Alain Bensoussan

Alexandre Brouste
email: alexandre.brouste@univ-lemans.fr

Youssef Esstafa

Asymptotically efficient one-step stochastic gradient descent

Keywords: Estimation, Inference, Numerical Optimization, Machine Learning, Simulation

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Asymptotically efficient one-step stochastic gradient descent

Introduction

The stochastic gradient descent [START_REF] Robbins | A stochastic approximation method[END_REF] for finding the root of a given functional is a widely used method in statistical learning. In the parametric estimation setting, this method leads to a (strongly) consistent estimator [START_REF] Blum | Approximation methods which converge with probability one[END_REF][START_REF] Wolfowitz | On the stochastic approximation method of Robbins and Monro[END_REF] but which is not asymptotically efficient in term of converging rate or in term of asymptotic variance depending on the conditions retained [START_REF] Chung | On a stochastic approximation method[END_REF][START_REF] Hodges | Two approximations to the Robbins-Monro process[END_REF][START_REF] Ruppert | Efficient estimations from a slowly convergent Robbins-Monro process[END_REF][START_REF] Sacks | Asymptotic distribution of stochastic approximation procedures[END_REF]. This method has been later improved by the stochastic gradient with averaging [START_REF] Polyak | Acceleration of stochastic approximation by averaging[END_REF][START_REF] Ruppert | Efficient estimations from a slowly convergent Robbins-Monro process[END_REF] or the adaptative stochastic gradient [START_REF] Lai | Adaptive design and stochastic approximation[END_REF][START_REF] Venter | An extension of the Robbins-Monro procedure[END_REF] which present an optimal asymptotic rate and variance.

In this paper, we propose a fast and asymptotically efficient alternative to averaging or adaptivity. It is based on the one-step procedure.

The one-step procedure was initially considered in (Le [START_REF] Cam | On the asymptotic theory of estimation and testing hypotheses[END_REF] for the estimation of parameters in independent and identically distributed (i.i.d.) samples. In this procedure, an initial guess estimator is proposed which is fast to be computed but not asymptotically efficient. Then, a single step of the gradient descent method is done on the log-likelihood function in order to correct the initial estimation and reach asymptotic efficiency. With some recent developments, the one-step procedure has been successfully generalized to more sophisticated statistical experiments as diffusion processes [START_REF] Gloter | Adaptive estimation for degenerate diffusion processes[END_REF][START_REF] Kamatani | Hybrid multi-step estimators for stochastic differential equations based on sampled data[END_REF], ergodic Markov chains [START_REF] Kutoyants | On multi-step MLE-process for Markov sequences[END_REF], inhomogeneous Poisson and Hawkes counting processes [START_REF] Brouste | Fast and asymptotically efficient estimation in the Hawkes processes[END_REF][START_REF] Dabye | Method of moments estimators and multi-step MLE for Poisson processes[END_REF], fractional Gaussian and stable noises observed at high frequency [START_REF] Brouste | Efficient estimation of stable Lévy process with symmetric jumps[END_REF][START_REF] Brouste | One-step estimation for the fractional Gaussian noise at high-frequency[END_REF].

In the following, Section 2 is dedicated to notations and known results of convergence rates for stochastic gradient descent (SGD), stochastic gradient descent with averaging (AVSGD), adaptative gradient descent (ADSGD) and maximum likelihood estimation (MLE). The main result on (strong) consistency and asymptotic normality of the one-step procedure in the multidimensional parameter setting is given in Section 3. Monte Carlo simulations are done in Section 4 to assess the performance of the proposed statistical procedure (OSSGD) in comparison with SGD, AVSGD, ADSGD and MLE in terms of computation time and asymptotic variance for samples of finite size.

Notations

In our parametric estimation problem, the observation sample is denoted X n = (X 1 , . . . , X n ) and is composed of independent and identically distributed random variables. The probability density f (•, u) (with respect to some σ-finite measure) of X 1 is parametrized by u ∈ Θ ⊂ R p where Θ is an open set. The true parameter ϑ ∈ Θ is to be estimated.

The estimation problem of the unknown parameter ϑ can be seen as finding the minimum of an unknown function

G(u) = E ϑ (-log f (X 1 , u)) or the root of its gradient M (u) = E ϑ (-∇ u log f (X 1 , u)) . (1)
Consequently, the Robins-Monro algorithm [START_REF] Robbins | A stochastic approximation method[END_REF]) can be directly used to find the root and is defined recursively by

ϑ i+1 = ϑ i -γ i (-∇ u log f (X i , ϑ i )), 1 ≤ i ≤ n -1, (2)
where (γ i ) i is the step sequence and ϑ 1 is the initial value (it may be random but square integrable) of the procedure. For instance, this procedure leads to a (strongly) consistent estimator [START_REF] Blum | Approximation methods which converge with probability one[END_REF][START_REF] Dvoretzky | On stochastic approximation[END_REF][START_REF] Wolfowitz | On the stochastic approximation method of Robbins and Monro[END_REF][START_REF] Pelletier | On the almost sure asymptotic behaviour of stochastic algorithms[END_REF] for

γ i ≥ 0, i γ 2 i < ∞ and i γ i = ∞.
This algorithm is fast but is not asymptotically efficient, neither in terms of converging rate nor in terms of asymptotic variance. For the sequence γ i = i -r and r ∈ (1/2, 1), it leads to an asymptotically normal estimator for which

n r 2 (ϑ n -ϑ) =⇒ N 0, 1 2 I p (3)
where I p stands for the p × p identity matrix. This result is proved in Proposition 1 of Section 3 for the multidimensionnal parameter setting following mainly [START_REF] Chung | On a stochastic approximation method[END_REF][START_REF] Fabian | On asymptotic normality in stochastic approximation[END_REF][START_REF] Hodges | Two approximations to the Robbins-Monro process[END_REF][START_REF] Ruppert | Efficient estimations from a slowly convergent Robbins-Monro process[END_REF]. It is worth mentioning that the asymptotic variance does not depend on ϑ in the i.i.d. setting.

In order to fasten the estimation convergence rate, two methods are classically used: averaging and adaptivity.

Averaging

The averaging method was proposed (see [START_REF] Polyak | Acceleration of stochastic approximation by averaging[END_REF][START_REF] Ruppert | Efficient estimations from a slowly convergent Robbins-Monro process[END_REF]) to reach variance efficiency with

ϑ * n = 1 n n i=1 ϑ i .
This estimator is consistent, asymptotically normal with efficient rate and variance, namely

√ n (ϑ * n -ϑ) =⇒ N (0, I(ϑ) -1 )
where

I(ϑ) = -E ϑ ∇ 2 u,u log f (X 1 , ϑ) (4)
stands to the Fisher information matrix (or the Hessian of the functional G).

For these reasons, it can be compared to the maximum likelihood estimator defined by

ϑ n = arg max u∈Θ 1 n n i=1 log f (X i , u).
(

) 5
The MLE is generally not in a closed form and its approximation can be time consuming for large samples.

Adaptivity

It had been shown, in the multidimensional setting, that the stochastic gradient descent with

γ i = ci -1 and c > 1 2λ min (I(ϑ)) , (6)
where λ min (A) is the lowest eigenvalue of the matrix A, is asymptotically rate efficient but is still not asymptotically variance efficient (see [START_REF] Duflo | Random iterative models[END_REF][START_REF] Pelletier | Asymptotic almost sure efficiency of averaged stochastic algorithms[END_REF]) with

√ n (ϑ n -ϑ) =⇒ N 0, c 2 I(ϑ)(2cI(ϑ) -I p ) -1 .
The constraint (6) depends on the unknown parameter and cannot be used in practice. For this reason, adaptative methods have been developed. In the simple setting of i.i.d. samples, the adaptative stochastic gradient descent writes

ϑ i+1 = ϑ i -i -1 I( ϑ i ) -1 -∇ u log f (X i , ϑ i ) , 1 ≤ i ≤ n -1.
It leads also to a consistent and asymptotical normal estimators with optimal limit variance [START_REF] Amari | Natural gradient works efficiently in learning[END_REF], namely

√ n ϑ n -ϑ =⇒ N (0, I(ϑ) -1 ).
To fasten the computation, we propose in the following the one-step procedure starting from an initial guess estimator taken from the stochastic gradient algorithm (2). This algorithm is shown to be faster than the classical computation of the MLE but still asymptotically efficient. It is an interesting alternative to the stochastic gradient algorithm with averaging or adaptative gradient descent and shows nice properties also on samples of finite size.

One-step estimation procedure

The one-step estimation procedure is proposed in this section to reach asymptotic efficiency. The estimation given at step n by the stochastic gradient descent (see Equation ( 2)) is corrected by

ϑ n = ϑ n + I(ϑ n ) -1 • 1 n n i=1 ∇ u log f (X i , ϑ n ). (7)
It leads to a consistent, asymptotically normal and asymptotically efficient estimator of ϑ.

In the following, we state the slowly convergence of the stochastic gradient descent in the multidimensional setting. We suppose that the statistical experiment is regular, i.e.

M (u) = I(ϑ)(u -ϑ) + o (∥u -ϑ∥) , (8)
where I(ϑ) is positive definite and stands for the Fisher information matrix defined in (4).

Proposition 1. For the sequence γ i = i -r and r ∈ (1/2, 1), the stochastic gradient descent provides an asymptotically normal estimator for which

n r 2 (ϑ n -ϑ) =⇒ N 0, 1 2 I p . (9)
Proof. The proof is postponed in Appendix A.

It is worth mentioning that the asymptotic variance does not depend on ϑ in this simple setting. This algorithm is fast but is not asymptotically efficient, neither in terms of converging rate nor in terms of asymptotic variance.

We suppose that the matrix valued function I(ϑ) is Lipschitz continuous, i.e. there exists a constant L > 0 such that

∥I(x) -I(y)∥ m ≤ L∥x -y∥, x, y ∈ Θ,
where ∥ • ∥ m and ∥ • ∥ stand for Euclidean norms in the space of matrices and vectors respectively. With this condition, we can state the main result:

Theorem 1. The sequence (ϑ n , n ≥ 1) of one-step estimators of ϑ defined by (7) is consistent and asympotically normal, i.e.

