Circular summation formulas for theta function ϑ4(z|τ)

R E S E A R C H

Open Access

Circular summation formulas for theta

function

ϑ

_

(z

|

τ

)

Yun Zhou

_{and Qiu-Ming Luo}

*_{Correspondence:}

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

Chongqing Normal University, Chongqing Higher Education Mega Center, Huxi Campus, Chongqing, 401331, People’s Republic of China Full list of author information is available at the end of the article

Abstract

In this paper, we obtain the circular summation formulas of the theta function

ϑ

τ

MSC: Primary 11F27; secondary 33E05

Keywords: elliptic functions; Ramanujan’s circular summation; circular summation; theta functions; theta function identities

1 Introduction

Throughout this paper we takeq=eπiτ_{,Im(τ) > ,z∈C. The classical Jacobi’s four theta} functionsϑi(z|τ),i= , , , , are defined as follows, with the notation of Tannery and

Molk (see,e.g., [–]):

ϑ(z|τ) = –iq  

∞

n=–∞

(–)nqn(n+)e(n+)iz, (.)

ϑ(z|τ) =q  

∞

n=–∞

qn(n+)e(n+)iz, (.)

ϑ(z|τ) =

∞

n=–∞

qneniz, (.)

ϑ(z|τ) =

∞

n=–∞

(–)nqneniz. (.)

From the Jacobi theta functions (.)-(.), via the direct calculation and applying the in-duction, we have the following properties, respectively:

ϑ(z+nπ|τ) = (–)nϑ(z|τ), ϑ(z+nπ τ|τ) = (–)nq–n 

e–nizϑ(z|τ), (.)

ϑ(z+nπ|τ) = (–)nϑ(z|τ), ϑ(z+nπ τ|τ) =q–n 

e–nizϑ(z|τ), (.)

ϑ(z+nπ|τ) =ϑ(z|τ), ϑ(z+nπ τ|τ) =q–n 

e–nizϑ(z|τ), (.)

ϑ(z+nπ|τ) =ϑ(z|τ), ϑ(z+nπ τ|τ) = (–)nq–n 

e–nizϑ(z|τ). (.)

From the Jacobi theta functions (.)-(.), via a direct calculation, we also have the fol-lowing transformation formulas:

ϑ

From the above equations we easily obtain

ϑ

From (.)-(.), we can obtain the following lemmas.

Lemma . For n any positive integer,we have

ϑ

Lemma . For n any positive integer,we have

ϑ

z+nπ τ 

τ

=

⎧ ⎨ ⎩

q–n_e–niz_ϑ(z|_{τ), n is even,}

q–n



_e–niz_ϑ(z|_{τ), n is odd,}

(.)

ϑ

z+nπ τ 

τ

=

⎧ ⎨ ⎩

in_q–n__e–niz_ϑ

(z|τ), n is even,

in_q–n_e–niz_ϑ(z|_{τ), n is odd.}

(.)

On p. in Ramanujan’s lost notebook (see [, p., Entry ..], or [, p.]), Ramanu-jan recorded the following claim (without proof ), which is now well known as RamanuRamanu-jan’s circular summation. The appellation of ‘circular summation’ is due to Son (see [, p.]).

Theorem .(Ramanujan’s circular summation) For each positive integer n and|ab|< ,

–n/<r≤n/

_∞

k=–∞

k≡r(modn)

ak(k+)/(n)bk(k–)/(n)

n

=f(a,b)Fn(ab), (.)

where

Fn(q) :=  + nq(n–)/+· · ·, n≥. (.)

Ramanujan’s theta functionf(a,b) is defined by

f(a,b) = ∞

n=–∞

an(n+)/bn(n–)/, |ab|< . (.)

By the definition of Ramanujan’s theta function above and routine calculations, we can rewrite Ramanujan’s circular summation (.) as follows (see, for details [, p.]).

Theorem .(Ramanujan’s circular summation) Let Uk=ak(k+)/(n)and Vk=bk(k–)/(n).

For each positive integer n and|ab|< ,

n–

k=

U_knfn

Un+k

Uk

,Vn–k Vk

=f(a,b)Fn(ab), (.)

where

Fn(q) :=  + nq(n–)/+· · ·, n≥. (.)

If we are going to apply the transformationτ−→–_τ to Ramanujan’s identity, it will be convenient to convert Ramanujan’s theorem into one involving the classical theta func-tionϑ(z|τ) defined by (.). Surprisingly Chanet al.[] prove that Theorem . below is

equivalent to Theorem ..

Theorem .(Ramanujan’s circular summation) For any positive integer n≥,

n–

k=

When n≥,

Fn(q) =  + nqn–+· · ·. (.)

Chanet al.[] also showed that Theorem . below is equivalent; we have Theorem . by applying the Jacobi imaginary transformation formula [, p.].

Theorem .(Ramanujan’s circular summation) For any positive integer n,there exists a

quantity Gn(τ)such that n–

k=

ϑ_n

z+kπ n

τ

=Gn(τ)ϑ(nz|nτ), (.)

where

Gn(τ) =

√

n(–iτ)(–n)/Fn

–  nτ

. (.)

