Discrete Fourier transform and Jacobi function identities

JOURNAL OF MATHEMATICAL PHYSICS 51, 023511 共2010兲

Discrete Fourier transform and Jacobi ␪ function identities R. A. Malekara兲 and H. Bhateb兲 Department of Mathematics, University of Pune, Ganeshkhind, Pune 411007, India 共Received 12 June 2009; accepted 13 November 2009; published online 2 February 2010兲

Classical theta function identities are derived using properties of eigenvectors corresponding to the discrete Fourier transform ⌽共2兲. In particular, an extended Watson addition formula is obtained corresponding to ⌽共2兲, and the corresponding result for the discrete Fourier transform ⌽共3兲 is obtained. © 2010 American Institute of Physics. 关doi:10.1063/1.3272005兴

I. INTRODUCTION

The discrete Fourier transform 共DFT兲 is an important tool for applications in engineering and physics, and it is also a source of interesting mathematical problems. Mehta1 has given constructions of eigenfunctions of the DFT in terms of the discrete analog of the Hermite functions that arise in the continuous cases. Various identities among these eigenvectors are also investigated. The derivatives of Jacobi theta functions as eigenfunctions of the DFT first appeared in Ref. 2. The properties of Jacobi theta functions and their derivatives under DFT are further investigated in Ref. 3. Matveev4 has proven beautiful consequences of the fact that the DFT ⌽ is a fourth root of unity, i.e., ⌽4 = I. Given any absolutely summable series gn, he has constructed eigenfunctions of the DFT from the series gn. This is then applied for the case when the series arises as the summands of a ␯ theta function with characteristic 共a , b兲, namely, ␪a,b共x , ␶ , ␯兲. This reduces to usual theta function when ␯ = 1. We extend the result of Matveev to derive identities of theta functions and, in particular, obtain an extended Watson addition formula corresponding to ⌽共2兲 and ⌽共3兲. The applications of Watson’s addition formula have appeared in Ref. 5. In Ref. 6 another addition formula for theta function is given which, in particular, implies the Winquist identity as well as other identities. The method we use is conceptually simple and does not depend on the properties of zeros of theta functions and their infinite product representation. The basic notations adopted in this paper and some preliminary results are presented in Sec. II. The identities of theta functions corresponding to the DFT ⌽共2兲 are discussed in Sec. III. Section IV contains identities corresponding to ⌽共3兲.

II. PRELIMINARY RESULTS

The matrix ⌽共n兲 corresponding to the DFT of size n is given by

⌽ jk共n兲 =

1

冑n q

jk

,

j,k = 0, . . . ,n − 1.

q=

2␲i . n

共1兲

a兲

Electronic mail: [email protected]. Electronic mail: [email protected].

b兲

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51, 023511-1

© 2010 American Institute of Physics

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R. A. Malekar and H. Bhate

Definition: For f = 共f 0 , . . . , f n−1兲t 苸 Cn we define the DFT ˜f 苸 Cn by ˜f = ⌽f = 共f˜0 ,˜f 1 , . . . , f n−1兲, where n−1

˜f = 1 兺 f e2␲ijk/n . k 冑n j=0 j It is clear that ⌽4 = I. The multiplicities corresponding to eigenvalues of the DFT ⌽共n兲 are given by 共see Ref. 4兲 n = 4m + 2 ⇒ m1 = m, n = 4m ⇒ m1 = m,

m2 = m + 1, m2 = m,

n = 4m + 1 ⇒ m1 = m, n = 4m + 3 ⇒ m1 = m + 1,

m3 = m,

m3 = m − 1,

m2 = m,

m3 = m,

m2 = m + 1,

m0 = m + 1, m0 = m + 1, m0 = m + 1,

m3 = m,

m0 = m + 1,

where mk is the multiplicity of ik. Let ␶ be a complex number with a positive imaginary part. The theta function ␪a,b共x , ␶兲 with characteristics 共a , b兲 is defined by 共see Ref. 7兲 m=⬁

␪a,b共x, ␶兲 =

兺

exp关␲i␶共m + a兲2 + 2␲i共m + a兲共x + b兲兴.