√

n(ϑ n -ϑ) =⇒ N (0, I(ϑ) -1 ).

(10)

Proof. The proof is postponed in Appendix B.

Simulations

The joint estimation of the shape parameter α and scale parameter β is considered in the statistical experiment generated by a sample (X 1 , X 2 , . . . , X n ) of i.i.d. Gamma random variables whose probability density function is given by

f (x) = β α Γ(α) x α-1 exp(-βx), x > 0.
Let us denote ϑ = (α, β). In this statistical experiment, the sequence of maximum likelihood estimators ( ϑ n ) n≥1 of ϑ is not in a closed-form. The sequence of MLE satisfies

√ n ϑ n -ϑ → N 0, I(ϑ) -1 ,
where

I(ϑ) = ψ (2) (α) -1 β -1 β α β 2 .
Here, ψ (n) is the polygamma functions (see (Abramowitz & Stegun, 1992, section 6.4.1, page 260)) defined by ψ (n) (α) = ∂ n ∂α n log Γ(α). The different estimators (MLE, SGD, OSSGD, AVSGD, ADSGD) have been compared in terms of variance and computation time on B = 2 × 10 3 Monte Carlo simulations for samples of size n = 10 4 . The SGD is done with γ i = i -r where r is chosen to be equal to 0.6. It is worth mentioning that the result are similar for all values of 1 2 < r < 1. We can see on Figure 1 that the optimal variance is reached for the OSSGD (as for the MLE, AVSGD and ADSGD) that naturally overperforms the non-optimal variance of the slowly converging SGD. It is worth noting the relative bias of the AVSGD.

In terms of computation time, the OSSGD (as the AVSGD) is more than 3 times faster than the MLE. In comparison, the ADSGD is more than two times faster.

For these reasons, the fast and asymptotically efficient OSSGD is a proper alternative to the averaged and the adapted stochastic gradient descent methods. MLE SGD OSSGD AVSGD ADSGD time (s) 98.52 30.23 30.40 30.41 43.48 It is worth noting that, for the specific case of the estimation of the parameters in the Gamma distribution, moment estimators [START_REF] Brouste | The R journal: OneStep : Le Cam's one-step estimation procedure[END_REF] or other original explicit estimators [START_REF] Ye | Closed-form estimators for the gamma distribution derived from likelihood equations[END_REF]) could have been considered as initial guess estimation in the one-step procedure instead of the SGD.

Conclusion

In this paper, we propose to apply the one-step procedure to the slowly converging stochastic gradient descent in order to fasten the convergence rate and reach asymptotical efficiency. It is a fast and asymptotically efficient alternative to averaging or adaptivity.

The one-step procedure for the stochastic gradient descent is considered here in the i.i.d. setting but it will be extended in a further work to the regression setting (linear regression, logistic regression (see also [START_REF] Bercu | An efficient stochastic Newton algorithm for parameter estimation in logistic regressions[END_REF] for an adaptative procedure), generalized linear models) for larger applications.

A Robbins-Monro's algorithm

The proof of Proposition 1 which shows asymptotic normality for the slowly converging stochastic gradient descent method (γ i = i -r , 1/2 < r < 1) in the multidimensional setting follows [START_REF] Sacks | Asymptotic distribution of stochastic approximation procedures[END_REF] and [START_REF] Ruppert | Efficient estimations from a slowly convergent Robbins-Monro process[END_REF]. The Robbins-Monro algorithm can be rewritten as noisy version of the classical gradient descent method, namely

ϑ i+1 = ϑ i -γ i M (ϑ i ) -γ i ξ i , i ≥ 1, (11)
where

ξ i = -∇ ϑ log f (X i , ϑ i ) -M (ϑ i ).
From condition (8), one has

ϑ i+1 -ϑ = ϑ i -ϑ -γ i (I(ϑ)(ϑ i -ϑ) + δ i ) -γ i ξ i = (I p -γ i I(ϑ))(ϑ i -ϑ) -γ i δ i -γ i ξ i , (12)
where δ i = o(∥ϑ i -ϑ∥). A straightforward induction based on the recursive equation (12) yields

ϑ i+1 -ϑ = α 0,i (ϑ 1 -ϑ) - i j=1 γ j α j,i δ j - i j=1 γ j α j,i ξ j ( 13
)
with matrices α j,i = i k=j+1 (I p -γ k I(ϑ)) for j ̸ = i and α i,i = I p .

The covariance matrix I(ϑ) is a symmetric positive definite real matrix, it can be diagonalized such that D(ϑ) = P T (ϑ)I(ϑ)P (ϑ) is a diagonal matrix and P (ϑ) is an orthogonal matrix. Let us denote by b 1 , b 2 , . . . , b p the eigenvalues of I(ϑ) in the deacreasing order. First, we can remark that

K 1 exp b p (j 1-r -i 1-r ) 2(1 -r) ≤ ∥α j,i ∥ m ≤ K 2 exp b p (j 1-r -i 1-r ) 2(1 -r) ,
for some positive constants K 1 and K 2 .

Let us define

v i = i j=1 γ 2 j ∥α j,i ∥ 2 m -1/2 .
It can be shown that v i ∼ i r/2 . In order to prove the proposition, we have to show successively, when i → ∞,

v i α 0,i (ϑ 1 -ϑ) -→ 0 in probability (Step 1) v i i j=1 γ j α j,i ξ j =⇒ N 0, 1 2 I p in law (Step 2)
and

v i i j=1 γ j α j,i δ j -→ 0 in probability (Step 3)
Slutsky's theorem concludes the proof of the Proposition. Here are the details for the three steps:

⋄ Step 1: Using the relation α j,i -α j-1,i = γ j I(ϑ)α j,i
we show as in [START_REF] Sacks | Asymptotic distribution of stochastic approximation procedures[END_REF] that, for a sequence W i → W (which could be ∞), we have i j=m

γ j I(ϑ)α j,i W j -→ W (14)
as i tends to infinity and finally lim i→∞ v i α j,i = 0. (15)

Since the random variable ϑ 1 is square integrable, one can easily obtain

E [∥α 0,i (ϑ 1 -ϑ)∥] ≤ K 3 ∥α 0,i ∥ m ,
where K 3 is a positive constant. Markov's inequality and Equation ( 15) provide the expected convergence.

⋄

Step 2: The spectral decomposition of I(ϑ) introduced above allows to write

P T (ϑ) i j=1 γ j α j,i ξ j = i j=1 γ j β j,i Υ j ,
where Υ j = P T (ϑ)ξ j and β j,i = i k=j+1 (I p -γ k D(ϑ)). Moreover, the random vectors Υ j are centered, independent and identically distributed with variance matrix

E Υ j Υ T j = P T (ϑ)E ξ j ξ T j P (ϑ) = P T (ϑ)I(ϑ)P (ϑ) = D(ϑ) = diag (b 1 , . . . , b p )
which is diagonal. We can use the same arguments developed in (Ruppert, 1988, Corollary 3.2) to obtain for each coordinate the proper convergence. For instance, one can obtain for the first component

i j=1 γ j (β j,i Υ j ) 1 = i j=1 γ j (β j,i ) 1,1 (Υ j ) 1 = -i -r i j=r(i) c(j, i)(Υ j ) 1 + o(1)
with r(i) = ⌊i -K 4 i r log i⌋ and c(j, i) = exp -b 1 i k=j+1 k -r . Hence, i r/2 i j=1 γ j (β j,i Υ j ) 1 =⇒ N (0, 1/2) in law when i → ∞. Similar arguments for the other coordinates give the result.

⋄

Step 3: The upper bound E ∥ϑ i -ϑ∥ 2 ≤ K 5 i 1-2r (K 5 > 0) can be found in (Chung, 1954, Theorem 1), which implies that, for r > 1 2 , E ∥δ i ∥ 2 -→ 0 as i goes to infinity. The expected result of this step follows from convergence in (14).

B One-step procedure

For an observation sample (X 1 , . . . , X n ), let us denote ℓ n (u) = n i=1 log f (X i , u). Recall that ϑ is the true parameter and

ϑ n = ϑ n + I(ϑ n ) -1 • 1 n ∇ u ℓ n (ϑ n ), n ≥ 1. ( 16
)
Consistency: The consistency of the sequence of initial guess estimators gives, as n → ∞, ϑ n -→ ϑ in probability. Since E ϑ ∇ u ℓ n (ϑ) = 0, the uniform law of large number gives, as n → ∞,

1 n ∇ u ℓ n (ϑ n ) -→ 0 R p
in probability. The uniform continuity of the Fisher information matrix gives the result. Since the initial stochastic gradient descent is also strongly consistent [START_REF] Blum | Approximation methods which converge with probability one[END_REF], we can also obtain the strong consistence with the strong law of large number.

Asymptotic normality: From ( 16), we have

√ n ϑ n -ϑ = √ n (ϑ n -ϑ) + I(ϑ n ) -1 • 1 √ n ∇ u ℓ n (ϑ n ).

Figure 1 :

 1 Figure 1: Statistical errors renormalized by √ n for MLE, SGD, OSSGD, AVSGD and ADSGD for n = 10 4 and B = 2 × 10 3 Monte-Carlo simulations. Theoretical optimal variance (in red) and variance of SGD (in blue) are superimposed.

Acknowledgments This research partially benefited from the support of the ANR project 'Efficient inference for large and high-frequency data' (ANR-21-CE40-0021) and the NSF project NSF-DMS 220 47 95.

The mean-value theorem gives

and

where I p is the p × p identity matrix.

The central limit theorem gives, as n → ∞,

in law and the proper convergence of the second term in the r.h.s. of Equation ( 17).