Ramanujan’s circular summation is an interesting subject in his notebook. On the sub-ject of Ramanujan’s circular summation and related identities of theta functions and their various extensions, a remarkably large number of investigations have appeared in the lit-erature (see, for example, Andrews, Berndt, Rangachari, Ono, Ahlgren, Chua, Murayama, Son, Chan, Liu, Ng, Chan, Shen, Cai, Zhu, and Xuet al.[–]).

Recently, Zeng [] extended Ramanujan’s circular summation in the following form.

Theorem . For any nonnegative integers n,k,a, and b with a+b=k,there exists a

quantity R(a,b;_aby,_knτ)such that

kn–

s=

ϑ_a

z kn+

y a+

πs kn

τ

kn

ϑ_b

z kn–

y b+

πs kn

τ

kn

=R

a,b; y ab,

τ kn

ϑ(z|τ). (.)

Chan and Liu [] further extended Theorem . in the following general form.

Theorem . Suppose y,y, . . . ,ynare n complex numbers such that y+y+· · ·+yn= ;

there exists a quantity Gm,n(y,y, . . . ,yn|τ)such that mn–

k=

n

j=

ϑ

z+yj+

kπ mn

τ

=Gm,n(y,y, . . . ,yn|τ)ϑ

mnz|mnτ, (.)

In the present paper, motivated by [, ], and [], by applying the theory and method of elliptic functions, we obtain the circular summation formulas of theta functionsϑ(z|τ)

2 Ramanujan’s circular summation formula for theta functions

ϑ

τ

In the present section, we obtain Ramanujan’s circular summation formula for the theta functionsϑ(z|τ). We now state our main result as follows.

Theorem . Suppose that m,n are any positive integers,p is any integer;y,y, . . . ,ynare

any complex numbers.

• Wheny+y+· · ·+yn=p_mπ +n_π,we have

Comparing (.) and (.), we have

=

mn–

k=

n

j=

(–)mq–n_e–im(mnz +yj+mnkπ)_ϑ 

z mn+yj+

kπ mn

τ

m_n

= (–)mnq–e–ize–im(y+y+···+yn)_{f(z). (.)}

• Wheny+y+· · ·+yn=p_mπ +n_π in(.), we have

f(z+π τ) =q–e–iz_{f(z). (.)}

We construct the function_ϑf(z)

(z|τ), by (.) forn= , (.) and (.), we find that the function

f(z)

ϑ(z|τ) is an elliptic function with double periodsπ andπ τ and hasonlya simple pole

at z= π_ +π τ_ in the period parallelogram. Hence the function _ϑf(z)

(z|τ) is a constant, say

C_,()(y,y, . . . ,yn;τ), and we have

f(z) =C_,()(y,y, . . . ,yn;τ)ϑ(z|τ),

or, equivalently,

mn–

k=

n

j=

ϑ

z mn+yj+

kπ mn

τ

m_n

=C()_,(y,y, . . . ,yn;τ)ϑ(z|τ). (.)

Letting

z−→mnz and τ−→mnτ

in (.), and then setting

R()_,(y,y, . . . ,yn;m,n,p;τ) =C,()

y,y, . . . ,yn;mnτ

,

we arrive at (.). Setting

z−→z+yj+

kπ mn

in (.), via the direct calculation, we obtain

n

j=

ϑ

z+yj+

kπ mn

τ

= ∞

r,...,rn=–∞

(–)r+···+rn_qr+···+rn

×e(r+···+rn)iz_ei(ry+···+rnyn)_emnkπi(r+···+rn)_{. (.)}

Setting

z−→mnz and τ−→mnτ

in (.), we get

ϑ

mnz|mnτ= ∞

r=–∞

Substituting (.) and (.) into (.), we find that

∞

r,...,rn=–∞

(–)r+···+rn_qr+···+rn_e(r+···+rn)iz_ei(ry+···+rnyn)

mn–

k=

emnkπi(r+···+rn)

=R()_,(y,y, . . . ,yn;m,n,p;τ)

∞

r=–∞

qmnremnriz. (.)

Equating the constants of both sides of (.), we get (.).

• Wheny+y+· · ·+yn=(p_+)_mπ +n_π in(.), we have

f(z+π τ) = –q–e–izf(z). (.)

We construct the function _ϑf(z)

(z|τ), by (.) forn= , (.) and (.), we find that the

func-tion _ϑf(z)

(z|τ) is an elliptic function with double periodsπ andπ τ and has onlya simple

pole atz=π τ_ in the period parallelogram. Hence the function _ϑf(z)

(z|τ) is a constant, say

C_,()(y,y, . . . ,yn;τ), and we have

f(z) =C_,()(y,y, . . . ,yn;τ)ϑ(z|τ),

or, equivalently,

mn–

k=

n

j=

ϑ

z mn+yj+

kπ mn

τ

m_n

=C()_,(y,y, . . . ,yn;τ)ϑ(z|τ). (.)

Letting

z−→mnz and τ−→mnτ

in (.), and then setting

R()_,(y,y, . . . ,yn;m,n,p;τ) =C,()

y,y, . . . ,yn;mnτ

,

we arrive at (.).

Similarly, we may obtain

R()_,(y,y, . . . ,yn;m,n,p;τ) =mn

∞

r+···+rn=

r,...,rn=–∞

qr+···+rn_ei(ry+···+rnyn)_{. (.)}

Clearly,

R,(y,y, . . . ,yn;m,n,p;τ) =R(),(y,y, . . . ,yn;m,n,p;τ) =R(),(y,y, . . . ,yn;m,n,p;τ).