m=−⬁

␪a,b共x , ␶兲 is connected with ␪0,0共x , ␶兲 = ␪共x , ␶兲 by ␪a,b共x, ␶兲 = ␪共x + a␶ + b, ␶兲exp关␲ia2␶ + 2␲a共x + b兲兴. The four classical Jacobi theta functions may be represented as m=⬁

␪0,0共x, ␶兲 =

兺

exp共␲im2␶ + 2␲imx兲,

m=−⬁

␪1/2,1/2共x, ␶兲 = 共− 1兲兺 exp关␲i␶共m + 1/2兲2 + 2␲i共m + 1/2兲共x + 1/2兲兴, m苸Z

␪1/2,0共x, ␶兲 =

exp关␲i␶共m + 1/2兲2 + 2␲i共m + 1/2兲x兴, 兺 m苸Z

␪0,1/2共x, ␶兲 =

exp关␲i␶m2 + 2␲im共x + 1/2兲兴. 兺 m苸Z

The Jacobi theta functions which are related to eigenfunctions of the DFT ⌽共n兲 are given by

␪ j/n,0共x, ␶兲 =

exp关␲i␶共m + j/n兲2 + 2␲i共m + j/n兲x兴兺 m苸Z

for j = 0,1,2, . . . ,n − 1.

Matveev4 has proven the following theorem which will be crucial to the following sections.


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DFT and Jacobi theta function identities

Theorem 2.1: (Matveev) For any ␶ with positive imaginary part the vector v共x , ␶ , k兲 with components v j共x , ␶ , k兲 , j = 0 , 1 , 2 , . . . , n − 1 given by v j共x, ␶,k兲 = ␪ j/n,0共x, ␶兲 + 共− 1兲k␪−j/n,0共x, ␶兲 +

1

冑n

冋冉共− i兲k␪

冊

冉

j+x ␶ x−j ␶ , 2 + 共− i兲3k␪ , , n n n n2

冊册

, 共2兲

is an eigenvector of the DFT with an eigenvalue ik , ⌽共n兲v共x, ␶,k兲 = ikv共x, ␶,k兲. 䊏 The proof of the above theorem follows from the fact that ⌽4 = I. It has been suggested in Ref. 4 that this may have applications to determine the identities of theta functions. For a given value of n, take eigenvectors of the form 共2兲 corresponding to the eigenvalue ik,with different values of x and ␶. At the most mk of these are linearly independent. Thus minors of the matrix consisting of the eigenvectors v共x , ␶ , k兲 of order greater than mk vanish. This may lead to new identities among theta functions. This idea is explored for the DFT ⌽共2兲 in the following section to obtain an extended Watson addition formula. In Sec. IV, the corresponding result for the DFT ⌽共3兲 is derived. III. IDENTITIES CORRESPONDING TO THE DFT ⌽„2…

The DFT ⌽共2兲 has only two eigenvalues +1 and ⫺1. The eigenvector corresponding to eigenvalue +1 is given by

v共x, ␶,0兲 =

冤

2␪共x, ␶兲 + 冑2␪ 2␪1/2,0共x, ␶兲 + 冑2␪

冉冊冉冊 x ␶ , 2 4

x+1 ␶ , 2 4

冥

.

共3兲

冥

.

共4兲

The eigenvector corresponding to eigenvalue of ⫺1 is given by

v共x, ␶,2兲 =

冤

2␪共x, ␶兲 − 冑2␪ 2␪1/2,0共x, ␶兲 − 冑2␪

冉冊冉冊 x ␶ , 2 4

x+1 ␶ , 2 4

We have ⌽共2兲关v共x, ␶,0兲 + v共x, ␶,2兲兴 = v共x, ␶,0兲 − v共x, ␶,2兲. This gives the following two identities:

␪共x, ␶兲 + ␪1/2,0共x, ␶兲 = ␪

x ␶ , , 2 22

共5兲

冊

共6兲

␪共2x,4␶兲 + ␪1/2,0共2x,4␶兲 = ␪共x, ␶兲,

共7兲

␪共x, ␶兲 − ␪1/2,0共x, ␶兲 = ␪

冉

冉冊

x+1 ␶ . , 2 22

共5兲 and 共6兲 are equivalent to


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␪共2x,4␶兲 − ␪1/2,0共2x,4␶兲 = ␪0,1/2共x, ␶兲. Using 共3兲,