Considering the first right-hand term, we have that (ϑ n ) n≥1 is n δ 2 -consistent by assumption and n 1 2 -δ → 0, as n → ∞, for 1 2 < δ ≤ 1. Then, we need to show that n δ 2

is bounded in probability as n → ∞ with

The second terms in the r.h.s. converges to zero at rate √ n. The Lipschitz continuity of the Fisher information allows to control the first and third terms by C∥ϑ n -ϑ∥ where C is a generic constant. Since (ϑ n ) n≥1 is n δ 2 -consistent, the quantity n δ 2 A is bounded in probability as n → ∞. The Slutsky theorem gives the final result.


D Duchesneau

Experimental aspects of neutrino oscillation physics

The experimental review of this paper is mainly focussed on a particular aspect of the neutrino physics which is the observation and understanding of the neutrino oscillation process. This quantum mechanical mechanism involves several parameters linked to the neutrino properties which are described through a lepton mixing matrix. The detailed knowledge of those parameters may provide a key tool to answer several fundamental questions including the existence or not of flavour violation in the leptonic sector. The discovery of neutrino flavour oscillation 10 years ago was followed by an intense experimental activity with several key experiments which provided many clues to start to unravel the neutrino oscillation puzzle. The program is still active and impressive experimental challenges are underway to pursue this goal. The aim of the talk is to recall the main neutrino historical path and to give an overview of the many experimental aspects of the neutrino oscillation studies performed since10 years and the main results obtained.

Introduction: some history and neutrino basis

The proof of existence of the particle called neutrino has not been a trivial exercice. Before detailing the experimental measurements it is worth to quote some anecdotes about their birth and existence to understand better the scientific approach and appreciate the evolution of ideas through time. W. Pauli [START_REF] Duchesneau | s letter to L[END_REF] introduced for the first time in 1930 the concept of a new spin half light neutral particle as a remedy to explain the observed violation of the principle of energy-momentum conservation in radioactive β decays. The electron energy spectrum was found to be continuous and not discrete and the new hypothetic neutral particle which could be emitted at the same time as the electron was a good solution to solve this anomaly. However this new idea was even not thought as a serious one by its author. Pauli, in a discussion with one of his friends the german astronomer W. Baade, apparently confessed [START_REF] Pietschmann | George Marx Memorial Lecture[END_REF]: 'Today I have done something which no theoretical physicist should ever do in his life: I have predicted something which shall never be detected experimentally'. In 1933, exploiting the concept of this additional neutral particle, E. Fermi built a coherent theory of beta decays [START_REF] Fermi | [END_REF] which later became the basis of the weak interaction theory. However this theory was not accepted by everybody as one can see from one referee's comment when he submitted the article to Nature: 'Abstract speculations too far from physical reality to be of any interest to the readers'. To complete this quite strange picture, Bethe and Peierls computed in 1934 [4] the interaction cross section of the neutrinos based on the available theory. Unfortunately, using too simplistic hypothesis they found that few MeV neutrinos resulted to have an interaction length of about one light year of lead and formulate the conclusion: '...this meant that one obviously would never be able to see a neutrino'. However despite those claims some physicists succeeded in building specific experiments to track and catch those quite 'elusive' particles called neutrinos.

Neutrino detection and discovery

The discovery path started 30 years after the introduction of the neutrino concept when F. Reines and R. Cowan observed in 1956 for the first time [5] an interaction coming from anti-neutrino produced in the core of the Savannah River nuclear plant. Three large tanks filled with liquid scintillator were used to detect the anti-neutrino using the inverse beta decay reaction (ν e p → e + n). They saw a number of events corresponding to the process in which a delayed coincidence signature between two flashes of light, one from an electron-positron annihilation and the other one from a neutron capture, separated by a few microseconds is measured. It was the ν e discovery. Six years later the second neutrino (ν μ ) was discovered by Lederman et al. [6] at the Brookhaven National Laboratory using for the first time an intense beam of neutrino produced in a pion beam. They found that the neutrino produced in the pion decays interact in the detector differently than would have done a electron neutrino. They saw in their spark chambers several tens of interactions where a muon was produced instead of an electron. The idea of using a neutrino beam was proposed by M. Schwartz in 1960 [7] to provide a tool to investigate the weak interactions. The basis is to use a high intensity proton beam to hit a target and produce an intense source of pions which produce neutrinos when decaying. This principle is the one used in what is called conventional neutrino beams. Finally the third neutrino type (ν τ ) was only discovered in 2000 by the Donut Collaboration at Fermilab [8]. The collaboration uses an intense beam dump in which charmed mesons decaying into taus and ν τ are produced. Its existence was expected since the beginnning of the eighties and it was suggested after the discovery of the charged tau lepton in 1974 for which a new flavour neutrino has been theoretically associated. The proof of the existence of the ν τ is based on the detection of the tau particle produced in charged current interaction in the detector. To detect this short lived particle it has been necessary to use a tracking detector with micrometer position resolution. The technology used for this goal was based on nuclear emulsion cloud chamber principle. This technology is also the one used in the current OPERA experiment [9].

Neutrino as laboratory tools

In the early time, the neutrino has been extensively used as a probe to study the weak interaction. The first important result came with the discovery of the neutral current in 1973 [10] at CERN with the Garagamelle heavy liquid bubble chamber well suited for the study of neutrino interactions. Figure 1 shows one of those events. Neutrino beams were also intensively used to probe the nucleon structure and to derive the structure functions of the neutrons and protons. Starting from this period it was possible to measure ν interaction cross sections on various target elements and over a large energy range extending from a few hundred MeV to a few hundred GeV. Figure 2 shows a compilation of the measured cross sections for neutrino and anti-neutrinos [11]. The corresponding values are for neutrinos Figure 1: Picture of an event seen in the Gargamelle bubble chamber. It corresponds to an elastic ν μ -e scattering. The observation of such events were essential to confirm the existence of neutral current processes.

Figure 2: Compilation of neutrino and anti-neutrino interaction cross section from a few hundred MeV to a few hundred GeV [11].

σ νN /E ν ≈ 0.67x10 -38 cm 2 /GeV and for anti-neutrinos σ νN /E ν ≈ 0.34x10 -38 cm 2 /GeV . It is interesting to note that those values are 1 million times larger than the cross section computed in 1934 by Bethe and Peierls. The major activities around the neutrinos in the 80's and 90's were essentially the measurement of the weak interaction parameters (up to a few % precision) and the study of the quark structure of the nucleon. To succeed in measuring those interactions in a large energy range it was essential to build high energy intense pure neutrino beams together with massive detectors. The detector technology has evolved in mainly three categories coming with the electronic development and the computer assisted technology development. The first category of detectors included heavy liquid bubble chambers like the Gargamelle chamber shown in Figure 3 able to offer a large target mass, a long interaction length and a precise tracking reconstruction. The second generation of detectors used from the late 80's were essentially calorimetric detectors similar to the CHARM detector shown in Figure 4 with a few 100 tons of target with gaseous or plastic sintillator planes sometimes in alternance with layers of absorber material.

The neutrinos and the Standard Model

In the minimal Standard Model (SM) of particle physics the neutrinos are defined as neutral spin 1/2 fermions subject to Weak interactions only. LEP exeperiments have determined from the width of the Z resonance lineshape that the number of active light neutrino flavours is N ν = 2.9840±0.0082 [START_REF]Precision Electroweak Measurements on the Z Resonance[END_REF]. There is clearly no room for an additional light ν species. In the Standard Model the ν are always Left Handed (LH) and their mass equals 0. Since 1998 there is evidence for flavour changing process in the neutrino sector. This oscillation mechanism implies that ν are massive which can be translated in the first hints for physics beyond the Standard Model. Obviously the SM should be extended to be reconciled with massive ν and the Higgs mechanism.

In order to find a solution, two main approaches exist. One is to add a new particle called Dirac ν. It is minimal extension with a new Dirac mass term. However it is not very satisfactory since the Right Handed (RH) ν interacts with the Higgs too weakly (10 12 times weaker than that of the top) to acquire mass. The second idea is to introduce a different type of particle which could be the Majorana ν. In this case heavy RH neutrinos are created for a brief moment (via the See-Saw mechanism) from LH ν interaction with Higgs. The consequence is that there is no fundamental distinction between matter and anti-matter. The question is not yet solved and more studies are needed to find the key for this issue.

It is clear that many fundamental questions concerning the neutrino sector remain open. Here is a brief list of them with the possible experimental steps needed to answer them.

• What is the absolute neutrino mass scale? This question is fundamental for cosmology and unification scheme of interactions. Possible experimental methods to answer is to measure ν time of flight (ex: Supernova 1987A gave the limit m< 20 eV) or to measure precisely the end point of electron beta decay spectrum. The actual limit obtained is m< 2.5 eV. Another method consists of inferring it from fluctuations of Cosmological Microwave Background (ex: WMAP gives m<0.23 eV). • Are neutrinos their own antiparticle (Majorana neutrinos) or not (Dirac neutrinos)? The answer can be obtained through the search for neutrinoless double beta decay which can also provide an additional clue to the absolute mass scale. • What are the relations between neutrino flavor eigenstates and mass eigenstates? Is there CP violation in the neutrino sector (leptogenesis)?

The study to answer those questions are described in the next section. The idea is to exploit all possible neutrino sources which are the Sun, nuclear reactors, atmospheric showers, beam accelerators of various energies.