The proof is complete.

Corollary .(Chan and Liu [, p., Theorem ]) Suppose that m,n are any positive integers;y,y, . . . ,ynare any complex numbers and y+y+· · ·+yn= .We have

mn–

k=

n

j=

ϑ

z+yj+

kπ mn

τ

=R,(y,y, . . . ,yn;m,n;τ)ϑ

mnz|mnτ, (.)

where

R,(y,y, . . . ,yn;m,n;τ) =mn

∞

r+···+rn=

r,...,rn=–∞

qr+···+rn_ei(ry+···+rnyn)_{. (.)}

Proof First, settingy+y+· · ·+yn=  in (.), we havep= –mn_ , noting thatpis any

integer, hence we say thatmnis even. Setting

z−→z+π 

in (.) of Theorem . and applying the properties (.) and (.), we have

mn–

k=

n

j=

ϑ

z+yj+

kπ mn

τ

=R,(y,y, . . . ,yn;m,n;τ)ϑ

mnz|mnτ. (.)

Next takingy+y+· · ·+yn=  in (.), we havep= –mn_+, noting thatpis any integer,

hence we say thatmnis odd. Setting

z−→z+π 

in (.) of Theorem . and applying the properties (.) and (.), this leads to the fol-lowing:

mn–

k=

n

j=

ϑ

z+yj+

kπ mn

τ

=R,(y,y, . . . ,yn;m,n;τ)ϑ

mnz|mnτ. (.)

Obviously, by (.) and (.), for any positive integersmn, we obtain (.).

We easily see that theR()_,(y,y, . . . ,yn;m,n,p;τ) andR(),(y,y, . . . ,yn;m,n,p;τ) are

in-dependent ofz, therefore, wheny+y+· · ·+yn= , we find that

R,(y,y, . . . ,yn;m,n;τ) =R(),(y,y, . . . ,yn;m,n;τ) =R(),(y,y, . . . ,yn;m,n;τ).

The proof is complete.

Remark . Obviously, Theorem . implies the well-known result (.), but we see that

from (.) we do not obtain Theorem .. However, we can reformulate the results of Chan and Liu as Theorem ..

Suppose thatm,nare any positive integers,pis any integer;y,y, . . . ,ynare any complex

• Wheny+y+· · ·+yn=p_mπ, we have mn–

k=

n

j=

ϑ

z+yj+

kπ mn

τ

=R,(y,y, . . . ,yn;m,n,p;τ)ϑ

mnz|mnτ. (.)

• Wheny+y+· · ·+yn=(p_+)_mπ, we have mn–

k=

n

j=

ϑ

z+yj+

kπ mn

τ

=R,(y,y, . . . ,yn;m,n,p;τ)ϑ

mnz|mnτ. (.)

Here

R,(y,y, . . . ,yn;m,n,p;τ) =mn

∞

r+···+rn=

r,...,rn=–∞

qr+···+rn_ei(ry+···+rnyn)_{. (.)}

3 The imaginary transformations formulas for Theorem 2.1

In the present section, we derive the corresponding imaginary transformations formulas for Theorem ..

Theorem . Suppose that m,n are any positive integers,p is any integer;y,y, . . . ,ynare

any complex numbers.

• Wheny+y+· · ·+yn=mnp+m

_n

 ,we have

mn–

k=

qk+kp+mnke(k+p+mn)iz

n

j=

ϑ

mz+yjπ τ+mkπ τ|mnτ

=F_,()(y,y, . . . ,yn;m,n,p;τ)ϑ(z|τ). (.)

• Wheny+y+· · ·+yn=(p+)_mn+m

_n

 ,we have

mn–

k=

qk+k(p+)+mnke(k+p+mn+)iz

n

j=

ϑ

mz+yjπ τ+mkπ τ|mnτ

=F_,()(y,y, . . . ,yn;m,n,p;τ)ϑ(z|τ). (.)

Here

F_,()(y,y, . . . ,yn;m,n,p;τ)

=(–iτ)

–n



(m_n)n

q– y_{ +}···+yn

mn _R() ,

yπ

m_n,

yπ

m_n, . . . ,

ynπ

m_n;m,n,p; –

 m_nτ

,

(.)

F_,()(y,y, . . . ,yn;m,n,p;τ)

=

mn–

k=

q–(k+mn )+m _n 

∞

r,...,rn=–∞

m(r+···+rn)=mn+k+p

F_,()(y,y, . . . ,yn;m,n,p;τ)

Applying the imaginary transformation formulas (see,e.g., [–])

ϑ

 , and simplifying, we thus obtain (.) and (.). Applying the series

expressions (.) and (.) in (.), via the direct calculation, we obtain (.). In a similar manner, we use the imaginary transformation formula:

ϑ

(.). Therefore we complete the proof of Theorem ..

4 The alternating Ramanujan’s circular summation formula

In this section, we will obtain the corresponding alternating Ramanujan’s circular sum-mation formula from Theorem ..

Theorem . Suppose that m,n are any positive integers,p is any integer;y,y, . . . ,ynare

any complex numbers.

mn–

k=

(–)k

n

j=

ϑ

z+yj+

kπ mn

τ

= (–)pR,(y,y, . . . ,yn;m,n,p;τ)ϑ

mnz|mnτ, mis odd. (.)

