Identities of arithmetic type between values of the theta function

Ramanujan J (2008) 16: 271–284 DOI 10.1007/s11139-007-9098-5

Identities of arithmetic type between values of the theta function associated to a lattice in Rd and its derivatives Allal Ghanmi · Youssef Hantout · Ahmed Intissar · Changgui Zhang · Azzouz Zinoun

Received: 23 June 2005 / Accepted: 12 December 2005 / Published online: 19 July 2008 © Springer Science+Business Media, LLC 2008

Abstract In this paper, we introduce a notion of similarly self dual lattice in a d-dimensional Euclidean space and a classical Jacobi theta function is associated to such a lattice. We establish identities of arithmetic type between values of this theta function and its successive derivatives. This work can be related to the spectral theory of the Landau operators. Keywords Dual lattice · Similarly self dual lattice · Poisson summation formula · Theta functions · Confluent hypergeometrique function Mathematics Subject Classification (2000) Primary 14K25 · Secondary 33C15

1 Introduction and statement of main result Let θZ (t) =

e−tm , 2

t ∈ R> := (0, ∞)

m∈Z

A. Ghanmi () · A. Intissar Department of Mathematics, Faculty of Sciences, P.O. Box: 1014, Mohammed V University—Agdal, 10 000 Rabat, Morocco e-mail: [email protected] Y. Hantout · C. Zhang Laboratoire Paul Painlevé, UMR-CNRS 8524, UFR de Mathématiques, USTL, Cité Scientifique, 59655 Villeneuve d’Ascq Cedex, France A. Zinoun Laboratoire de Physique des Lasers, Atomes et Molécules, UMR-CNRS 8523, UFR de Physique, USTL, Cité Scientifique, 59655 Villeneuve d’Ascq, France

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be the theta “constant” function associated to the standard discrete subgroup Z of the additive group (R, +). Then, the classical Jacobi identity for θZ (t) is known to be given by 1/2 2 π π . (1.1) θZ θZ (t) = t t The identity (1.1) is very significant for every t > 0 except for the (only) fixed point of the involution π2 . t In fact for t ∗ = π , the identity (1.1) is trivial and reads simply τ : R> → R> ,

t → τ (t) =

θZ (π) = θZ (π). However, the value t ∗ = π becomes relevant when taking the k-derivative of θZ (t). Namely, by differentiating one time both sides of (1.1), we get, for any t > 0, 2 5/2 2 π π 1 π 1/2 π = − tθZ (t) + , (1.2) θZ tθZ 2 t t t t and evaluating (1.2) at t ∗ = π , we get θZ (π) = −4πθZ (π).

(1.3)

That is, we have the following identity 2 2 e−πm = 4π m2 e−πm . m∈Z

(Z)1

m∈Z

More generally, by continuing the process of differentiation, k-times, we get an arith(j ) metic identity between θZ and its successive derivatives θZ , j = 1, 2, . . . , k, when evaluated at the time t ∗ = π . Namely, for any positive integer k ≥ 1, we have the following identity (Z)1 k−1 k−1 2j k!

2k (j ) θZ (π) = − k−1 (1 + 2q) π j θZ (π) j !(k − j )! (1 + 2q) q=0 j =1

q=j

(k)

+ (1 − (−1)k )π k θZ (π) .

(Z)k

The identity (Z)k reads explicitly as k−1 k (−1)j 2j k! k−1

2 2 2 e−πm = − k−1 (1 + 2q) (πm2 )j e−πm j !(k − j )! q=0 (1 + 2q) j =1

m∈Z

+ ((−1)k − 1)

(πm2 )k e−πm

m∈Z

2

q=j

.

m∈Z

Identities of arithmetic type between values of the theta function

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In this paper, we will be considering an arbitrary given lattice of the additive group (Rd , +), d ≥ 1, to which we associate the theta function θ (t), t > 0, defined by 2 θ (t) = e−t|γ | , (1.4) γ ∈

where |γ |2 is the square length of γ ∈ with respect to the usual Euclidean scalar product on (Rd , , ), and we are looking for a class (large enough) of lattices in Rd , d ≥ 1, for which there exist analogous identities such as in (Z)k above, that relate values of θ (t) in (1.4) at a certain time t∗ , to those of its successive derivatives for any given in such class. For doing this, let be an arbitrary fixed lattice in Rd , d ≥ 1, let

