gives us Theorem . in [], so we have

F(, –, –;a,b,c; , , ) =s

ab,c+s

ca,b– s

bc,a

or equivalently

F(, –, –;a,b,c; , , ) =

 –

a

bc,

whereaandaare even andcc≡ (moda).

Remark  If we substituteu=w= ,v= – andk=k= ,k=  into (), we get

F(, –, ;a,b,c; , , )

= a–

x=

(–)[bxa] cx a

+ 

b–

y=

(–)y ay

b cy

b

+

c–

z=

(–)[bzc] az c

.

From (), we see that

F(, –, ;a,b,c; , , ) = (a– )(c– )(–)b–.

Therefore

a–

x=

(–)[bxa] cx a

+ 

b–

y=

(–)y ay

b cy

b

+

c–

z=

(–)[bzc] az c

= (a– )(c– )(–)b–,

which gives us Theorem . in [], so we have

F(, –, ;a,b,c; , , ) =s

bc,a– s

ca,b+s

ab,c

or equivalently

F(, –, ;a,b,c; , , ) =

–

 

a bc+

c ab

,

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Remark  If we substituteu=v=w= , andk=k=k= , into (), we get

F(, , ;a,b,c; , , ) = (a– )(b– )(c– ).

(cf. [, (.)], and the references cited in each of these earlier works).

By the mathematical induction method, we shall generalize Theorem . But ﬁrst we need a new deﬁnition.

Deﬁnition  Let u, . . . ,un be indeterminants. Leta, . . . ,an be positive integers,

rela-Now we can give the generalization of Theorem  as follows.

Theorem  Let u,u, . . . ,unbe indeterminants.Let a,a, . . . ,anbe positive integers,

rel-atively prime in pairs,and let k,k, . . . ,kn be positive integers. For n≥, the following

identity holds true:

F(u, . . . ,un;a, . . . ,an;k, . . . ,kn) =

Remark  If we substituten=  in Theorem , then Theorem  reduces to Theorem .

Corollary  Substituting u=u=· · ·=un= and k=k=· · ·=kn= into Deﬁnition

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a–

A proof of Corollary  was given by Beck [, Corollary .].

Remark  Substitutingn=  into (), we easily have

a–

 into the above equation, one can arrive at the reciprocity

law of the Dedekind-Rademacher sums:

a–

(cf.[, ], and the references cited in each of these earlier works).

Substitutingu=u=· · ·=un=  into Deﬁnition  and Theorem , then we arrive at the following result.

Corollary  Let a,a, . . . ,an be positive integers, relatively prime in pairs, and let tity, which was proved by Berndt and Dieter [, p., Corollary .]:

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Substituting u=u =· · ·=un= – into Deﬁnition  and Theorem , we obtain the following corollary.

Corollary 

n

l=

al–

xl=

(xl– )kl+kl(xl– )kl–

n

m=,m=l

amxl

al

km

(–)xl+[amxl_al ]

= n

m=

(–)am–₍_a

m– )km. ()

Remark  Substitutingn=  andk=k= , in Corollary , then we arrive at

a–

x=

(x– )(–)x+[

ax

a_ ] ax a

+

a–

x=

(x– )(–)x+[

ax

a_ ] ax a

= (–)a+a₍_a

– )(a– ).

We are now ready to give a proof of Theorem .

Proof of Theorem Substitutingk=k=· · ·=kn=  into (), we obtain n

l=

al–

xl=

(xl– ) n

m=,m=l

(–)xl+[amxl_al ] amxl al

=

n

m=

(–)am–₍_a

m– ).

Combining the above equation with (), we get

Yn–(a,a, . . . ,an;a) +Yn–(a,a, . . . ,an;a) +· · ·+Yn–(a,a, . . . ,an–;an) =

n

m=

(–)am–₍_a

m– ).

Hence, we arrive at the desired result.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors contributed equally to the writing of this paper. All authors read and approved the ﬁnal manuscript.

Author details

1_{Department of Mathematics, Uludag University, Gorukle, Bursa 16059, Turkey.}2_{Department of Mathematics, Akdeniz}

University, Antalya, 07058, Turkey.

Acknowledgements

The authors are supported by the research funds of Akdeniz University and Uludag University (Uludag University project numbers are 201424 and 201220).

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References

1. Berndt, BC: Analytic Eisenstein series, Theta-functions, and series relations in the spirit of Ramanujan. J. Reine Angew. Math.303/304, 332-365 (1978)

2. Goldberg, LA: Transformation of Theta-functions and analogues of Dedekind sums. Thesis, University of Illinois Urbana (1981)

3. Simsek, Y: Relations between theta-functions Hardy sums Eisenstein series and Lambert series in the transformation formula of logηg,h(z). J. Number Theory99, 338-360 (2003)

4. Apostol, TM: Modular Functions and Dirichlet Series in Number Theory. Springer, Berlin (1976) 5. Apostol, TM, Vu, TH: Elementary proofs of Berndt’s reciprocity laws. Pac. J. Math.98, 17-23 (1982)

6. Beck, M: Geometric proofs of polynomial reciprocity laws of Carlitz, Berndt, and Dieter. In: Diophantine Analysis and Related Fields. Sem. Math. Sci., vol. 35, pp. 11-18 (2006)

7. Berndt, BC, Dieter, U: Sums involving the greatest integer function and Riemann Stieltjes integration. J. Reine Angew. Math.337, 208-220 (1982)

8. Berndt, BC, Goldberg, LA: Analytic properties of arithmetic sums arising in the theory of the classical Theta-functions. SIAM J. Math. Anal.15, 143-150 (1984)

9. Carlitz, L: Some polynomials associated with Dedekind sums. Acta Math. Acad. Sci. Hung.26, 311-319 (1975) 10. Hardy, GH: On certain series of discontinues functions connected with the modular functions. Q. J. Math.36, 93-123

(1905)

11. Pettet, MR, Sitaramachandrarao, R: Three-term relations for Hardy sums. J. Number Theory25, 328-339 (1989) 12. Rademacher, H: Generalization of the reciprocity formula for Dedekind sums. Duke Math. J.21, 391-397 (1954) 13. Simsek, Y: On generalized Hardy sumss5(h,k). Ukr. Math. J.56(10), 1434-1440 (2004)

14. Simsek, Y: Theorems on three-term relations for Hardy sums. Turk. J. Math.22, 153-162 (1998) 15. Simsek, Y: A note on Dedekind sums. Bull. Calcutta Math. Soc.85, 567-572 (1993)

16. Simsek, Y: On analytic properties and character analogs of Hardy sums. Taiwan. J. Math.13, 253-268 (2009) 17. Sitaramachandrarao, R: Dedekind and Hardy sums. Acta Arith.48, 325-340 (1987)

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