• Wheny+y+· · ·+yn=(p_+)_mπ +n_π,we have mn–

k=

(–)k

n

j=

ϑ

z+yj+

kπ mn

τ

= (–)p+R,(y,y, . . . ,yn;m,n,p;τ)ϑ

mnz|mnτ, mis even, (.)

mn–

k=

(–)k

n

j=

ϑ

z+yj+

kπ mn

τ

= (–)p+R,(y,y, . . . ,yn;m,n,p;τ)ϑ

mnz|mnτ, mis odd. (.)

Here

R,(y,y, . . . ,yn;m,n,p;τ) =mn

∞

r+···+rn=

r,...,rn=–∞

qr+···+rn_ei(ry+···+rnyn)_{. (.)}

Proof Let

z−→z+mπ τ 

in Theorem ..

By using (.), (.), and (.) we compute

n

j=

ϑ

z+yj+

kπ mn+

mπ τ 

τ

=

⎧ ⎨ ⎩

(–)k_imn_q–m_n_e–mniz_e–mi(y+···+yn)n

j=ϑ(z+yj+_mnkπ|τ), mis even,

(–)k_imn_q–mn_e–mniz_e–mi(y+···+yn)n

j=ϑ(z+yj+k_mnπ|τ), mis odd,

(.)

ϑ

mnz+m

_{nπ τ}



mnτ

=q–m

_n  _e–mniz_ϑ



mnz|mnτ, (.)

ϑ

mnz+m

_{nπ τ}



mnτ

=iq–m

_n

 _e–mniz_ϑmnz|_m_{nτ. (.)}

Substituting (.) and (.) into (.) of Theorem . and simplifying, we get (.) of Theorem ..

Substituting (.) and (.) into (.) of Theorem . and simplifying, we arrive at (.)

of Theorem .. This proof is complete.

Corollary . Suppose that m,n are any positive integers,p is any integer;y,y, . . . ,ynare

• Wheny+y+· · ·+yn= ,we have mn–

k=

(–)k

n

j=

ϑ

z+yj+

kπ mn

τ

=i–mn_R

,(y,y, . . . ,yn;m,n,p;τ)ϑ

mnz|m_{nτ, mis even, (.)}

mn–

k=

(–)k

n

j=

ϑ

z+yj+

kπ mn

τ

=i–mnR,(y,y, . . . ,yn;m,n,p;τ)ϑ

mnz|mnτ,

mis odd andnis even. (.)

• Wheny+y+· · ·+yn=_π_m,we have mn–

k=

(–)k+

n

j=

ϑ

z+yj+

kπ mn

τ

=i–mn_R

,(y,y, . . . ,yn;m,n,p;τ)ϑ

mnz|mnτ, mis even, (.)

mn–

k=

(–)k+

n

j=

ϑ

z+yj+

kπ mn

τ

=i–mnR,(y,y, . . . ,yn;m,n,p;τ)ϑ

mnz|mnτ,

mis odd andnis even. (.)

• Wheny+y+· · ·+yn= andm,nare odd,we have mn–

k=

(–)k

n

j=

ϑ

z+yj+

kπ mn

τ

=imn–R,(y,y, . . . ,yn;m,n,p;τ)ϑ

mnz|mnτ. (.)

Here

R,(y,y, . . . ,yn;m,n,p;τ) =mn

∞

r+···+rn=

r,...,rn=–∞

qr+···+rn_ei(ry+···+rnyn)_{. (.)}

Proof Ifmnis even, settingp= –mn_ iny+y+· · ·+yn=p_mπ +n_π, then we havey+y+ · · ·+yn=  and (–)p=i–mn. We directly deduce (.) of Corollary ..

Ifmnis an even, settingp= –mn_ iny+y+· · ·+yn=p_mπ+nπ+

π

, we havey+y+· · ·+yn=

π

m. We get (.) of Corollary ..

Ifmnis an odd, settingp= –mn_+ iny+y+· · ·+yn=p_mπ +n_π +π_ and (–)p=i–mn–,

we havey+y+· · ·+yn= . We get (.) of Corollary .. The proof is complete.

Corollary . Suppose that m,n are any positive integers.We have

mn–

k=

(–)kϑ_n

z+kπ mn

τ

=i–mnR,(m,n;τ)ϑ

mn–

k=

(–)kϑ_n

z+kπ mn

τ

=i–mnR,(m,n;τ)ϑ

mnz|mnτ, m is odd and nis even, (.)

mn–

k=

(–)kϑ_n

z+kπ mn

τ

=imn–R,(m,n;τ)ϑ

mnz|mnτ, m,n are odd, (.)

mn–

k=

(–)k+ϑ_n

z+ π mn+

kπ mn

τ

=i–mnR,(m,n;τ)ϑ

mnz|mnτ, m is even, (.)

mn–

k=

(–)k+ϑ_n

z+ π mn+

kπ mn

τ

=i–mnR,(m,n;τ)ϑ

mnz|mnτ, m is odd and n is even, (.)

where

R,(m,n;τ) =mn

∞

r+···+rn=

r,...,rn=–∞

qr+···+rn_ei(ry+···+rnyn)_{. (.)}

Proof We sety=y=· · ·=yn=  in (.), (.), and (.) of Corollary .. We take

y=y=· · ·=yn=_π_mnin (.) and (.) of Corollary ..