∗ = γ ∗ ∈ Rd ; γ ∗ , γ ∈ 2πZ, ∀γ ∈ (1.5) be the dual lattice of and denote by Spg (), respectively Spg ( ∗ ), the set of the square lengths |γ |2 for γ varying in , i.e.,

Spg () = |γ |2 , γ ∈ , Spg ( ∗ ) = |γ ∗ |2 , γ ∗ ∈ ∗ . Then the class of lattices in Rd for which we will establish the analogues of the identities to (Z)k are those that satisfy the following condition-definition (SSD). Definition 1.1 (SSD) Let d ≥ 1, a lattice in Rd and ∗ its dual as defined in (1.5). Then, the lattice will be said similarly self dual (SSD) if there exists λ > 0 such that Spg ( ∗ ) = λSpg ().

(SSD)

Within the above notation, we can state the main result of this paper. Namely, we have Main Theorem Let d ≥ 1 and a lattice of Rd and assume that is similarly self dual (SSD) in the sense of Definition 1.1. Then, the value of the theta function θ (j ) at the time t ∗ = t∗ = π/(vol(Rd / ))2/d is related to its successive derivatives θ , j = 1, 2, . . . , k, by the following identity of arithmetic type k−1 −1 ∗ d ∗ j (j ) ∗ k ∗ k (k) ∗ θ (t ) = akj t θ (t ) + ((−1) − 1)t θ (t ) , (1.6) (d/2)k j =1

or equivalently γ ∈

e

−t ∗ |γ |2

k−1 −1 ∗ 2 d = (−1)j akj (t ∗ |γ |2 )j e−t |γ | (d/2)k γ ∈

j =1

+ ((−1) − 1) k

γ ∈

∗

2 k −t ∗ |γ |2

(t |γ | ) e

,

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where (d/2)0 = 1,

(d/2)k =

k−1

((d/2) + j )

j =0 d are universal coefficients given explicitly by and akj

d akj

k−1 2−k+j k!

= (d + 2q), j !(k − j )!

j = 0, 1, . . . , k − 1.

q=j

In particular, for k = 1, the identity (1.6) reads simply θ (t ∗ ) = (−4/d) · t ∗ θ (t ∗ ) or equivalently γ ∈

−π|γ |2

e (vol(Rd / ))2/d =

−π|γ |2 π 4 2 (vol(Rd / ))2/d · |γ | e . d (vol(Rd / ))2/d

(1.7)

γ ∈

Here vol(Rd / ) stands for the Lebesgue volume of the d-dimensional torus Td = Rd / . Remark 1.2 The above theorem can be reworded in terms of identities between values of the traces of the heat semigroup et∗ and the operator ( ∗ )k et∗ , where ∗ is the usual Laplacian Rd acting on the Hilbert space L2 ∗ (Rd ) := L2 (Rd / ∗ ). Namely, we have Trace(et∗ ) = θ (t) and for every fixed integer k ≥ 1, (k) Trace(( ∗ )k et∗ ) = θ (t). Note that the obtained identities come essentially from spectral theory of the usual Euclidean Laplacian acting on torus. Henceforth, one may look for identities that come from the spectral theory of perturbed Laplacians such as the Landau operators that arise in even dimension d = 2n given in the real coordinates (x, y) ∈ Rd = Rn × Rn by n ∂ ∂ yj − ν 2 (|x|2 + |y|2 ) ν = R2n + 2iν − xj ∂xj ∂yj j =1

when acting on sections of an appropriate line bundle Lν over torus in Rd = R2n . For physicists, Landau operators are Hamiltonians of a charged particle in a uniform magnetic field with intensity ν. We hope to furnish in the near future related arithmetic identities that encompass those of the above theorem corresponding to the case ν = 0. Remark 1.3 For d = 1 and d = 2, the assumption (SSD) in the above main theorem is superfluous. As this will be shown, any lattice in Rd , d = 1 or d = 2, is similarly self dual. Hence the identities (1.6) hold for any lattice in R1 at t ∗ = π and any lattice in R2 at the time t ∗ = π/vol(R2 / ). Here vol(R2 / ) stands for the area of a fundamental cell of in R2 with respect to the Lebesgue measure on R2 .