Mixing matrix and 3 massive ν oscillation 2.1 Formalism

There are three known neutrino flavor eigenstates ν α = (ν e , ν μ , ν τ ). Since 1998 there is clear evidence of the existence of transitions between the flavor eigenstates suggesting that neutrinos have non-zero masses. This oscillation mechanism can be formulated in a way that the mass eigenstates ν i = (ν 1 , ν 2 , ν 3 ) with masses m i = (m 1 , m 2 , m 3 ) are related to the flavor eigenstates by a 3 × 3 unitary mixing matrix U ν called Maki-Nakagawa-Sakata-Pontecorvo (MNSP) matrix,

|ν α = i (U ν αi ) * |ν i (1)
From this formulation, four numbers are needed to specify all of the matrix elements, namely three mixing angles (θ 12 , θ 23 , θ 13 ) and one complex phase (δ). The parametrisation of the mixing matrix is usually represented following the form proposed by Chau and Keung [13] defined by:

U ν = ⎛ ⎝ c
where c jk ≡ cos θ jk and s jk ≡ sin θ jk . Neutrino oscillations are driven by the splittings between the neutrino mass eigenstates. It is useful to define the differences between the squares of the masses of the mass eigenstates

Δm 2 ij ≡ m 2 i -m 2 j .
With this parametrization the probability that a neutrino of energy E and initial flavor α will "oscillate" into a neutrino of flavor β is given by

P αβ ≡ P (ν α → ν β ) = | ν β | exp(-iHt)|ν α |
2 , which in vacuum gives

P αβ = 3 j=1 U * αj U βj exp(-iE j t) 2 = 3 j=1 3 k=1 U αj U * αk U * βj U βk exp -i Δm 2 kj 2E t ( 3
)
If neutrinos of energy E travel a distance L then a measure of the propagation time t is given by L/E. Nonzero Δm 2 ij will result in neutrino flavor oscillations that have maxima at given values of L/E, and oscillation amplitudes that are determined by the matrix elements U ν αi , and hence by θ 12 , θ 23 , θ 13 , and the CP violation phase δ. The oscillation frequency is governed by the ratio L/E and any experiment dedicated to oscillation studies will be designed to control this ratio as much as possible to infer the neutrino oscillation parameters.

Matter effects-the MSW effect

It is important to understand that the propagation in vacuum is the simplest case. However neutrino sources, like the sun, can be far from the detection point and the neutrinos have to travel also matter densities before reaching the detectors.The induced effect is understood as the Mikheyev, Smirnov, Wolfenstein (MSW) effect [14] which was first confirmed by the solar neutrino observation. The effect is to add an additional term in the hamiltonian for describing the neutrino propagation in matter. The extra term arises because ν e have an extra interaction via W boson exchange with electrons in the Sun or Earth. This is not the case for the two other neutrino flavours. The MSW effect can produce an energy spectrum distortion and flavor regeneration in Earth giving a day-night effect. If observed, an important consequence is that the matter interactions depend on the mass hierarchy defined by the sign of Δm 2 31 . If one includes the matter effects in the 3ν transition probability described above, Eq. 3 becomes much more complicated. As an example here is the probability of transition betweem ν μ and ν e including dominant, subdominant oscillations, matter and CP violation terms:

where the matter densities is embedded in a

= ±2 √ 2G f N e E ν .

Experimental status

Several experiments exploiting all the possible sources of neutrino gave important and coherent results to determine the various oscillation parameters. The determination of Δm 2 31 and θ 23 came initially from the observation of a clear disappearance of the ν μ produced in the atmospheric particle showers. This observation was made by the Kamiokande, Soudan II and MACRO experiments in the 90's. However the first compelling evidence for neutrino oscillation in the atmospheric shower came in 1998 with the measurement by Super-Kamiokande [15] of a dependance of this deficit as a function of the zenith angle which is directly related to the distance the neutrinos are traveling through the Earth. While the ν e were not showing such distortions the preferred hypothesis fitting well all the data is the ν μ → ν τ oscillation origin. Applying this hypothesis the best fit result gives Δm 2 31 = 2.1x10 -3 eV 2 and a maximal mixing with θ 23 = 1. This result was confirmed by the K2K experiment [16] which used a home made neutrino source with the first long baseline neutrino beam in Japan. The beam was sent from KEK laboratory to Super-Kamiokande detector 250 km away. The mean neutrino energy was 1.3 GeV and they observed a slight deficit of neutrino interaction compared to what was expected from the extrapolation of the interactions seen in a near detector without oscillation. The resulting values of Δm 2 31 and θ 23 were fully compatible with the parameter space measured with the atmospheric neutrinos. The determination of Δm 2 21 and θ 12 is coming essentially from the solar and reactor neutrinos. Since more than 30 years experiments ( 37 Cl, Gallex/GNO, SAGE, SuperK etc...) got a puzzling fact with the observation of a deficit of more than 30-50% of the solar neutrino arriving on the earth compared to the prediction given by the standard model of the sun. The real breakthrough arrived with the SNO detector which observed not only the same deficit but unlike the others was sensitive to all neutrino flavours and not only to ν e interactions. Thanks to the use of heavy water, it was possible to measure the neutral current process via the breaking of the tritium nucleus liberating a neutron which gives a neutron capture signal in the detector. From their measurements shown in Fig. 6 they found that the 'all flavour' rate is compatible with the model of the sun and that the flux of ν e is compatible with an oscillation hypothesis favouring also a large mixing angle [17]. The solar neutrino solution was tested and confirmed by the Kamland experiment [19] which performed the first observation of neutrino oscillations from reactor sources by measuring the energy spectrum of neutrinos produced in about 55 nuclear reactors in Japan. The mean distance for the neutrino to reach the Kamioka mine were the detector is located is about 180 km. The result of the KamLAND measurement, shown in Figure 7, exhibits the expected oscillatory behaviour and constitutes compelling evidence for neutrino oscillations. By combining all solar and Kamland results the oscillation hypothesis fit gives the following values for the second mass difference and the second angle: Δm 2 21 = (7.59 ± 0.21)x10 -5 eV 2 and a large mixing with tan 2 θ 12 = 0.47 +0.06 -0.05 . The third mixing angle θ 13 is still not measured. There exist only an experimental constraint from the CHOOZ reactor experiment [20] giving sin 2 (2θ 13 ) < 0.20 at 90% CL. Table 1 summarises the present knowledge of the oscillation parameters exploiting all the possible data available. The numbers shown are obtained from a global 3-ν analysis performed in summer 2008 [21].

Running accelerator oscillation projects

The goals of the presently running and future experiments are to go deeper in the understanding of the MNSP mixing matrix and oscillation mechanism. The main items are: more precise measurements of Δm 2 31 and θ 23 , the quest of θ 13 , studies of the mass hierarchy with the sign of Δm 2 31 through matter effects and study possible CP violation in leptonic sector by comparing the transition probabilities of neutrino with the ones of anti-neutrinos. All this list represents a lenghty experimental and theoretical program with several challenging steps. The current phase corresponds to the first long baseline generation using conventional muon neutrino beam. Two projects are running. The Minos/NUMI project in the USA and the OPERA/CNGS european project. They have both a baseline of about 730 km. Minos is performing detailed studies [22] of ν μ disappearance to improve the parameter precisions while OPERA is designed to provide a direct proof of the existence of ν μ → ν τ transition looking at direct ν τ appearance [9,23]. Both projects aim to measure also θ 13 looking at the appeance channel ν μ → ν e . First results on this subject from Minos have been presented during 2009 winter conferences [START_REF] Sanchez | Initial results on ν e appearance in MINOS, Talk given at the Rencontres de Moriond[END_REF]. Figure 8 shows the ratio of the MINOS Far Detector data energy spectrum to the energy spectrum expected in the absence of neutrino disappearance where a clear distortion and deficit is visible at low energies compatible with the oscillation hypothesis. Figure 9 shows the reconstruction in emulsions of a neutrino interaction vertex recorded in OPERA during the 2008 run. The vertex tracks show a kink on one of them similar to what is expected from a short lived particle like a charm meson or a tau particle decaying in the target emulsions.

Figure 8: Ratio of the MINOS Far Detector data energy spectrum to the energy spectrum expected in the absence of neutrino disappearance (black points) [22].

Figure 9: Reconstruction of a neutrino interaction vertex in a emulsion target element of OPERA. One track shows a clear kink similar to the ones expected from tau or charm particle decays.

Future experiments

The second experimental phase corresponds to the quest of θ 13 which remains the missing mixing angle. For this search two approaches are confronted. As it was shown previously the transition ν μ → ν e combines effects from θ 13 and the CP violation phase δ. The two parameters are correlated in the ν e appearance channel. In addtion, matter effects could modify the transition probabilities depending on the mass hierarchy. Each experiment should carefully take into account all those effects. A positive result with accelerator experiment will give a bi-parameter contour solution while a disappearance experiment like in reactor experiment will give access to the θ 13 value.

Long Baseline Experiments

There are two projects going on. The T2K experiment [START_REF]Terri and the T2K Collaboration[END_REF] which is well advanced using a ν μ beam from Tokai to Super-Kamiokande with a baseline of 295 km. The mean neutrino energy is 0.7 GeV with an ν e contamination of about 0.4%. The second project called NOνA [START_REF]The NOνA Experiment at Fermilab (E929[END_REF] is under study and is using a baseline of 810 km with mean ν μ energy of about 2.2 GeV. The very long distance will give possibility to study matter effects on the neutrino oscillation rate. The neutrino beams are conventional beams but with an increased power going from 0.4-0.8 MW exploiting also the 'off-axis' technique which allows a much narrower neutrino energy spectrum with reduced background contamination as well as a better control of the L/E ratio to maximise the ν e appearance. Details about the T2K project and the advancement can be found in Ref. [START_REF]Terri and the T2K Collaboration[END_REF]. The beam has been already commissioned in May 2009 and a gradual beam power increase is expected from 0.2 MW to 1.0 MW in 4 years. There are two near detectors at 280m from the neutrino source, one at 0 degree angle and one at 2.0 o . The Super-Kamiokande detector will be used as the far detector and is ready for data taking. A full running period should start at the end of this year. The NOνA experiment will search for ν μ → ν e oscillations in the existing NuMI neutrino beam using a 15 kiloton liquid scintillator detector. Funding has been recently approved and construction is about to start on the far detector site.