冉

冤

冊

␶

v x + , ␶,0 = 2

where a = exp共−␲i␶ / 4 − ␲ix兲,

v共x + 1, ␶,0兲 =

冤

冉冊冉冊

2a␪1/2,0共x, ␶兲 + a冑2␪ 2a␪共x, ␶兲 − a冑2␪

2␪共x, ␶兲 + 冑2␪

冉

共8兲

x ␶ , 2 4

x+1 ␶ , 2 4

冊冉冊

x+1 ␶ , 2 4

− 2␪1/2,0共x, ␶兲 + 冑2␪

x ␶ , 2 4

冥

冥

,

.

共9兲

共10兲

Since v共x + 1 , ␶ , 0兲 , v共x + ␶ / 2 , ␶ , 0兲 are eigenvectors corresponding to the same eigenvalue of 1, which has multiplicity of 1 we have det共v共x + 1 , ␶ , 0兲 , v共x + ␶ / 2 , ␶ , 0兲兲 = 0. This gives

冉

冊冉冊

x+1 ␶ x ␶ + ␪2 , . , 2 4 2 4

共11兲

2 2 共2x,4␶兲 = ␪0,1/2 共x, ␶兲 + ␪2共x, ␶兲. 2␪2共2x,4␶兲 + 2␪1/2,0

共12兲

␪共2x,2␶兲␪共0,2␶兲 = 兺 exp共␲i共m2 + n2兲2␶ + 2␲im2x兲.

共13兲

2 共x, ␶兲 = ␪2 2␪2共x, ␶兲 + 2␪1/2,0

共11兲 is equivalent to

Now consider

m,n

Let m + n = n1 and m − n = n2, so that n1 and n2 are of the same parity. Rewriting 共13兲 in terms of n1 and n2 we have

␪共2x,2␶兲␪共0,2␶兲 =

exp共␲i共n21 + n22兲␶ + 2␲i共n1 + n2兲x兲兺 n ⬅n 共mod 2兲 1

=

2

exp共␲i共n21 + n22兲␶ + 2␲i共n1 + n2兲x兲 + exp共␲i共n21 + n22兲␶ + 2␲i共n1 + n2兲x兲兺兺 n ,n are even n ,n are odd 1 2

1 2

2 = ␪2共2x,4␶兲 + ␪1/2,0 共2x,4␶兲.

From 共12兲 we have 2 ␪2共x, ␶兲 + ␪0,1/2 共x, ␶兲 = 2␪共2x,2␶兲␪共0,2␶兲.

共14兲

This is a well known transformation of the Landen type. All other transformations of the Landen type can be derived using a similar method. Now the det共v共x , ␶ , 0兲 , v共x + 1 , ␶ , 0兲 = 0兲. This gives − 4␪共x, ␶兲␪1/2,0共x, ␶兲 + 2冑2␪共x, ␶兲␪ − 2冑2␪1/2,0共x, ␶兲␪

冉

冊

冉冊

冉冊冉冊冊冉冊

x ␶ x ␶ x ␶ − 2冑2␪1/2,0共x, ␶兲␪ , + 2␪2 , − 4␪共x, ␶兲␪1/2,0共x, ␶兲 , 2 4 2 4 2 4

冉

x+1 ␶ x+1 ␶ x+1 ␶ − 2冑2␪共x, ␶兲␪ − 2␪2 = 0. , , , 2 4 2 4 2 4

Using Eqs. 共5兲 and 共6兲 the terms with coefficients of 2冑2 cancel each other out. This gives


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冉冊冉

冊

x+1 ␶ x ␶ , = 4␪共x, ␶兲␪1/2,0共x, ␶兲. , − ␪2 2 4 2 4

共15兲

2 ␪2共x, ␶兲 − ␪0,1/2 共x, ␶兲 = 4␪共2x,4␶兲␪1/2,0共2x,4␶兲.

共16兲

␪2 共15兲 is equivalent to

Applying the same argument as in the derivation of 共14兲 we have 2␪共2x,4␶兲␪1/2,0共2x,4␶兲 = ␪1/2,0共2x,2␶兲␪1/2,0共0,2␶兲.