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The paper is organized as follows. In Sect. 2, we recall some needed background on lattices of Rd and we write the Poisson summation formula for their associated theta function θ (t). Section 3 deals with the proof of the main result for lattices of Rd that are similarly self dual in Rd for d ≥ 1. In Sect. 4, we collect some related remarks for likewise theta functions in R1 and R2 . We conclude the paper by an appendix in which we give the proof of Lemma 3.2.

2 Background on lattices of Rd , the Poisson summation formula and its application to the θ -function Let (Rd , , ) be the Euclidean space of dimension d ≥ 1 endowed with its standard scalar product: x, y = dj =1 xj yj for x, y ∈ Rd and set |x| = x, x 1/2 . Then a lattice of Rd is a given discrete subgroup of maximal rank of the additive group (Rd , +). That is can be written as

(2.1) = γ ∈ Rd ; γ = m1 a1 + · · · + md ad ; mj ∈ Z, j = 1, . . . , d , where a1 , . . . , ad are R-linearly independent vectors in Rd . The nonsingular d × d matrix A whose the columns are the vectors a1 , . . . , ad , is called a generator matrix of the lattice . The (positive definite symmetric) Gram-Schmidt matrix G = AT A, where AT is the transpose of A, whose entries are ai , aj , is called a metric tensor of . Associated to such given lattice is its dual lattice ∗ defined by (1.5). Note that −1 if A is a generator matrix of , then the matrix (2π)AT is a generator matrix of ∗ and so a metric tensor G∗ of ∗ is given by G∗ = 4π 2 G−1 . Also, let () denote a fundamental cell of and |()| := vol(()) its Lebesgue volume. Then, we have √ |()| = det G = vol(Rd / ), where Rd / is the d-dimensional torus associated to . Also, recall that if ( ∗ ) is a fundamental cell of ∗ , then we have the identity |( ∗ )| =

(2π)d . |()|

For additional background on lattices, one can refer to [4] and the references therein. With the above data, we can reformulate Definition 1.1, Sect. 1 of similarly self dual lattice in terms of the tensor metric matrix G of the lattice as follows: Definition 2.1 Let a lattice of Rd with G as a metric tensor and let ∗ be its dual lattice with G∗ as metric tensor. Then is said to be similarly self dual if there exist λ > 0 and X ∈ GL(d, Z), the group of nonsingular d × d matrices whose entries are in Z, such that G∗ = λX T GX.

(2.2)

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Remark 2.2 Note that in view of (2.2), the factor λ is completely determined and it is given by λ=

4π 2 . (det G)2/d

Since a metric tensor G∗ of ∗ can be given by G∗ = 4π 2 G−1 the identity (2.2) can be reworded as G−1 =

1 X T GX (det G)2/d

(2.3)

for some X ∈ GL(d, Z). Further, we should note that the square length |γ ∗ |2 of γ ∗ ∈ ∗ is given by |γ ∗ |2 = G∗ m, m for some m ∈ Zd and in view of (2.2) we have |γ ∗ |2 =

4π 2 |γ |2 (det G)2/d

(2.4)

for some γ ∈ and vice versa. Hence Definition 2.1 is equivalent to Definition 1.1 cited in the introduction. Remark 2.3 Throughout this paper, we will be using only Definition 2.1 as in (2.2). Examples of similarly self dual lattices in Rd , d ≥ 1 1. All lattices of R1 are similarly self dual. This is obvious for any lattice of (R, +) is generated by δ > 0, i.e., = δZ, and its dual lattice ∗ is given by ∗ = (2π/δ)Z. Hence, Definition 2.1 holds trivially with λ = 4π 2 /δ 4 and X = 1. 2. All lattices of R2 are also similarly self dual. Indeed, this follows from the following lemma. Lemma 2.4 Let A be a nonsingular 2 × 2 matrix. Then, the inverse matrix A−1 of A is given by the formula A−1 = where J =

0 1 . −1 0

1 J T AT J, det A

Hence, for each Gram-Schmidt matrix G = AT A, we have G−1 =

1 1 J T GT J = J T GJ. det G det G

That is Definition 2.1 holds for λ=

4π 2 4π 2 = , det G (det A)2

X = J ∈ GL(2, Z).