Reactor experiments

The alternative method is to measure the survival rate of ν e close to nuclear reactor like it was done with the CHOOZ experiment. Up to second order in sin 2θ 13 and α = the survival probability can be expressed as:

P νe→νe 1 -sin 2 2θ 13 sin 2 (Δm 2 31 L/4E) + α 2 (Δm 2 31 L/4E) 2 cos 4 θ 13 sin 2 2θ 12 , (4)
The last term of this expression can be easily neglected if the ratio L/E is chosen close to the atmospheric maximum. Reactor experiments thus provide a clean measurement of the mixing angle θ 13 , free from contamination coming from matter effects and other parameter correlations or degeneracies. They are essentially dominated by statistical and systematic errors. In order to reduce the systematic uncertainties the principle of the new generation of reactor experiment is to use two detectors. One near at about 100 to 200 m from the reactor core and one far at about 1 to 2km. This garantees that the L/E ratio is close to the atmospheric maximum value. The comparison of the measured νe between the two sites will cancel part of the systematical uncertainties from the reactor flux and cross sections. The target mass of the two detectors vary from 8 tons to about 100 tons. The real challenge is to be able to reduce the relative normalisation uncertainty below 1%. There are several projects under way. The Double-Chooz experiment is the most advanced one [START_REF] Chooz | A Search for the Neutrino Mixing Angle θ 13[END_REF]. The far (1km) detector site is constructed and the 11.2 tons detector should start taking data beginning of 2010. The near site (400m) is under construction and will be completed to host a second identical detector in 2011. The full setup should take data in 2011. Figure 10 shows the θ 13 sensitivity limit as a function of the year which can be obtained by Double-Chooz running in two phases. Figure 11: World perspectives for θ 13 sensitivity limit [START_REF] Mezzetto | Workshop on "Neutrino Telescopes[END_REF].

The second main project is called Daya Bay [START_REF]Daya Bay proposal[END_REF] and it can be considered as a second generation experiment which is more amibitious since it aims to reduce the systematic uncertainty by a factor 2 and be able to give a limit on sin 2 2θ 13 < 0.01 at 90% CL. The principle will be to use 6 to 10 mobile detectors which can be interchanged to compare their efficiencies. The project may start after 2011.

Figure 11 shows the world limit which can be achieved as a function of time by combining all the experiments which aim to contribute to θ 13 measurements. The Double-Chooz, T2K and Daya Bay projects will allow to reduce the limit on sin 2 2θ 13 by more than one order of magnitude in a decade if the angle stays too small to be detected.

Conclusions and perspectives

The reactor and accelerator experiments are complementary; At the 2016 horizon there should be confirmation of the transition ν μ → ν τ , more precise measurements of θ 23 and Δm 2 31 and more constraints on θ 13 if not a discovery. And why not maybe the first indication for mass hierarchy choice. It is the first step to pinpoint the true nature of the leptonic flavour transformation. It will not allow to see CP violation but it will help in defining what should be the path to follow beyond 2015. This next step will be the quest for δ CP and precision measurements of the neutrino parameters. The technology and experiments foreseen are various. They include more powerful superbeams of second generation together with BetaBeam coupled to large volume detectors like Megatonne Water cherenkov, liquid argon, liquid scintillator detectors and a facility called Neutrino Factory.

Neutrino physics is a very active field. Since 10 years several new results changed our view of the field and comforted us to revise our current knowledge within the Standard Model. A lot of experimental and theoretical challenges are in front of us and worth being pursued.

Figure 3 :

 3 Figure 3: Gargamelle bubble chamber at CERN.Figure 4: CHARM experiment at CERN.

Figure 4 :

 4 Figure 3: Gargamelle bubble chamber at CERN.Figure 4: CHARM experiment at CERN. In more recent times another type of detectors looking at neutrinos appeared. They are of spherical or cylindrical shapes and correspond to large containers of liquid surrounded by large numbers of photodetectors. The principle is to track the light coming from Cherenkov emission of fast particles in the liquid medium or light coming from annihilation or scintillation processes. Three types of such detectors are shown in Figure 5. The left picture shows the internal tank of the Super-Kamiokande detector in Japan filled with 22 ktons of water. The picture in the middle shows the SNO detector in Canada which contained about 1 kton of heavy water. The picture on the right shows the inner vessel of the Borexino detector in Italy which contains 300 tons of pseudocumen liquid scintillator. All those detectors offer large volumes for interaction and necessitate a large number of photodetectors going from 1000 to 10000.

Figure 5 :

 5 Figure 5: a) Super-Kamiokande water Cherenkov detector; b) SNO heavy water detector; c) BOREXINO liquid scintillator detector.

Figure 6 :

 6 Figure 6: Flux of 8 B solar neutrinos that are μ or τ flavor vs flux of electron neutrinos deduced from the three neutrino reactions in SNO [18].

Figure 7 :

 7 Figure 7: Ratio of the background subtracted anti-neutrino spectrum to the expectation for nooscillation as a function of L/E. L is the effective baseline taken as a flux-weighted average (L=180 km) [19].

Figure 10 :

 10 Figure 10: Double CHOOZ prospects for θ 13 sensitivity limit.

  13 c 12 c 13 s 12 s 13 e -iδ -c 23 s 12 -s 13 s 23 c 12 e iδ c 23 c 12 -s 13 s 23 s 12 e iδ c 13 s 23 s 23 s 12 -s 13 c 23 c 12 e iδ -s 23 c 12 -s 13 c 23 s 12 e iδ c 13 c 23

⎞ ⎠

Table 1 :

 1 Global 3ν oscillation analysis (2008) extracted from[21]: best-fit values and allowed n σ ranges for the mass-mixing parameters.Parameter δm 2 /10 -5 eV 2 sin 2 θ 12 sin 2 θ 13 sin 2 θ 23 Δm 2 /10 -3 eV 2

	Best fit	7.67	0.312	0.016	0.466	2.39
	1σ range	7.48 -7.83	0.294 -0.331 0.006 -0.026 0.408 -0.539	2.31 -2.50
	2σ range	7.31 -8.01	0.278 -0.352	< 0.036	0.366 -0.602	2.19 -2.66
	3σ range	7.14 -8.19	0.263 -0.375	< 0.046	0.331 -0.644	2.06 -2.81


Raouf Abd

Chouikha
email: chouikha@math.univ-paris13.fr.

FUNCTIONS RELATED TO JACOBI THETA FUNCTIONS AND APPLICATIONS I

Keywords: innite product expansions, theta functions, elliptic functions, trigonometric expansions

In this paper we highlight trigonometric holomorphic functions of two variables f j (v, τ ), j = 1, 2, 3, 4 where Imτ > 0, and | Imv |< 1 2 Imτ . These functions are related to the four theta functions :

We propose to describe some properties of these functions. In particular the quotient v+1,τ ) which makes it possible to deduce remarkable identities. Various aspects will be considered, expressions in the form of innite products and their convergence, their development in Fourier series, the convergence of their derivatives, and so on. We of course nd again the classical expansions of theta functions in trigonometric forms or innite products.

Introduction

Throughout this paper we assume Imτ > 0 and take q = e iπτ . Jacobi theta functions for j = 1, 2, 3, 4 are dened as ( [START_REF] Erdelyi | Higher transcendental functions Vol. I-II[END_REF] or [START_REF] Magnus | Formulas and Theorems for the Special Functions of Mathematical Physics Bd[END_REF])

θ 1 (v, τ ) = 2 n≥0
(-1) n q (n+ 1 2 ) 2 sin((2n + 1)πv) θ 2 (v, τ ) = 2 n≥0 q (n+ 1 2 ) 2 cos((2n + 1)πv)

θ 3 (v, τ ) = 1 + 2 n≥1 q n 2 cos(2nπv) θ 4 (v, τ ) = 1 + 2 n≥1 (-1) n q n 2 cos(2nπv)
These four theta functions can be extended to complex values for v and q such that | q |< 1. All four theta functions are entire and periodic functions of v, [START_REF] Appell | Fonctions elliptiques et applications Gauthiers-Villard[END_REF], [START_REF] Chouikha | On Properties of Elliptic Jacobi Functions and Applications[END_REF].

Using the Jacobi triple product identity, we can nd the innite product representation ( [START_REF] Erdelyi | Higher transcendental functions Vol. I-II[END_REF], [START_REF] Whittaker | A course of Modern Analysis Cambridge[END_REF]) for θ j (v, τ ), namely:

θ 1 (v, τ ) = 2q 1 4 sin v n≥1
(1 -q 2n )(1 -q 2n e 2iπv )(1 -q 2n e -2iπv ) = 2q 1 4 sin v(q 2 ; q 2 ) ∞ (q 2 e 2iπv ; q 2 ) ∞ (q 2 e -2iπv ; q 2 ) ∞ θ 2 (v, τ ) = n≥1 (1 -q 2n )(1 + q 2n e 2iπv )(1 + q 2n e -2iπv ) = 2q 1 4 cos v(q 2 ; q 2 ) ∞ (-q 2 e 2iπv ; q 2 ) ∞ (-q 2 e -2iπv ; q 2 ) ∞ θ 3 (v, τ ) = n≥1 (1 -q 2n )(1 + q 2n-1 e 2iπv )(1 + q 2n-1 e -2iπv ) = (q 2 ; q 2 ) ∞ (-qe 2iπv ; q 2 ) ∞ (-qe -2iπv ; q 2 ) ∞ θ 4 (v, τ ) = n≥1 (1 -q 2n )(1 -q 2n-1 e 2iπv )(1 -q 2n-1 e -2iπv ) = (q 2 ; q 2 ) ∞ (qe 2iπv ; q 2 ) ∞ (qe -2iπv ; q 2 ) ∞ where q = e iπτ , v ∈ C, τ ∈ H + , and (a; b) ∞ = n≥0 (1 -ab n ). Trigonometric expansions of theta functions can be proved by the methods of residue calculus as described by [5, p.358] and [START_REF] Magnus | Formulas and Theorems for the Special Functions of Mathematical Physics Bd[END_REF]. It is known that from these products we may deduce the Fourier expansions of log(θ j (v, τ ) and θ j θj . The four Jacobi theta functions are naturally related. Starting from one of them we may obtain the other three by simple calculation. Dierentiating θ 1 (v, τ ) with respect to v and putting v → 0 yields θ 1 (0, τ ) = 2q 1 4 (q 2 ; q 2 ) 3 ∞ .