共17兲

2 共0,2␶兲. 2␪共0,4␶兲␪1/2,0共0,4␶兲 = ␪1/2,0

共18兲

2 ␪2共x, ␶兲 − ␪0,1/2 共x, ␶兲 = 2␪1/2,0共2x,2␶兲␪1/2,0共0,2␶兲.

共19兲

At x = 0 we have

From 共16兲 and 共17兲 we have

This is again an identity of the Landen type. From 共14兲 at x = 0 we have 2 ␪2共0, ␶兲 + ␪0,1/2 共0, ␶兲 = 2␪2共0,2␶兲.

共20兲

2 2 ␪2共0, ␶兲 − ␪0,1/2 共0, ␶兲 = 2␪1/2,0 共0,2␶兲.

共21兲

4 2 ␪4共0, ␶兲 − ␪0,1/2 共0, ␶兲 = 4␪2共0,2␶兲␪1/2,0 共0,2␶兲.

共22兲

4 4 ␪4共0, ␶兲 − ␪0,1/2 共0, ␶兲 = ␪1/2,0 共0, ␶兲.

共23兲

Similarly from 共19兲 at x = 0,

From 共20兲 and 共21兲 we have

Using 共18兲 we have

This is a well known Jacobi identity between the null values of theta functions. Many of the classical identities involving the squares of theta functions can be obtained by the method illustrated above. In the next theorem we illustrate the technique to extend the classical Watson’s addition formula for theta functions 共see Ref. 8兲. Theorem 3.1: (Extended Watson’s addition formula)

␪0,1/2共x1 + x2, ␶兲␪共x1 − x2, ␶兲 − ␪0,1/2共x1 − x2, ␶兲␪共x1 + x2, ␶兲 = 2␪共2x1 + 2x2,4␶兲␪1/2,0共2x1 − 2x2,4␶兲 − 2␪共2x1 − 2x2,4␶兲␪1/2,0共2x1 + 2x2,4␶兲.

共24兲

Proof: Using 共5兲 and 共6兲

␪

␪

冉

冉

冊

x1 ⫾ x2 ␶ = ␪共x1 ⫾ x2,2␶兲 + ␪1/2,0共x1 ⫾ x2,2␶兲, , 2 2

冊

x1 ⫾ x2 + 1 ␶ , = ␪共x1 ⫾ x2,2␶兲 − ␪1/2,0共x1 ⫾ x2,2␶兲. 2 2

共25兲

共26兲

From 共3兲 and 共4兲, we have


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v共x1 + x2,2␶,0兲 =

v共x1 − x2,2␶,0兲 =

冤冤

2␪共x1 + x2,2␶兲 + 冑2␪

2␪1/2,0共x1 + x2,2␶兲 + 冑2␪

冉冉

2␪共x1 − x2,2␶兲 + 冑2␪ 2␪1/2,0共x1 − x2,2␶兲 + 冑2␪

x1 + x2 ␶ , 2 2

冊

冊冥冊冊冥

,

x1 + x2 + 1 ␶ , 2 2

冉冉

x1 − x2 ␶ , 2 2

x1 − x2 + 1 ␶ , 2 2

are eigenvectors of ⌽共2兲 corresponding to eigenvalues +1 which has multiplicity of 1. Therefore, det共v共x1 + x2,2␶,0兲, v共x1 − x2,2␶,0兲兲 = 0. This gives 4␪共x1 + x2,2␶兲␪1/2,0共x1 − x2,2␶兲 + 2冑2␪共x1 + x2,2␶兲␪ − x2,2␶兲␪

冉

冊冉冊冉冉冊冑冊冉冊

冉

冊

x1 − x2 + 1 ␶ , + 2冑2␪1/2,0共x1 2 2

冊

x1 + x2 ␶ x1 + x2 ␶ x1 − x2 + 1 ␶ , , ␪ , + 2␪ − 4␪1/2,0共x1 + x2,2␶兲␪共x1 − x2,2␶兲 2 2 2 2 2 2