(2.5)


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3. For d ≥ 3 there is no hope that the inverse of a nonsingular matrix G can be related to G through a formula like (2.2) in the case of d = 2. Nevertheless, lattices with generator matrix A satisfying AT A = k 2 Id for some k = 0 are obviously similarly self dual in the sense of Definition 2.1. Also, we can construct, for any dimension d ≥ 3, generator matrices of lattices not necessarily orthogonal that are similarly self dual. See some concrete examples in Sect. 4 that arise in the theory of quasicrystallography as those given in [2]. Now, recall that the Poisson summation formula related to an arbitrary fixed lattice , not necessarily rectangular or orthogonal in Rd , is given by: Lemma 2.5 (Poisson Summation Formula) Let be a lattice in Rd and ∗ be its dual as defined by (1.5). Then, for every smooth function ϕ in the Schwartz space on Rd , we have the following Poisson summation formula ∗ ϕ(x + γ ) = |( ∗ )| ϕ (γ ∗ )ei x,γ , (2.6) γ ∗ ∈ ∗

γ ∈

where x ∈ Rd and

ϕ (ξ ) =

Rd

e−i x,ξ ϕ(x)dx

is the Fourier transform of ϕ. In particular, when x = 0, the equality (2.6) reads as ϕ(γ ) = |( ∗ )| ϕ (γ ∗ ).

(2.7)

γ ∗ ∈ ∗

γ ∈

As an application, let t > 0 and let ϕt (y) = e−t|y| be the Gaussian function on Rd , whose Fourier transform ϕt (ξ ) is known to be given by 2

ϕt (ξ ) =

1 4πt

d/2

e−

|ξ |2 4t

.

Then, by applying the above Lemma 2.5, we get the following corollary. Corollary 2.6 For every t > 0 and x ∈ Rd , we have the following identity

e

−t|x+γ |2

=

γ ∈

1 4πt

d/2

|( ∗ )|

e−

|γ ∗ |2 ∗ 4t +i x,γ

.

(2.8)

γ ∗ ∈ ∗

In particular for x = 0, we have γ ∈

e−t|γ | = 2

1 4πt

We need the following definition.

d/2

|( ∗ )|

γ ∗ ∈ ∗

e−

|γ ∗ |2 4t

.

(2.9)

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Definition 2.7 We call the theta function associated to and we denote θ (t) the following function in t ∈ R> defined by the series 2 θ (t) := e−t|γ | . (2.10) γ ∈

Note that the theta function θ (t) as defined in (2.10) can be interpreted as the trace of the heat semigroup et∗ on the torus Rd / ∗ with ∗ =

∂2 ∂2 + · · · + ∂x12 ∂xd2

the Euclidean Laplacian acting on ∗ -periodic functions of Rd . Remark 2.8 The identity (2.9) can be written in terms of the theta functions θ and θ ∗ as 1 1 d/2 , (2.11) θ (t) = |( ∗ )|θ ∗ 4πt 4t or in terms of traces as Trace(e

t ∗

)=

1 4πt

d/2

1

|( ∗ )|Trace(e 4t ),

t > 0.