Remark . Takingm=  in (.) and (.), and noting that

R,(n;τ) =R,(n;τ) =n

∞

r+···+rn=

r,...,rn=–∞

qr+···+rn_,

we have

n–

k=

(–)kϑ_n

z+kπ n

τ

=

i–n_R

,(n;τ)ϑ(nz|nτ), nis even,

i–n_R

,(n;τ)ϑ(nz|nτ), nis odd,

(.)

which is just Chan’s result (see [, p., Theorem .]). Hence (.) and (.) of Corol-lary . are the corresponding extension of Chan’s result.

Remark . We note that Chan’s result (see [, p., Theorem .]) has a misprint; we

have here corrected this point in (.).

Remark . Settingm=  in (.), we have

n–

k=

(–)kϑ_n

z+ π n+

kπ n

τ

Settingn=  in (.), we have

The above two formulas are the analogs of Boon’s results (see [, p.]).

Theorem . Suppose that m,n are any positive integers,p is any integer;y,y, . . . ,ynare

any complex numbers.

Here

R,(y,y, . . . ,yn;m,n,p;τ) =mn

∞

r+···+rn=

r,...,rn=–∞

qr+···+rn_ei(ry+···+rnyn)_{. (.)}

Proof Let

z−→z+π  +

mπ τ 

in Theorem ..

By using (.), (.), and (.) we have

n

j=

ϑ

z+yj+

kπ mn+

π  +

mπ τ 

τ

=

⎧ ⎨ ⎩

(–)k_q–mn_e–mniz_e–mi(y+···+yn)n

j=ϑ(z+yj+_mnkπ|τ), mis even,

(–)k+n_q–m_n_e–mniz_e–mi(y+···+yn)n

j=ϑ(z+yj+_mnkπ|τ), mis odd,

(.)

ϑ

mnz+m

_{nπ τ}

 +

mnπ 

mnτ

=

⎧ ⎨ ⎩

q–mn_e–mniz_ϑ(mnz|_m_{nτ), mnis even,}

–iq–m

_n

 _e–mniz_ϑ(mnz|_m_{nτ), mnis odd,}

(.)

ϑ

mnz+m

_{nπ τ}

 +

mnπ 

mnτ

=

⎧ ⎨ ⎩

iq–m

_n

 _e–mniz_ϑ(mnz|_m_{nτ), mnis even,}

q–m

_n

 _e–mniz_ϑ(mnz|_m_{nτ), mnis odd.}

(.)

Substituting (.) and (.) into (.) of Theorem . and simplifying, we get (.), (.), and (.) of Theorem ..

Substituting (.) and (.) into (.) of Theorem . and simplifying, we arrive at (.), (.), and (.) of Theorem .. This proof is complete.

Corollary . Suppose that m,n are any positive integers;y,y, . . . ,ynare any complex

numbers.

• Wheny+y+· · ·+yn= andmis even,we have mn–

k=

(–)k

n

j=

ϑ

z+yj+

kπ mn

τ

=R,(y,y, . . . ,yn;m,n;τ)ϑ

mnz|mnτ. (.)

• Wheny+y+· · ·+yn= andmis odd andnis even,we have mn–

k=

(–)k

n

j=

ϑ

z+yj+

kπ mn

τ

=R,(y,y, . . . ,yn;m,n;τ)ϑ

• Wheny+y+· · ·+yn= andmis odd andnis odd,we have mn–

k=

(–)k+

n

j=

ϑ

z+yj+

kπ mn

τ

=R,(y,y, . . . ,yn;m,n;τ)ϑ

mnz|mnτ. (.)

• Wheny+y+· · ·+yn=_π_m andmis even,we have

mn–

k=

(–)k

n

j=

ϑ

z+yj+

kπ mn

τ

=R,(y,y, . . . ,yn;m,n;τ)ϑ

mnz|mnτ. (.)

• Wheny+y+· · ·+yn=_π_m andmis odd andnis even,we have

mn–

k=

(–)k

n

j=

ϑ

z+yj+

kπ mn

τ

=R,(y,y, . . . ,yn;m,n;τ)ϑ

mnz|mnτ. (.)

• Wheny+y+· · ·+yn= andm,nare odd,we have mn–

k=

(–)k

n

j=

ϑ

z+yj+

kπ mn

τ

=R,(y,y, . . . ,yn;m,n;τ)ϑ

mnz|mnτ. (.)

Here

R,(y,y, . . . ,yn;m,n;τ) =mn

∞

r+···+rn=

r,...,rn=–∞

qr+···+rn_ei(ry+···+rnyn)_.

(.)

Proof Ifmnis an even, settingp= –mn

 iny+y+· · ·+yn=

pπ

m + nπ

 , then we havey+y+ · · ·+yn=  and (–)p=i–mn. We directly deduce (.) and (.) of Corollary . from

Theorem ..

Ifmnis an even, settingp= –mn_ iny+y+· · ·+yn=p_mπ +n_π +_π_m, we havey+y+ · · ·+yn=_π_m. We get (.), (.), and (.) of Corollary . from Theorem ..