Setting z = 0 in the other three expressions we obtain θ 2 (0, τ ) = 2q 1 4 (q 2 ; q 2 ) ∞ (-q 2 ; q 2 ) 2 ∞ , θ 3 (0, τ ) = (q 2 ; q 2 ) ∞ (-q; q 2 ) 2 ∞ , θ 4 (0, τ ) = (q 2 ; q 2 ) ∞ (q; q 2 ) 2 ∞ .

The aim of this paper is to represent θ j (v, τ ) as product

f j (v, τ ) f j (v + 1, τ )
and to give a complete description of these functions f j (v, τ ). We will see in the sequel that not only the products but also the quotients fj (v,τ ) fj (v+1,τ ) seem play a particular role. In particular, we derive their representation as innite products, their Fourier series expansions, as well as for log(f j (v, τ )) and for f j fj (v, τ ). Some other signicant properties and equalities will be derived, as the following identity valid for any integer n

1 -cos πv 1 + cos πv n k≥0 e iπv -q k e iπv + q k e -iπv + q k e -iπv -q k = k≥0 e iπv -q k-n e iπv + q k-n e -iπv + q k+n e -iπv -q k+n .
Moreover, using these innite products we prove

θ 4 (2 n+1 v, 2 n+1 τ ) θ 4 (0, 2 n+1 τ ) = θ 4 (v, τ ) θ 4 (0, τ ) p=n p=0 θ 3 (2 p v, 2 p τ ) θ 3 (0, 2 p τ ) , θ 1 (2 n+1 v, 2 n+1 τ ) θ 1 (0, 2 n+1 τ ) = θ 1 (v, τ ) θ 1 (0, τ ) p=n p=0 θ 2 (2 p v, 2 p τ ) θ 2 (0, 2 p τ ) ,
generalizing the known cases n = 0, n = 1 see [7,p.23].

Functions defined by an infinite product

Consider the complex functions of two variables f j , j = 1, 2, 3, 4 dened by the innite products

f 4 (v, τ ) = k≥0 1 - sin πv sin(k + 1 2 )πτ , f 3 (v, τ ) = k≥0 1 - (-1) k sin πv cos(k + 1 2 )πτ , f 1 (v, τ ) = k≥1 1 - sin πv sin kπτ , f 2 (v, τ ) = k≥1 1 - (-1) k sin πv cos kπτ
where v ∈ C and τ belongs to the half plane ( τ : Im(τ ) > 0).

Remarks: These products may also be rewritten under dierent forms. For example

f 4 (v, τ ) = 2 k≥0 cos ( k 2 + 1 4 )πτ + π v 2 sin ( k 2 + 1 4 )πτ -π v 2 sin(k + 1 2 )πτ = k≥0 cos ( k 2 + 1 4 )πτ + π v 2 cos( k 2 + 1 4 )πτ sin ( k 2 + 1 4 )πτ -π v 2 sin( k 2 + 1 4 )πτ . f 1 (v, τ ) = k≥0 cos ( k 2 )πτ + π v 2 cos( k 2 )πτ sin ( k 2 )πτ -π v 2 sin( k 2 )πτ .
The following ensures the convergence of these products.

Proposition 2-1 Consider the following sequences of functions

(U k,4 )(v, τ ) = sin πv sin(k + 1 2 )πτ , (U k,3 )(v, τ ) = sin πv cos(k + 1 2 )πτ , (U k,1 )(v, τ ) = sin πv sin kπτ , (U k,2 )(v, τ ) = sin πv cos kπτ .
For v such that 0 <| Imv |< 1 2 Imτ, the sums k≥0 | U k,j (v, τ ) |, j = 1, 2, 3, 4 converges uniformly. Equivalently the innite products k≥0 (1 -U k,j (v, τ )) are uniformly convergent.

Proof

We demonstrate that the innite products are absolutely and uniformly convergent by application of the M-test (see, e.g., [START_REF] Copson | Theory of Functions of a Complex Variable[END_REF] E.T. Copson, Theory of Functions of a Complex Variable, pp. 104-6) which states that an innite product (1 + g k (v)) converges uniformly and absolutely in a bounded closed region if :

g k (v) are such that | g k (v) |≤ M k and k M k is convergent. Indeed, write τ = τ 1 + iτ 2 where τ 1 ∈ , τ 2 > 0. Then | sin πv sin(k + 1 2 )πτ | 2 ≤ | sinh πτ2 2 sin(k + 1 2 )π(τ 1 + iτ 2 ) | 2 = (sinh πτ2 2 ) 2 [(sin(k + 1 2 )π(τ 1 )) 2 + (sinh(k + 1 2 )π(τ 2 )) 2 ] ≤ sinh πτ2 2 sinh(k + 1 2 )πτ 2 2 = M 2 k,4 < 1
Since τ 2 > 0 it is easy to see that the serie

k M k,4 = sinh πτ 2 2 k 1 sinh(k + 1 2 )πτ 2 converges.
By the same way, applying again the M-test for the other U k,j (v, τ ) we get

| sin πv cos(k + 1 2 )πτ |≤ M k,3 | sin πv sin kπτ |≤ M k,1 , | sin πv cos kπτ |≤ M k,2
Moreover, the series k M k,j , j = 1, 2, 3 converge.

Replacing v by v + 1, v + 2 and τ by τ + 1, ... we then obtain the following Proposition 2-2 The above function f j (v, τ ) veries the following properties

for Imτ > 0, Imv < 1 2 Imτ
1 -For j = 3, 4, one has

f j (v + 2, τ ) = f j (v, τ ) = f j (v, τ + 4), f j (v + 1, τ ) = f j (-v, τ ) = f j (v, τ + 2), f j (v, τ + 3) = f j (v + 1, τ + 1).
2 -For j = 1, 2 one has

f j (v + 2, τ ) = f j (v, τ ) = f j (v, τ + 2) f j (v + 1, τ + 2) = f j (-v, τ + 1),
Moreover,

f 4 (v, τ + 1) = f 3 (v, τ ), f 1 (v, τ + 1) = f 2 (v, τ ). f 1 (v, τ + 1 2 ) = f 4 (v, τ ), f 2 (v, τ + 1 2 ) = f 3 (v, τ ), f 4 (v + τ 2 , τ ) = f 1 (v, τ ) (e iπv -1) k≥1 1 -q 2k 1 -q 2k+1 = f 1 (v, τ ) (e iπv -1) f 4 (0, τ ) f 1 (0, τ ) .
Proof These identities are direct consequence of denitions of f j (v, τ ) and from the following relations

f 4 (v, τ + 1) = k≥0 1 -(-1) k sin πv cos(k + 1 2 )πτ = f 3 (v, τ ), f 1 (v, τ + 1) = k≥1 1 -(-1) k sin πv cos kπτ = f 2 (v, τ ).
The following gives a triple product identity for f j (v, τ )

Proposition 2-3 The functions f j (v, τ ) may also be expressed

f 4 (v, τ ) = k≥0 (e -iπv + q k+ 1 2 )(e iπv -q k+ 1 2 ) 1 -q 2k+1 = k≥0 (e -iπv +q k+ 1 2 )(e iπv -q k+ 1 2 )(1+q k ) f 1 (v, τ ) = k≥1 (e -iπv + q k )(e iπv -q k ) 1 -q 2k , f 4 (v+ τ 2 , τ ) = 1 e iπv -1 k≥0 1 -q 2k 1 -q 2k+1 f 1 (v, τ ) f 4 (v, τ ) f 4 (v + 1, τ ) = k≥0 e iπv -q k+ 1 2 e iπv + q k+ 1 2 e -iπv + q k+ 1 2 e -iπv -q k+ 1 2 , f 1 (v, τ ) f 1 (v + 1, τ ) = k≥1 e iπv -q k e iπv + q k e -iπv + q k e -iπv -q k , f 4 (v + τ, τ ) f 4 (v, τ ) = e iπv -q -1 2 e -iπv -q 1 2 , f 1 (v + τ, τ ) f 1 (v, τ ) = e iπv -q -1 e -iπv -q .
Theorem 2-4 For an integer n one has

f 4 (v + nτ, τ ) = k≥0 (e -iπv + q k+n+ 1 2 )(e iπv -q k-n+ 1 2 ) 1 -q 2k+1 = k≥0 e -iπv + q k+n+ 1 2 )(e iπv -q k-n+ 1 2 )(1 + q k ). f 1 (v + nτ, τ ) = k≥1 (e -iπv + q k+n )(e iπv -q k-n ) 1 -q 2k .
In particular, the following identities hold

1 -cos πv 1 + cos πv n k≥0 e iπv -q k e iπv + q k e -iπv + q k e -iπv -q k = k≥0 e iπv -q k-n e iπv + q k-n e -iπv + q k+n e -iπv -q k+n , 1 -cos π(v + τ ) 1 + cos π(v + τ ) n k≥1 e iπv -q k e iπv + q k e -iπv + q k e -iπv -q k = k≥1 e iπv -q k-n e iπv + q k-n e -iπv + q k+n e -iπv -q k+n .
Proof Incrementing v by v + τ one has

f 4 (v + τ, τ ) f 4 (v + 1 + τ, τ ) = k≥0 e iπv -q k-1 2 e iπv + q k-1 2 e -iπv + q k+ 3 2 e -iπv -q k+ 3 2 = e iπv -q -1 2 e iπv + q -1 2 k≥0 e iπv -q k+ 1 2 e iπv + q k+ 1 2 e -iπv -q 1 2 e -iπv + q 1 2 k≥0 e -iπv -q k+ 1 2 e -iπv + q k+ 1 2 = e iπv -q -1 2 e iπv + q -1 2 e -iπv -q 1 2 e -iπv + q 1 2 f 4 (v, τ ) f 4 (v + 1, τ ) .
Then we deduce for any integer n

f 4 (v + nτ, τ ) f 4 (v + 1 + nτ, τ ) = e iπv -q -1 2 e iπv + q -1 2 n e -iπv -q 1 2 e -iπv + q 1 2 n k≥0 e iπv -q k+ 1 2 e iπv + q k+ 1 2 e -iπv + q k+ 1 2 e -iπv -q k+ 1 2 = 1 -cos π(v + τ 2 ) 1 + cos π(v + τ 2 ) n k≥0 e iπv -q k+ 1 2 e iπv + q k+ 1 2 e -iπv + q k+ 1 2 e -iπv -q k+ 1 2 = k≥0 e iπv -q k-n+ 1 2 e iπv + q k-n+ 1 2 e -iπv + q k+n+ 1 2 e -iπv -q k+n+ 1 2 .
Equivalently, replacing v by v -τ 2 it yields

1 -cos πv 1 + cos πv n k≥0
e iπv -q k e iπv + q k e -iπv + q k e -iπv -q k = k≥0 e iπv -q k-n e iπv + q k-n e -iπv + q k+n e -iπv -q k+n .