(2.12)

3 Identities of arithmetic type between values of the theta function θ (t) and its derivatives at a “certain” time t∗ > 0 Let be an arbitrary lattice of Rd , d ≥ 1, and θ (t) its associated theta function as defined in (2.10). Then, the following functional relationship holds when is a similarly self dual in the sense of Definition 2.1. Namely, we have Proposition 3.1 Let be a similarly self dual lattice in Rd , d ≥ 1. Then, the theta function θ ∗ associated to the dual lattice ∗ is related to θ associated to by 4π 2 t , t > 0. (3.1) θ ∗ (t) = θ |()|4/d Furthermore, for every t > 0, we have the following identity of Jacobi type d/2 π2 π θ (t) = . θ t|()|2/d t|()|4/d

(3.2)

Proof Let be a similarly self dual lattice of Rd whose generator matrix is A and let ∗ be its dual lattice. Also let ∗ 2 e−t|γ | . θ ∗ (t) = γ ∗ ∈ ∗


279 −1

For any γ ∗ ∈ ∗ , there exists m ∈ Zd such that γ ∗ = 2πAT m, which implies that |γ ∗ |2 = 4π 2 G−1 m, m , G = AT A (G is a Gram-Schmidt matrix of ). It follows that 2 −1 e−4π G m,m t . θ ∗ (t) = m∈Zd

By Definition 2.1 and formula (2.3), one can find X ∈ GL(d, Z) such that G−1 = (1/|()|4/d )X T GX; the function θ ∗ (t) can be then written as θ ∗ (t) =

e

−

4π 2 |()|4/d

GXm,Xm t

m∈Zd

which, by the change n = Xm ∈ Zd , it can be rewritten as θ ∗ (t) =

e

−

4π 2 t |()|4/d

Gn,n

,

t > 0.

(3.3)

n∈Zd

On the other hand, for any t > 0, it follows 2 e−t Gn,n = e−t|γ | = θ (t). n∈Zd

γ ∈

Hence, from the equality (3.3), one can get that 4π 2 t θ ∗ (t) = θ . |()|4/d

(3.4)

The second identity (3.2) of the proposition follows from (2.11) combined with (3.4) and keeping in mind the fact |()||( ∗ )| = (2π)d . This finishes the proof of the proposition. Now, we proceed to the proof of the main result of this paper as cited in the introduction. Proof of the main Theorem Recall that from (3.2) of the above proposition, we have θ (t) =

π |()|2/d

d/2 t

−d/2

θ

π2 . t|()|4/d

Then, by differentiating one time the both sides of (3.5), we get d/2 π d π2 θ 2 t|()|2/d t|()|4/d

(3.5)

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= − tθ (t) +

π t|()|2/d

(2/d)+2

tθ

π2 t|()|4/d

.

(3.6)

Now let τ : R> → R> be the involution defined by τ (t) = π 2 /(t|()|4/d ) and t ∗ = t∗ be its unique fixed point in R> , which is given by t∗ =

π . |()|2/d

(3.7)

Then, by evaluating (3.6) at the time t ∗ given in (3.7), we get the identity θ (t ∗ ) = −

4 ∗ ∗ · t θ (t ). d

(3.8)

Next, in order to continue the differentiation many times, we need the following d k −c −x/t lemma to give the explicit expression of dt e ); see the Appendix for the k (t proof. Precisely, we have Lemma 3.2 Let c > 0 and x > 0. Then, for every integer k ≥ 1, we have d k −c −x/t (t e ) = (−1)k (c)k t −c−k 1 F1 (−k; c; x/t)e−x/t , (3.9) dt k where (c)k = kj =1 (c + j − 1) is the Pochhammer symbol and 1 F1 (a; c; x) = (a)j x j j ≥0 (c)j j ! is the usual confluent hypergeometric function. Therefore, by taking the k-derivative of both sides of (3.5) and using the above lemma with c = d/2 and x = (π/|()|2/d )2 |γ |2 = t ∗ 2 |γ |2 , we get θ(k) (t) = t ∗ d/2

dk t ∗2 2 (t −d/2 e− t |γ | ) k dt

γ ∈

= (−1)k t ∗ d/2

γ ∈

t ∗ 2 2 − t ∗2 |γ |2 |γ | e t (d/2)k t −(d/2)−k 1 F1 −k; d/2; . t

(3.10)

The above hypergeometric function 1 F1 (−k; d/2; z) is a polynomial of degree k that we can write as 1 F1 (−k; d/2; z) =

k j =0

(−1)j k! zj . (d/2)j (k − j )!j !