Ifmnis an odd, settingp= –mn_+ iny+y+· · ·+yn=p_mπ +n_π +_π_m and (–)p=i–mn–,

we havey+y+· · ·+yn= . We get (.) of Corollary . from Theorem .. The proof

is complete.

Remark . Equation (.) of Corollary . is just the main result of Zhu (see [,

j= , , . . . ,n, in (.); we readily deduce that

R,(y,y, . . . ,yn;m,n;τ) =mnq–

mn



∞

r+···+rn=mn

r,...,rn=–∞

qr+···+rn_ei(ry+···+rnyn)_{, (.)}

which just is the corresponding result of Zhu (see [, p., () of Theorem .]).

Corollary . Suppose that m,n are any positive integers.We have

mn–

k=

(–)kϑ_n

z+kπ mn

τ

=R,(m,n;τ)ϑ

mnz|mnτ, m is even, (.)

mn–

k=

(–)kϑ_n

z+kπ mn

τ

=R,(m,n;τ)ϑ

mnz|mnτ, m is odd and n is even, (.)

mn–

k=

(–)kϑ_n

z+kπ mn

τ

=R,(m,n;τ)ϑ

mnz|mnτ, m,n are odd, (.)

mn–

k=

(–)kϑ_n

z+ π mn+

kπ mn

τ

=R,(m,n;τ)ϑ

mnz|mnτ, m is even, (.)

mn–

k=

(–)kϑ_n

z+ π mn+

kπ mn

τ

=R,(y,y, . . . ,yn;m,n,p;τ)ϑ

mnz|mnτ, m is odd and n is even, (.)

mn–

k=

(–)kϑ_n

z+kπ mn

τ

=R,(y,y, . . . ,yn;m,n,p;τ)ϑ

mnz|mnτ, m,n are odd, (.)

where

R,(m,n;τ) =mn

∞

r+···+rn=

r,...,rn=–∞

qr+···+rn_{. (.)}

Proof We sety=y=· · ·=yn=  in (.), (.), and (.) of Corollary .. We take

y=y=· · ·=yn=_mnπ in (.) and (.) of Corollary .. We deduce Corollary ..

Remark . Equation (.) of Corollary . is an extension of Boon’s result. On taking n=  in (.), and noting thatR,(m, ;τ) =m, we have

m–

k=

(–)kϑ

z+kπ m

τ

=mϑ

mz|mτ, mis even, (.)

which is just Boon’s result (see [, p., the second identity of ()]).

Remark . Equation (.) of Corollary . is also an extension of Boon’s result. On

takingn=  in (.), and noting thatR,(m, ;τ) =m, we have

m–

k=

(–)k_ϑ



z+ π m+

kπ m

τ

=R,(m, ;τ)ϑ

mz|m_{τ, mis even, (.)}

which is just Boon’s result (see [, p., the first identity of ()]).

Remark . Equation (.) of Corollary . is an alternating summation of

Ramanu-jan’s circular summation. Takingm=  in (.), we have

n–

k=

(–)kϑ_n

z+kπ n

τ

=R,(n;τ)ϑ(nz|nτ). (.)

Remark . We note that Boon’s result (see [, p., the first identity of ()]) has a

misprint, we have here corrected this point in our formula (.).

Remark . If we make the transformationsz−→z+π

,z−→z+

mnπ τ

 ,z−→z+

π

 +

mnπ τ

 , and using the transformation formulas (.)-(.) for Theorem ., we can also

obtain the corresponding alternating circular summation formulas.

5 The inverse formulas for Ramanujan’s circular summation formula In [], Liu obtained the following two results.

Lemma .(see [, p., Theorem .]) Suppose that n is a positive integer and f(z|τ)

is an entire function of z satisfying the functional equations

f(z|τ) =f(z+π|τ) =qnenizf(z+π τ|τ). (.) Then for any positive integer m,there exists a constant a()m independent of z such that

n–

k=

e–mkinπ_f

z+kπ n

τ

=na()_memizϑ(nz+mπ τ|nτ), (.)

f(z|τ) =

n–

m=

a()_memizϑ(nz+mπ τ|nτ). (.)

Lemma .(see [, p., Theorem .]) Suppose that n is a positive integer and f(z|τ)

is an entire function of z satisfying the functional equations

Then for any positive integer m,there exists a constant a()m independent of z such that n–

k=

(–)ke–mkinπ_f

z+kπ n

τ

=na()_memizϑ(nz+mπ τ|nτ), (.)

f(z|τ) =

n–

m=

a()_memizϑ(nz+mπ τ|nτ). (.)

Liu also found many new identities of the theta functions from Lemma . and Lemma ..

Below we give some new inverse relations for theta functionϑ(z|τ) by using the results

of Liu.

Theorem . Suppose that m,n are any positive integers,p is any integer;y,y, . . . ,ynare

any complex numbers.

• Wheny+y+· · ·+yn=pπ+n_π,we have n–

k=

e–imknπ

n

j=

ϑ

z+yj+

kπ n

τ

=na()_memizϑ(nz+mπ τ|nτ), (.)

n

j=

ϑ(z+yj|τ) = n–

m=

a()_memizϑ(nz+mπ τ|nτ). (.)