By the same way one obtains

f 1 (v + nτ, τ ) f 1 (v + 1 + nτ, τ ) = e iπv -q -1 e iπv + q -1 n e -iπv -q e -iπv + q n k≥1 e iπv -q k e iπv + q k e -iπv + q k e -iπv -q k = 1 -cos π(v + τ ) 1 + cos π(v + τ ) n k≥1 e iπv -q k e iπv + q k e -iπv + q k e -iπv -q k = k≥1 e iπv -q k-n e iπv + q k-n e -iπv + q k+n e -iπv -q k+n .
The following yields trigonometric expansions of log (f j (v, τ )) , j = 1, 2, 3, 4

Proposition 2-5 The logarithmic expression log (f j (v, τ )) may be expressed as

log (f 4 (v, τ )) = k≥0 1 -e -iπv q k+ 1 2 1 -q 2k+1 = - p≥1 k≥0 1 p sin πv (sin(k + 1 2 )πτ ) p , log (f 3 (v, τ )) = - p≥1 k≥0 1 p sin πu (cos(k + 1 2 )πτ ) p log (f 1 (v, τ )) = - p≥1 k≥1 1 p sin πv (sin kπτ ) p , log (f 2 (v, τ )) = - p≥1 k≥1 1 p sin πv (cos kπτ ) p Proof Indeed, since we have k≥0 (1 -U k ) = k≥0 exp(log(1 -U k )) = exp   k≥0 log(1 -U k )   then f 4 (v, τ ) = exp   k≥0 log 1 - sin πv sin(k + 1 2 )πτ   log (f 4 (-v, τ )) =   - p≥1 k≥0 (-1) p p sin πv (sin(k + 1 2 )πτ ) p   .
On the other hand, since as we have seen by Proposition 1 :

sinh πτ 2 2 sinh(k+ 1 2 )πτ2 2 = M 2 k,4 < 1 then the following series is well dened log (f 4 (v, τ )) = - n≥1 k≥0 1 2n sin πv (sin(k + 1 2 )πτ ) 2n - n≥0 k≥0 1 2n + 1 sin πv (sin(k + 1 2 )πτ ) 2n+1 .
By the same way one obtains the other series expansions for log (f j (v, τ )) , j = 1, 2, 3.

In the next result we derive Fourier expansions of log (f j (v, τ )) , j = 1, 2, 3, 4 as well as for its derivative Theorem 2-6 A trigonometric expansion of the function log (f

4 (v, τ )) is log (f 4 (v, τ )) = n≥1 a n (τ ) cos(2nπv) + m≥1 b m (τ ) sin((2m + 1)πv)
where

a n (τ ) = q n n(1 -q 2n ) ; b m (τ ) = (-1) m q 2m+1 (2m + 1)(1 -q 4m+2 )
Proof Indeed, It suces to express cos(2nα) and sin((2m + 1)πv) in terms of sin α and to identify the coecients of the series

cos(2nα) = (-1) n 2 (2 sin α) 2n + n-1 k=0 (-1) n+k+1 2n k + 1 2n -k + 1 k (2 sin α) 2n-2k-2 = n 0≤p≤n (-1) p (n + p -1)! (2p)!(n -p)! (2 sin α) 2p = n p=0 b 2p,n (sin α) 2p , where b 2n,n = (-1) n 2 2n-1 ; b 0,n = 1; and b 2p,n = (-1) p 2 2p n n -p n + p -1 n -p -1 for n = p.
By the same way we rewrite

sin((2m + 1)πv) = (-1) n 2 (2 sin α) 2n-1 + n-1 k=0 (-1) n+k 2n k 2n -k k (2 sin α) 2n-2k-1 = n 0≤p≤n (-1) p (n + p)! (2p)!(n -p)! (2 sin α) 2p-1 .
After developing and identifying we nd the coecients of the Fourier expansions.

Proposition 2-7 A trigonometric expansion of the functions

d dv log (f j (v, τ )) , j = 1, 2, 3, 4 is 1 f 4 ∂f 4 ∂v (v, τ ) = k≥0 -π cos πv sin(k + 1 2 )πτ -sin πv , 1 f 3 ∂f 3 ∂v (v, τ ) = k≥0 -π cos πv cos(k + 1 2 )πτ -sin πv 1 f 1 ∂f 1 ∂u (u, τ ) = k≥0 -π cos πv sin(kπτ ) -sin πv , 1 f 2 ∂f 2 ∂v (v, τ ) = k≥0
-π cos πv cos(kπτ ) -sin πv f 3 and f 4 is dened in the "strip" | Imv |< 1 2 Imτ, while f 1 and f 2 is dened in the "strip" | Imv |< Imτ.

Link with Jacobi theta functions

Consider the Ramanujan theta function

φ(a, b) = k∈Z a k(k+1) 2 b k(k-1) 2 φ(a, b) = (-a, ab) ∞ (-b, ab) ∞ (ab, ab) ∞ with | ab |< 1.
We denote here (α, β) ∞ = i≥1 (1 -αβ i ). When a = -qe 2iπv , b = -qe -2iπv this function is related for example to the fourth theta function

θ 4 (v, q) = φ(-qe 2iπv , -qe -2iπv ).
From the knowledge of the zeros of θ 4 (v, q) it is possible to obtain innite products representing this function.

3.1. Link with θ 4 (v, τ ). There is a natural relation with θ 4 (v, τ )

Theorem 3-1 Let θ 4 (v, τ ) the fourth Jacobi theta function. Then we have

θ 4 (v, τ ) θ 4 (0, τ ) = f 4 (v, τ ) f 4 (-v, τ ) = f 4 (v, τ ) f 4 (v + 1, τ )
where f 4 and its logarithmic derivative satisfy the innite product

f 4 (v, τ ) = k≥0 1 - sin πv sin(k + 1 2 )πτ 1 θ 4 ∂θ 4 ∂v (v, τ ) = 1 f 4 ∂f 4 ∂v (v, τ ) + 1 f 4 ∂f 4 ∂v (-v, τ ). θ 4 is dened in the "strip" | Imv |< 1 2 Imτ.
Proof of Theorem 3-1 Write

log (f 4 (v, τ )) = 1 2 log θ 4 (v, τ ) θ 4 (0, τ ) - n≥1 k≥0 1 2n + 1 sin πv (sin(k + 1 2 )πτ ) 2n+1 .
Recall that ([5 p.358])

1 θ 4 ∂θ 4 ∂v (v, τ ) = 4π n≥1 q n 1 -q 2n sin 2nπv = -2π n≥1 sin 2nπv sin nπτ = -π k≥0 sin 2πv sin(k + 1 2 )πτ 2 -(sin πv) 2
On the other hand since θ4(v,τ ) θ4(0,τ

) = f 4 (v, τ ) f 4 (-v, τ ) then 1 θ 4 ∂θ 4 ∂v (v, τ ) = -π k≥0 cos πv sin(k + 1 2 πτ ) -sin πv + π k≥0 cos πv sin(k + 1 2 πτ ) + sin πv = 1 f 4 ∂f 4 ∂v (v, τ ) + 1 f 4 ∂f 4 ∂v (-v, τ ).
To state the convergence of 1 such that the series n g n converges on any compact set of Ω. Notice G the product of (1 + g n ) on Ω. Then the series of meromorphic functions n g n 1+gn is normally convergent to G G .

Link with other theta functions. Recall the others functions

f i (v, τ ), i = 1, 2, 3. f 3 (v, τ ) = k≥0 1 - (-1) k sin πv cos(k + 1 2 )πτ , f 1 (v, τ ) = k≥1 1 - sin πv sin kπτ , f 2 (v, τ ) = k≥1 1 - (-1) k sin πv cos kπτ .
Notice also the following relations between them

f 3 (v, τ ) = f 4 (v, τ + 1) = k≥0 1 - (-1) k sin πv cos(k + 1 2 )πτ f 2 (v, τ ) = f 1 (v, τ + 1) = k≥1 1 - (-1) k sin πv cos kπτ .
Theorem 3-4 The other theta functions may be expressed

θ 3 (v, τ ) = θ 3 (0, τ ) f 3 (v, τ ) f 3 (v + 1, τ ), θ 1 (v, τ ) = (π sin(πv) θ 1 (0, τ ) f 1 (v, τ ) f 1 (v + 1, τ ), θ 2 (v, τ ) = cos(πv) θ 2 (0, τ ) f 2 (v, τ ) f 2 (v + 1, τ ).
Moreover,

1 θ j ∂θ j ∂v (v, τ ) = 1 f j ∂f j ∂v (v, τ ) + 1 f j ∂f j ∂v (-v, τ ), j = 1, 2, 3.
Here θ 3 is dened in the "strip" | Imv |< 1 2 Imτ, while θ 1 and θ 2 is dened in the "strip" | Imv |< Imτ.