− 2冑2␪1/2,0共x1 + x2,2␶兲␪ − 2␪

冉

冉

x1 − x2 ␶ x1 + x2 + 1 ␶ , , − 2 2␪共x1 − x2,2␶兲␪ 2 2 2 2

冊

x1 + x2 + 1 ␶ x1 − x2 ␶ , ␪ , = 0. 2 2 2 2

共27兲

In this expression consider the terms with 2冑2 as coefficient. Using formulas 共25兲 and 共26兲, we have A = ␪共x1 + x2,2␶兲␪

冉

x1 − x2 + 1 ␶ , 2 2

冊

= ␪共x1 + x2,2␶兲␪共x1 − x2,2␶兲 − ␪共x1 + x2,2␶兲␪1/2,0共x1 − x2,2␶兲,

B = ␪1/2,0共x1 − x2,2␶兲␪

冉

x1 + x2 ␶ , 2 2

冊

= ␪1/2,0共x1 − x2,2␶兲␪共x1 + x2,2␶兲 + ␪1/2,0共x1 + x2,2␶兲␪1/2,0共x1 − x2,2␶兲,

C = ␪1/2,0共x1 + x2,2␶兲␪

冉

x1 − x2 ␶ , 2 2

冊

= ␪1/2,0共x1 + x2,2␶兲␪共x1 − x2,2␶兲 + ␪1/2,0共x1 + x2,2␶兲␪1/2,0共x1 − x2,2␶兲,

D = ␪共x1 − x2,2␶兲␪

冉

x1 + x2 + 1 ␶ , 2 2

冊

= ␪共x1 − x2,2␶兲␪共x1 + x2,2␶兲 − ␪共x1 − x2,2␶兲␪1/2,0共x1 + x2,2␶兲. It is clear that A + B − C − D = 0. Hence all the terms with coefficients 2冑2 cancel each other out. Equation 共27兲 becomes


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4␪共x1 + x2,2␶兲␪1/2,0共x1 − x2,2␶兲 − 4␪1/2,0共x1 + x2,2␶兲␪共x1 − x2,2␶兲 = 2␪

冉

冊冉

冊冉

冊冉

冊

x1 + x2 + 1 ␶ x1 − x2 ␶ x1 − x2 + 1 ␶ x1 + x2 ␶ , ␪ , , ␪ , . − 2␪ 2 2 2 2 2 2 2 2

Replacing x1 , x2 , ␶ by 2x1 , 2x2 , 2␶, respectively, we obtain 2␪共2x1 + 2x2,4␶兲␪1/2,0共2x1 − 2x2,4␶兲 − 2␪1/2,0共2x1 + 2x2,4␶兲␪共2x1 − 2x2,4␶兲 = ␪0,1/2共x1 + x2, ␶兲␪共x1 − x2, ␶兲 − ␪0,1/2共x1 − x2, ␶兲␪共x1 + x2, ␶兲. This proves 共24兲. Theorem 3.2: (Watson addition formula)

䊏

␪1/2,1/2共x1, ␶兲␪1/2,1/2共x2, ␶兲 = ␪共x1 + x2,2␶兲␪1/2,0共x1 − x2,2␶兲 − ␪共x1 − x2,2␶兲␪1/2,0共x1 + x2,2␶兲. 共28兲 Proof: Consider first term of the left hand side of 共24兲,

冋

冉

␪0,1/2共x1 + x2, ␶兲␪共x1 − x2, ␶兲 = 兺 exp ␲i共m2 + n2兲␶ + 2␲i 共m + n兲x1 + 共m − n兲x2 +

m 2

冊册

. 共29兲

Let m + n = n1, m − n = n2, where n1 and n2 are of same parity. Rewriting 共29兲 in terms of n1 and n2, =

冋冉冊冉冉冊册兺

兺 exp n ⬅n 共mod 2兲 1

2

␲i

+ 2 ␲ i n 1x 1 + n 2x 2 +

冊册冋冉冊

n21 + n22 n1 + n2 ␶ + 2 ␲ i n 1x 1 + n 2x 2 + 2 4 n1 + n2 4

exp ␲i

+

n1,n2 are odd

n21

+ 2

n22

=

冋冉冊冊册

兺 exp n ,n are even 1 2

冉

n21 + n22 ␶ 2

␲i

␶ + 2 ␲ i n 1x 1 + n 2x 2 +

n1 + n2 4

= ␪0,1/2共2x1,2␶兲␪0,1/2共2x2,2␶兲 + ␪1/2,1/2共2x1,2␶兲␪1/2,1/2共2x2,2␶兲. Similarly, using the same argument it is easy to show that