• Wheny+y+· · ·+yn=pπ+n_π+π_ andnis even,we have n–

k=

(–)k_e–imkπ

n

n

j=

ϑ

z+yj+

kπ n

τ

=na()_memiz_ϑ

(nz+mπ τ|nτ), (.)

n

j=

ϑ(z+yj|τ) = n–

m=

a()_memizϑ(nz+mπ τ|nτ). (.)

Here

a()_m = (–)m

∞

r,...,rn=–∞

r+···+rn=m

qr+···+rn_ei(ry+···+rnyn)_{, (.)}

a()_m = (–)min+q–m–n

∞

r,...,rn=–∞ r+···+rn=m+n

qr+···+rn_ei(ry+···+rnyn)_{. (.)}

Proof Let

f(z|τ) =

n

j=

ϑ(z+yj|τ).

We easily obtain by using the first and second identities of (.) forn= , respectively:

f(z+π|τ) =

n

j=

ϑ(z+π+yj|τ) = n

j=

f(z+π τ|τ) =

n

j=

ϑ(z+π τ+yj|τ) = (–)nq–ne–nize–i(y+y+···+yn)f(z|τ). (.)

• Wheny+y+· · ·+yn=pπ+n_π in(.). From(.)and(.), we have

f(z|τ) =f(z+π|τ) =qnenizf(z+π τ), (.)

which satisfies condition(.). By Lemma . we obtain(.)and(.)of Theorem . at once.

Next we compute the constanta()m independent ofz. We apply the definitions (.) and

(.). Setting

z−→nz+mπ τ and τ−→nτ

in (.), we get

ϑ(nz+mπ τ|nτ) =

∞

r=–∞

qnreri(nz+mπ τ). (.)

Setting

z−→z+yj+

kπ n

in (.), we get

n

j=

ϑ

z+yj+

kπ n

τ

= ∞

r,...,rn=–∞

(–)r+···+rn_qr_+···+rn_ei(ry+···+rnyn)_ekπ_ni(r+···+rn)_ei(r+···+rn)z_{. (.)}

Substituting (.) and (.) into (.), we find that

n–

k=

e–imknπ ∞

r,...,rn=–∞

(–)r+···+rn_qr+···+rn_ei(ry+···+rnyn)_ekπni(r+···+rn)_ei(r+···+rn)z

=namemiz

∞

r=–∞

qnreri(nz+mπ τ). (.)

Comparing the constants of both sides of (.) and noting thaty+y+· · ·+yn=pπ+n_π,

we obtain (.) of Theorem ..

• Wheny+y+· · ·+yn=pπ+n_π +π_ in(.)andnisonlyeven. From(.)and

(.), we have

which satisfies condition(.)of Lemma .. From Lemma . we obtain(.)and

(.)of Theorem . immediately. In a similar way we can prove(.). The proof is

complete.

Corollary . Suppose that m,n are any positive integers;y,y, . . . ,ynare any complex

numbers.

• Wheny+y+· · ·+yn= andnis even,we have n–

k=

n

j=

ϑ

z+yj+

kπ n

τ

=na()_ ϑ(nz|nτ), (.)

n

j=

ϑ(z+yj|τ) = n–

m=

a()_memizϑ(nz+mπ τ|nτ). (.)

• Wheny+y+· · ·+yn=π_ andnis even,we have n–

k=

(–)k

n

j=

ϑ

z+yj+

kπ n

τ

=na()_ ϑ(nz|nτ), (.)

n

j=

ϑ(z+yj|τ) = n–

m=

a()_memizϑ(nz+mπ τ|nτ). (.)

Here

a() =

∞

r,...,rn=–∞

r+···+rn=

qr+···+rn_ei(ry+···+rnyn)_{, (.)}

a()_ =in+q–n

∞

r,...,rn=–∞ r+···+rn=n

qr+···+rn_ei(ry+···+rnyn)_{, (.)}

a()_m = (–)m ∞

r,...,rn=–∞

r+···+rn=m

qr+···+rn_ei(ry+···+rnyn)_{, (.)}

a()_m = (–)m_in+_q–m–n_

∞

r,...,rn=–∞ r+···+rn=m+n

qr+···+rn_ei(ry+···+rnyn)_{. (.)}

Proof Ifnis an even, settingp= –n_ iny+y+· · ·+yn=pπ+n_π, then we havey+y+ · · ·+yn= . We directly deduce (.) (settingm= ) and (.) of Corollary . from

Theorem ..

Ifnis an even, settingp= –n_iny+y+· · ·+yn=pπ+n_π+π_, we havey+y+· · ·+yn=π_.

We get (.) (settingm= ) and (.) of Corollary . from Theorem .. The proof is

complete.

Settingn=  in (.) noting thaty+y=  anda() =ϑ(y|τ), we have

Setting n=  in (.), noting that y+y=  and a() =ϑ(y|τ), a() = –e–iy×

ϑ(y|τ), we have

ϑ(z+y|τ)ϑ(z–y|τ)

=ϑ(y|τ)ϑ(z|τ) –ei(z–y)ϑ(y|τ)ϑ(z+π τ|τ). (.)

Settingn=  in (.) noting thaty+y=π_ anda() =ϑ(y|τ), we have

n–

k=

(–)k

n

j=

ϑ

z+yj+

kπ n

τ

=na()_ ϑ(nz|nτ), (.)

n

j=

ϑ(z+yj|τ) = n–

m=

a()_memizϑ(nz+mπ τ|nτ), (.)

a()_m = (–)m_in+_q–m–n

∞

r,...,rn=–∞ r+···+rn=m+n

qr+···+rn_ei(ry+···+rnyn)_{. (.)}

Corollary . Assume m is any positive integer.