Hence substituting (3.11) in the previous equality (3.10), we obtain (−1)k

t∗ t

−d/2

t k θ(k) (t) =

k j =0

d akj

t ∗2 t

j

(j )

θ

t ∗2 , t

(3.11)


281

which we may write as θ

t ∗2 t

k−1 ∗2 ∗2 j −1 (j ) t d t akj θ = (d/2)k t t j =1

+

t ∗2 t

k

(k) θ

t ∗2 t

− (−1)

k

t∗ t

−d/2

(k) t k θ (t)

, (3.12)

where we have set d = akj

(d/2)k k! , (d/2)j (k − j )!j !

j = 1, . . . , k − 1.

(3.13)

Hence, by evaluating (3.12) at the unique fixed point of the involution t → π 2 /(t|()|4/d ) = t ∗ 2 /t, i.e., at the time t ∗ = t∗ := π/|()|2/d , we obtain the following identity k−1 −1 d ∗ j (j ) ∗ k ∗ k (k) ∗ akj t θ (t ) + (1 − (−1) )t θ (t ) . θ (t ) = (d/2)k ∗

j =1

The above obtained identities can be rewritten explicitly as k−1 −1 ∗ 2 ∗ 2 d e−t |γ | = (−1)j akj (t ∗ |γ |2 )j e−t |γ | (d/2)k γ ∈

γ ∈

j =1

+ ((−1)k − 1)

(t ∗ |γ |2 )k e

−t ∗ |γ |2

.

γ ∈

Thus the proof of the theorem is completed.

d as given in (3.13), up to 2k , Remark 3.3 Note here that the involved coefficients akj are universal integral and depend only in d. Indeed, using the fact that (d/2)j = j −1 2−j q=0 (d + 2q) so that k−1

(d/2)k = 2−k+j (d + 2q), (d/2)j q=j

we get that d = 2k akj

k−1 2j k!

(d + 2q), j !(k − j )!

j = 1, . . . , k − 1.

q=j

We conclude this paper by pointing out some related remarks.

(3.14)

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4 Concluding remarks In this concluding section, we will show how our procedure for deriving the identities of “arithmetic” type as stated in the main theorem for theta functions θ (t) can work well for other type of associated likewise theta functions. For this, we will be emphasizing only on some examples that arise in number theory in R1 and R2 . 4.1 Examples in R1 2 Besides the classical theta function θZ (t) = m∈Z e−tm , t > 0, introduced in the beginning of this paper, we may consider similar theta functions such as ηZ (t) defined here by: 2 ηZ (t) = (−1)m e−t (6m+1) /12 , t > 0, (4.1) m∈Z

which is known to satisfy the same functional equation as θZ (t) (see for instance [1] it )), namely where ηZ is given as θZ ( 12 )( 2π ηZ (t) =

1/2 2 π π . ηZ t t

(4.2)

Therefore, using the same procedure of differentiation and the technical Lemma 3.2, as we did for θZ (t), we can establish identities of arithmetic type between values of ηZ and its successive derivatives at the time t ∗ = π . For instance, we get directly, by differentiating the both sides of (4.2) and evaluating at the time t ∗ = π , the following identity ηZ (π) = −4πηZ (π).

(4.3)

The above identity reads explicitly in terms of series as 1 2 m −π(3m2 +m) m 2 e−π(3m +m) . (−1) e = 4π (−1) 3m + m + 12 m∈Z

m∈Z

4.2 Examples in R2 Let θQ (t) = m∈Z2 e−tQ(m) , t > 0, be the theta series associated to a positive definite quadratic form Q in R2 , say Q(x) = ax12 + bx1 x2 + cx22 , x = (x1 , x2 ) ∈ R2 , with a > 0, c > 0 and DQ = b2 − 4ac < 0 where DQ is the so-called discriminant of the quadratic form Q. Then the quadratic form Q can be written as Q(x) = Gx, x with G a positive definite symmetric matrix in R2 . Since the matrix G can be written in the form G = AT A for some A ∈ GL(2, R), we can consider the lattice A in R2 generated by A to which we apply (3.2) Proposition 3.1 for dimension d = 2 to see that the series θQ (t) satisfy the Jacobi fundamental equation −4π 2 2π . θQ θQ (t) = tDQ t −DQ