␪0,1/2共x1 − x2, ␶兲␪共x1 + x2, ␶兲 = ␪0,1/2共2x1,2␶兲␪0,1/2共2x2,2␶兲 − ␪1/2,1/2共2x1,2␶兲␪1/2,1/2共2x2,2␶兲. By using the above two results in 共24兲, we get

␪1/2,1/2共2x1,2␶兲␪1/2,1/2共2x2,2␶兲 = ␪共2x1 + 2x2,4␶兲␪1/2,0共2x1 − 2x2,4␶兲 − ␪共2x1 − 2x2,4␶兲␪1/2,0共2x1 + 2x2,4␶兲. Watson’s addition formula 共28兲 is obtained by replacing x1 , x2 , ␶ by x1 / 2 , x2 / 2 , ␶ / 2, respectively, in the above equation. 䊏 IV. IDENTITIES CORRESPONDING TO THE DFT ⌽„3…

The DFT ⌽共3兲 has three eigenvalues 1, i, and ⫺1 each of multiplicity of 1, 共see Ref. 4兲. The eigenvector corresponding to eigenvalue +1 is given by using Theorem 2.1,

冤

2␪共x, ␶兲 +

v共x, ␶,0兲 = ␪1/3,0共x, ␶兲 + ␪2/3,0共x, ␶兲 +

␪1/3,0共x, ␶兲 + ␪2/3,0共x, ␶兲 +

2

冑3 ␪

冑冋冉冑冋冉 1

3

1

3

冉冊 x ␶ , 3 32

冊冉冊冉

␪

x+1 ␶ x−1 ␶ , 2 +␪ , 3 3 3 32

␪

x+2 ␶ x−2 ␶ , 2 +␪ , 3 3 3 32

冊册冊册

冥

.


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The eigenvector corresponding to eigenvalue of ⫺1 is given by

冤

2␪共x, ␶兲 −

v共x, ␶,2兲 = ␪1/3,0共x, ␶兲 + ␪2/3,0共x, ␶兲 −

␪1/3,0共x, ␶兲 + ␪2/3,0共x, ␶兲 −

2

冑3 ␪

冑冋冉冑冋冉 1

3

1

3

冉冊 x ␶ , 3 32

冊冉冊冉

␪

x+1 ␶ x−1 ␶ , 2 +␪ , 3 3 3 32

␪

x+2 ␶ x−2 ␶ , 2 +␪ , 3 3 3 32

冊册冊册

冥

冊册冊册

冥

Similarly the eigenvector corresponding to eigenvalue i is given by

v共x, ␶,1兲 =

冤

0

␪1/3,0共x, ␶兲 − ␪2/3,0共x, ␶兲 − ␪2/3,0共x, ␶兲 − ␪1/3,0共x, ␶兲 −

冋冉冑冋冋

冊冉冊冉

冑3

␪

x+1 ␶ x−1 ␶ , 2 −␪ , 3 3 3 32

i

␪

x+2 ␶ x−2 ␶ −␪ , , 3 32 3 32

i

3

.

.

We have

␪共x + 1, ␶兲 = ␪共x, ␶兲,

␪1/3,0共x + 1, ␶兲 = ␻␪1/3,0共x, ␶兲,

␪2/3,0共x + 1, ␶兲 = ␻2␪2/3,0共x, ␶兲,

where ␻ = e2␲i/3 is the cube root of unity. We have ⌽共3兲关v共x, ␶,0兲 + v共x, ␶,1兲 + v共x, ␶,2兲兴 = v共x, ␶,0兲 + iv共x, ␶,1兲 − v共x, ␶,2兲. By equating the first component we obtain

␪共x, ␶兲 + ␪1/3,0共x, ␶兲 + ␪2/3,0共x, ␶兲 = ␪

冉冊

x ␶ , . 3 32

共30兲

Replacing x by x + 1 and x + 2 in 共30兲 we get the following identities:

␪共x, ␶兲 + ␻␪1/3,0共x, ␶兲 + ␻2␪2/3,0共x, ␶兲 = ␪

冉

x+1 ␶ , , 3 32

冊

共31兲

␪共x, ␶兲 + ␻2␪1/3,0共x, ␶兲 + ␻␪2/3,0共x, ␶兲 = ␪

冉

x+2 ␶ , . 3 32

冊

共32兲

␪共x, ␶兲 = ␪共3x,9␶兲 + ␪1/3,0共3x,9␶兲 + ␪2/3,0共3x,9␶兲,

共33兲

␪0,1/3共x, ␶兲 = ␪共3x,9␶兲 + ␻␪1/3,0共3x,9␶兲 + ␻ ␪2/3,0共3x,9␶兲,

共34兲

␪0,2/3共x, ␶兲 = ␪共3x,9␶兲 + ␻2␪1/3,0共3x,9␶兲 + ␻␪2/3,0共3x,9␶兲.

共35兲

The above identities are equivalent to

2

Identities 共30兲–共35兲 are extensions of the identities 共5兲–共8兲. Theorem 4.1: [Extended Watson addition formula corresponding to ⌽共3兲 ] 3␪共3x + 3y,9␶兲␪1/3,0共3x − 3y,9␶兲 + 3␪共3x + 3y,9␶兲␪2/3,0共3x − 3y,9␶兲 − 3␪共3x − 3y,9␶兲␪1/3,0共3x + 3y,9␶兲 − 3␪2/3,0共3x + 3y,9␶兲␪共3x − 3y,9␶兲 = ␪0,1/3共x + y, ␶兲␪共x − y, ␶兲 + ␪0,2/3共x + y, ␶兲␪共x − y, ␶兲 − ␪共x + y, ␶兲␪0,1/3共x − y, ␶兲 − ␪共x + y, ␶兲␪0,2/3共x − y, ␶兲.

共36兲


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J. Math. Phys. 51, 023511 共2010兲


Proof: Consider the eigenvectors v共x + y , 3␶ , 0兲, v共x − y , 3␶ , 0兲,

冤冤

冑冉冑冋冉冑冋冉 2

2␪共x + y,3␶兲 +

v共x + y,3␶,0兲 = ␪1/3,0共x + y,3␶兲 + ␪2/3,0共x + y,3␶兲 +

␪1/3,0共x + y,3␶兲 + ␪2/3,0共x + y,3␶兲 +

3

1

3

1

3

␪1/3,0共x − y,3␶兲 + ␪2/3,0共x − y,3␶兲 +

1

3

冊冉冊冉

冊册冊册

冊冉冊冉

冊册冊册

␪

x+y+2 ␶ x+y−2 ␶ +␪ , , 3 3 3 3

3

3

冊

x+y+1 ␶ x+y−1 ␶ , , +␪ 3 3 3 3

冑冉冑冋冉冑冋冉 1

x+y ␶ , 3 3

␪

2

2␪共x − y,3␶兲 +

v共x − y,3␶,0兲 = ␪1/3,0共x − y,3␶兲 + ␪2/3,0共x − y,3␶兲 +

␪

␪

x−y ␶ , 3 3

冊

␪

x−y+1 ␶ x−y−1 ␶ , , +␪ 3 3 3 3

␪

x−y+2 ␶ x−y−2 ␶ +␪ , , 3 3 3 3

冥冥

,

.

These are eigenvectors corresponding to the same eigenvalue of 1 which has multiplicity of 1. Therefore, any 2 ⫻ 2 minor of 关v共x + y , 3␶ , 0兲 , v共x − y , 3␶ , 0兲兴 is zero. We consider the minor formed by the first two components in the eigenvectors v共x + y , 3␶ , 0兲, v共x − y , 3␶ , 0兲, i.e.,

冏

v0共x + y,3␶,0兲 v0共x − y,3␶,0兲 v1共x + y,3␶,0兲 v1共x − y,3␶,0兲

冏

= 0.