• Whennis even,we have

n–

k=

ϑ_n

z+kπ n

τ

=na()_ϑ(nz|nτ), (.)

ϑ_n(z|τ) =

n–

m=

a()_memizϑ(nz+mπ τ|nτ). (.)

• Whennis even,we have

n–

k=

(–)kϑ_n

z+ π n+

kπ n

τ

=na()_ ϑ(nz|nτ), (.)

ϑ_n

z+ π n

τ

=

n–

m=

a()_memizϑ(nz+mπ τ|nτ). (.)

Here

a()_ = ∞

r,...,rn=–∞

r+···+rn=

qr+···+rn_{, (.)}

a()_ =inq–n

∞

r,...,rn=–∞ r+···+rn=n

qr+···+rn_{, (.)}

a()_m = (–)m ∞

r,...,rn=–∞

r+···+rn=m

qr+···+rn_{, (.)}

a()_m = (–)minq–m–n_emnπi ∞

r,...,rn=–∞ r+···+rn=m+n

Proof We sety=y=· · ·=yn=  in (.) and (.) of Corollary .. We takey=y= · · ·=yn=_π_nin (.) and (.) of Corollary .. We obtain Corollary ..

6 Some applications

The well-known cubic theta functiona(τ) is defined (see []) by

a(τ) = ∞

r,r=–∞

qr+rr+r_{. (.)}

We define the multiple theta seriesa(y,y|τ) as

a(y,y|τ) =

∞

r,r=–∞

qr+rr+r_ei(ry+ry)_{. (.)}

Obviously,a(τ) =a(, |τ).

We give some applications of Theorem . below.

Corollary . Suppose that m,n are any positive integers;y,y, . . . ,ynare any complex

numbers.

• Wheny+y+· · ·+yn= ,we have mn–

k=

n

j=

ϑ

z+yj+

kπ mn

τ

=

R,(y,y, . . . ,yn;m,n;τ)ϑ(mnz|mnτ), mnis even,

R,(y,y, . . . ,yn;m,n;τ)ϑ(mnz|mnτ), mnis odd.

(.)

• Wheny+y+· · ·+yn=_π_m andmnis even,we have mn–

k=

n

j=

ϑ

z+yj+

kπ mn

τ

=R,(y,y, . . . ,yn;m,n;τ)ϑ

mnz|mnτ, (.)

where

R,(y,y, . . . ,yn;m,n;τ) =mn

∞

r+···+rn=

r,...,rn=–∞

qr+···+rn_ei(ry+···+rnyn)_{. (.)}

Proof Whenmnis even, takingp= –mn_ in (.); whenmnis odd, takingp= –mn_+in (.), we easily get (.).

Whenmnis even, takingp= –mn_ in (.), we easily obtain (.).

Corollary . Suppose that m,n are any positive integers,a,b are nonnegative integers

and a+b=n;y is any complex numbers.

mn–

k=

ϑ_a

z+by+ kπ mn

τ

ϑ_b

z–ay+kπ mn

τ

=

R,(y;m,n,a,b;τ)ϑ(mnz|mnτ), mn is even,

R,(y;m,n,a,b;τ)ϑ(mnz|mnτ), mn is odd.

• Whenmnis even,we have

mn–

k=

ϑ_n

z+ π mn+

kπ mn

τ

=R,(m,n;τ)ϑ

mnz|mnτ. (.)

Here

R,(y;m,n,a,b;τ) =mn

∞

r+···+rn=

r,...,rn=–∞

qr+···+rn_eniy(r+···+ra)_{, (.)}

R,(m,n;τ) =mn

∞

r+···+rn=

r,...,rn=–∞

qr+···+rn_{. (.)}

Proof Wheny+y+· · ·+yn= , takingy=y=· · ·=ya=byandya+=ya+=· · ·=yn=

–ay,a+b=nin Corollary ., we have (.).

Wheny+y+· · ·+yn=_π_m, takingy=y=· · ·=yn=_π_mn, in Corollary ., we have

(.).

Remark . Letz−→z+π_in (.) and using the transformation formulas (.) and (.),

we have

mn–

k=

ϑ_n

z+ π mn+

kπ mn

τ

=R,(m,n;τ)ϑ

mnz|mnτ, (.)

which is a new formula; it is also an extension of Boon’s result. If we taken=  in (.) and (.), we deduce

m–

k=

ϑ

z+(k+ )π m

τ

=mϑ

mz|mτ, (.)

which is just Boon’s result (see [, p.]).

Remark . We here have corrected a misprint concerning the formulas ofϑ(nz|nτ) and

ϑ(nz|nτ) in [, p.].

Takingy=y=· · ·=yn=  in (.), we have the following.

Corollary . Suppose that m,n are any positive integers.We have

mn–

k=

ϑ_n

z+ kπ mn

τ

=

R,(m,n;τ)ϑ(mnz|mnτ), mn is even,

R,(m,n;τ)ϑ(mnz|mnτ), mn is odd.

(.)

Here

R,(m,n;τ) =mn

∞

r+···+rn=

r,...,rn=–∞