(4.4)


283

Using again the procedure of differentiation of both sides of (4.4) as well as the ∗ = 2π/ −D , we can obtain idenLemma 3.2 and evaluating at the time t ∗ = tQ Q tities of arithmetic type relating the value of θQ at t = t ∗ to those of its successive k-derivatives at the time t = t ∗ . For instance, for k = 1, we have the following identity ∗ θQ (t ∗ ) = −2t ∗ θQ (t ),

(4.5)

which reads explicitly as m∈Z2

e

√−2π Q(m) −D Q

=

√−2π Q(m) 4π Q(m)e −DQ . −DQ 2

(4.6)

m∈Z

Remark 4.1 Note that the above procedure can be applied to a finite linear sums of theta series θQ1 (t ∗ ), θQ2 (t ∗ ), θQ3 (t ∗ ), . . . , in R2 , provided that the positive definite quadratic forms Q1 , Q2 , Q3 , . . . , have the same discriminant, i.e., DQ1 = DQ2 = DQ3 = · · ·. For instance for the function E(t) := θQ1 (t) − 2θQ2 (t), where Q1 (x) = x12 + x1 x2 + 6x22 and Q2 (x) = 2x12 + x1 x2 + 3x22 and for which we have DQ1 = √ DQ2 = −23, see [5]. Then, in view of (4.6), we see that at the time t ∗ = 2π/ 23 we have for instance 2π −4π 2π = √ E √ . (4.7) E √ 23 23 23 Remark 4.2 Finally, we should mention to the reader that by taking the positive definite quadratic form Q(x) = ax12 + cx22 (a > 0 and c > 0), we can apply our main result in the case d = 2 to obtain a lot of quadratic relationships of the usual theta function θZ (t), defined in the introduction, with its successive k-derivatives at the points t > 0, s > 0 such that ts = π 2 , i.e., s = τ (t). For instance for k = 1, we can show that we have the following first quadratic relationship θZ (t)θZ (s) = −2 tθZ (t)θZ (s) + sθZ (s)θZ (t) . (4.8) In particular for t = s = π , the above identity reduce to the identity (1.3) cited in the introduction θZ (π) = −4πθZ (π).

(4.9)

Acknowledgements A part of this paper was written when the third author was visiting University of Sciences and Technologies of Lille 1. He is grateful to all members and staff of the Department of Mathematics for their generous hospitality.

Appendix Proof of Lemma 3.2 We prove the identity (3.9) by induction. For k = 1 the equality (3.9) holds by direct derivation. Indeed, we have x −x x −x d −c − x −c−1 −c−1 t t (t e ) = −ct e t. 1 − c e = −ct 1 F1 −1; c; dt t t

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Hence, assume that (3.9) holds up to k ≥ 1 and show that it holds for k + 1. To do this, d k+1 d dk we use the fact that dt k+1 = dt dt k and the induction assumption to have the equality x −x d −c−k d k+1 −c − x k t t . t e t e = (−1) (c)k 1 F1 −k; c; dt t dt k+1 Now, we use the change of variable z =

x t

(∗)

to rewrite (∗) as

z2 d −z c+k d k+1 −c − x t = −(−1)k (c)k e z 1 F1 (−k; c; z) e t k+1 c+k−1 dz dt x

(∗∗)

and by applying the following differentiation formula for the confluent hypergeometric function 1 F1 (a; c; z) (see [3], p. 264) d n −z c−a+n−1 −z c−a−1 e z 1 F1 (α; c; z) = (c − a)n e z 1 F1 (a − n; c; z) n dz with n = 1 and a = −k, and replacing it in (∗∗) for z = x/t, it follows at once that the equality (3.9) holds for k + 1. This ends the proof of the lemma.

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