In the following expansion we have used the temporary shorthand notation ␪ j/3,0共x ⫾ y , 3␶兲 = ␪ j/3,0共x ⫾ y兲 and ␪共共x ⫾ y ⫾ j兲 / 3 , ␶ / 3兲 = ␪共共x ⫾ y ⫾ j兲 / 3兲 for j = 0 , 1 , 2, 2␪共x + y兲␪1/3,0共x − y兲 + 2␪共x + y兲␪2/3,0共x − y兲 + +

冉冊冉冊冉冉冑冊冑冉 2

冑3 ␪

冉冊

2

冑3

␪共x + y兲␪

冉

冊冉冊冉

2 2 x+y x−y+1 x+y x+y ␪1/3,0共x − y兲 + ␪ ␪2/3,0共x − y兲 + ␪ ␪ 冑3 3 3 3 3 3

冊冊

冉

2 x−y+1 x−y−1 + ␪共x + y兲␪ 冑3 3 3

冊

冊

2 x+y x−y−1 − 2␪共x − y兲␪1/3,0共x + y兲 − 2␪共x − y兲␪2/3,0共x + y兲 + ␪ ␪ 3 3 3 − −

2

3

2

3

␪共x − y兲␪ ␪

冉冉冊冉

冊冉冊冊冉冊冉

2 2 x+y+1 x+y−1 x−y − ␪共x − y兲␪ − ␪ ␪1/3,0共x + y兲冑冑 3 3 3 3 3

冊

2 x−y 2 x−y x−y x+y+1 x+y−1 ␪2/3,0共x + y兲 − ␪ ␪ − ␪ ␪ =0 3 3 3 3 3 3 3

共37兲

Label the summands on the left hand side of 共37兲 successively as A1 , A2 , A3 , . . . , A16. Then 16 兺k=1 Ak = 0. By using formulas 共30兲–共32兲 we have A3 + A4 + A5 + A6 + A11 + A12 + A13 + A14 = 0. 共37兲 becomes 3␪共x + y兲␪1/3,0共x − y兲 + 3␪共x + y兲␪2/3,0共x − y兲 − 3␪共x − y兲␪1/3,0共x + y兲 − 3␪共x − y兲␪2/3,0共x + y兲 =␪

冉冊冉冉冊冉

冊冉冊冉冊

冊冉冊冉

x−y x+y+1 x−y x+y+2 x+y x−y+1 ␪ +␪ ␪ −␪ ␪ 3 3 3 3 3 3

−␪

x+y x−y+2 ␪ , 3 3

冊

共38兲


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J. Math. Phys. 51, 023511 共2010兲

replacing x , y , ␶ by 3x , 3y , 3␶ we get the required formula in the theorem. Corollary 4.1.1:

3␪共3x,9␶兲␪1/3,0共0,9␶兲 + 3␪共3x,9␶兲␪2/3,0共0,9␶兲 − 3␪共0,9␶兲␪1/3,0共3x,9␶兲 − 3␪2/3,0共3x,9␶兲␪共0,9␶兲 = ␪0,1/3共x, ␶兲␪共0, ␶兲 + ␪0,2/3共x, ␶兲␪共0, ␶兲 − ␪共x, ␶兲␪0,1/3共0, ␶兲 − ␪共x, ␶兲␪0,2/3共0, ␶兲. Proof: Put y = x in the Theorem 4.1 and replace x by x / 2 to get 共39兲.

共39兲䊏

ACKNOWLEDGMENTS

The authors are grateful to the referee in bringing Ref. 2 to our notice. We thank the referee for various suggestions regarding the presentation of the article. M. L. Mehta, J. Math. Phys. 28, 781 共1987兲. D. Galetti and M. A. Marchiolli, Ann. Phys. 249, 454 共1996兲. 3 M. Ruzzi, J. Math. Phys. 47, 063507 共2006兲. 4 V. B. Matveev, Inverse Probl. 17, 633 共2001兲. 5 S. Cooper and P. C. Toh, J. Math. Anal. Appl. 347, 1 共2008兲. 6 Z.-G. Liu, Adv. Math. 212, 389 共2007兲. 7 E. T. Whittaker and G. N. Watson, A Course of Modern Analysis, 4th ed. 共Cambridge University Press, Cambridge, 1927兲. 8 H. Mckean and V. Moll, Elliptic Curves 共Cambridge University Press, Cambridge, 1999兲. 1 2