We then deduce expression of Jacobi theta functions as innite products Corollary 3-5 Let q = e iπτ , | q |< 1. The functions θ j , j = 1, 2, 3, 4 may also be expressed as innite products

θ 4 (v, τ ) θ 4 (0, τ ) = k≥0 1 - sin πv sin(k + 1 2 )πτ 2 = k≥0 cos 2πv -cos(2k + 1)πτ 1 -cos(2k + 1)πτ (1)
θ 3 (v, τ ) θ 3 (0, τ ) = k≥0 1 - sin πv cos(k + 1 2 )πτ 2 = k≥0 cos 2πv + cos(2k + 1)πτ 1 + cos(2k + 1)πτ (2) θ 1 (v, τ ) (π sin πv) θ 1 (0, τ ) = k≥1 1 - sin πv sin kπτ 2 = k≥1 cos 2πv -cos 2kπτ 1 -cos 2kπτ
(3) Proof Indeed, we will only prove for θ 4 (v, τ ) the others will be deduced by the same way. Starting from [START_REF] Lawden | Elliptic functions and applications[END_REF] and notice that

θ 2 (v, τ ) (cos πv) θ 2 (0, τ ) = k≥1 1 - sin πv cos kπτ 2 = k≥1 cos 2πv + cos 2kπτ 1 + cos 2kπτ (4)
log θ 4 (v, τ ) θ 4 (0, τ ) =   - p≥1 k≥0 1 p sin πv (sin(k + 1 2 )πτ ) 2p   = -2 k≥0 p≥1 1 2p (X k ) 2p
where

X k = sin πv (sin(k + 1 2 )πτ )
On the other hand we may calculate

-2 p≥1 1 2p (X k ) 2p = log(1 -(X k ) 2 ).
it follows that

log θ 4 (v, τ ) θ 4 (0, τ ) = k≥0 log(1 -(X k ) 2 ) = k≥0 log 1 - sin πv (sin(k + 1 2 )πτ ) 2
Therefore we obtain the innite product of θ 4

θ 4 (v, τ ) = θ 4 (0, τ ) k≥0 1 - sin πv sin(k + 1 2 )πτ 2 .
Similarly, the other expressions of theta functions θ j (v, τ ), j = 1, 2, 3 are obtained by the same manner (details omitted).

Remarks 3-5 Observe that we may derive

θ 4 ( 1 2 , τ ) θ 4 (0, τ ) = θ 3 (0, τ ) θ 4 (0, τ ) = k≥0 1 - 1 sin(k + 1 2 )πτ 2 , θ 3 ( 1 2 , τ ) θ 3 (0, τ ) = θ 4 (0, τ ) θ 3 (0, τ ) = k≥0 1 - 1 cos(k + 1 2 )πτ 2 .
Thus,

k≥0 1 - 1 sin(k + 1 2 )πτ 2 1 - 1 cos(k + 1 2 )πτ 2 = 1.
By the same way we have also

k≥1 1 - 1 sin kπτ 2 1 - 1 cos(kπτ 2 = 1.
Corollary 3-8 Let q = e iπτ , | q |< 1. The functions θ j , j = 1, 2, 3, 4 may also be expressed as From Corollary 3-5 we also deduce the following Corollary 3-9 Let q = e iπτ , | q |< 1. The logarithmic derivative of functions θ j , j = 1, 2, 3, 4 with respect to v may be expressed Proof Indeed, by logarithmic dierentiation (11) and ( 13) with respect to v we obtain 1 θ4 ∂θ4 ∂v and 1 θ1 ∂θ1 ∂v . The others are deduced in incrementing v by 1 2

θ 4 (v, τ ) θ 4 (0, τ ) = k≥0 1 - sin πv sin(k + 1 2 )πτ 2 = k≥0 sin[(k + 1 2 )πτ + πv] sin[(k + 1 2 )πτ -πv] (sin[(k + 1 2 )πτ ]) 2 (5)
θ 3 (v, τ ) θ 3 (0, τ ) = k≥0 1 - sin πv cos(k + 1 2 )πτ 2 = k≥0 cos[(k + 1 2 )πτ + πv] cos[(k + 1 2 )πτ -πv] (cos[(k + 1 2 )πτ ]) 2 (6)
θ 1 (v, τ ) (π sin πv) θ 1 (0, τ ) = k≥1 1 - sin πv sin kπτ 2 = k≥1 sin[kπτ + πv] sin[kπτ -πv] (sin[kπτ ]) 2 (7)
θ 2 (v, τ ) (cos πv) θ 2 (0, τ ) = k≥1 1 - sin πv cos kπτ 2 = k≥1 cos[kπτ + πv] cos[kπτ -πv] (cos[kπτ ]) 2 (8)
1 θ 4 ∂θ 4 ∂v (v, τ ) = -π k≥0 sin 2πv sin(k + 1 2 )πτ 2 -(sin πv) 2 1 θ 3 ∂θ 3 ∂v (v, τ ) = π k≥0 sin 2πv cos(k + 1 2 )πτ 2 -(sin πv) 2 1 θ 2 ∂θ 2 ∂v (v, τ ) = -(tan πv) + π k≥1 sin 2πv (cos kπτ ) 2 -(sin πv) 2 1 θ 1 ∂θ 1 ∂v (v, τ ) = (cot
1 θ 1 ∂θ 4 ∂v (v + 1 2 , τ ) = 1 θ 3 ∂θ 3 ∂v (v, τ ), 1 θ 1 ∂θ 1 ∂v (v + 1 2 , τ ) = 1 θ 2 ∂θ 2 ∂v (v, τ ).
For the convergence of 1 θj ∂θj ∂v we use again the above Lemma 3-2. Thanks to this expression in innite products of the Jacobi theta functions we are able to nd again several known theta identities. In particular, the identities derived from the Landen transformation, [7, p.18] we then deduce

θ 1 (2v, 2τ ) θ 4 (0, 2τ ) = θ 1 (v, τ ) θ 3 (0, τ ) θ 2 (v, τ ) θ 4 (0, τ ) = (π sin πv) k≥0 1 - sin 2πv sin 2kπτ 2 θ 2 (2 p v, 2 p τ ) θ 2 (0, 2 p τ ) .
Proof of Theorem Recall the expansions (Corollary 3-8)

θ 4 (v, τ ) θ 4 (0, τ ) = k≥0 sin[(k + 1 2 )πτ + πv] sin[(k + 1 2 )πτ -πv] (sin[(k + 1 2 )πτ ]) 2 θ 1 (v, τ ) π(sin πv)θ ( 0, τ ) = k≥0 sin[kπτ + πv] sin[kπτ -πv] (sin[kπτ ]) 2 .
Then we deduce θ 4 (4v, 4τ ) θ 4 (0, 4τ ) (sin 2(2k + 1)πτ ) 2 = 16 θ 4 (v, τ ) θ 4 (0, τ )

θ 3 (v, τ ) θ 3 (0, τ ) θ 3 (2v, 2τ ) θ 3 (0, 2τ ) ×(sin(k + 1 2 )πτ ) 2 (cos(k + 1 2 )πτ ) 2 (sin(2k + 1)πτ ) 2 ,
The last part of Theorem 3-11 may be proved by induction on n.

3.3. Link with elliptic functions. Consider now the zeta function of Jacobi.

It is dened by

Z(z, k) = 1 2K d dz log θ 4 (v, τ ), where v = z 2K and K = 2 π 2 0 dx √ 1-k 2 sin 2 x
is the complete elliptic integral of the rst kind and the modulus is such that 0 < k < 1. We have The zeta function of Jacobi has the following form

Z(z, k) = Z 1 (z, k) -Z 2 (z, k) = π 2K k≥0 sin(π2v) sin 2 (πv) -sin 2 (k + 1 2 πτ ) Z 1 (z, k) = 1 f 4 (v, τ ) ∂f 4 (-v, τ ) ∂u (u, τ ), Z 2 (z, k) = 1 f 4 ∂f 4 ∂u (-u, τ ) where v = z 2K satises | sin πv |<| (sin( 1 2 )πτ ) | .
In particular, the logarithmic derivatives of theta functions can be written under the forms

θ 4 (v, τ ) θ 4 (v, τ ) = 4π sin(π2v) k≥0 q 2k+1 1 -2q 2k+1 cos 2πv + q 4k+2 θ 3 (v, τ ) θ 3 (v, τ ) = -4π sin(π2v) k≥0 q 2k+1 1 + 2q 2k+1 cos 2πv + q 4k+2 θ 2 (v, τ ) θ 2 (v, τ ) = -tan(πv) -4π sin(π2v) k≥0 q 2k+2 1 + 2q 2k+2 cos 2πv + q 4k+4 θ 1 (v, τ ) θ 1 (v, τ ) = cot(πv) + 4π sin(π2v) k≥0 q 2k+2 1 -2q 2k+2 cos 2πv + q 4k+4 .
Moreover, the equations for θ 1 and θ 2 are valid in the strip | Imv |< Imτ, those for θ 3 and θ 4 are valid in the strip

| Imv |< 1 2 Imτ. Indeed, Z(z, k) = 1 2K d dz log θ 4 (v, τ ) = π 2K sin(π2v) k≥0 p≥1 sin πv (sin(k + 1 2 )πτ ) 2p .
Suppose the variable v satises | sin πv (sin( 12 )πτ )

|< 1.

We then obtain

sin πv (sin(k + 1 2 )πτ ) 2 p≥0 sin πv (sin(k + 1 2 )πτ ) 2p = sin πv (sin(k+ 1 2 )πτ ) 2 1 - sin πv (sin(k+ 1 2 )πτ ) 2 = (sin πv) 2 (sin(k + 1 2 )πτ ) 2 -(sin πv) 2 .
Therefore, the result follows. The domain of convergence for these series may be extended to the strip | Imv |< 1 2 Imτ (see for example [7 p.489]). Notice that the zeta function of Jacobi also has a Fourier expansion

Z(z, k) = 2π K n≥1 q n 1 -q 2n sin nπz K .
3.4. Link with classical innite products. From Corollaries 3-4 and 3-5 we nd again by another way the classical innite products for θ j (v, τ )

We then deduce the following classical expressions of theta functions as innite products (see [9, p. (1 -q 2k+1 ) 2 θ 3 (v, τ ) θ 3 (0, τ ) = k≥0 1 + 2q 2k+1 cos 2πv + q 4k+2 (1 + q 2k+1 ) 2 θ 1 (v, τ ) θ 1 (0, τ ) = sin πv π k≥1 1 -2q 2k cos 2πv + q 4k (1 -q 2k ) 2 θ 2 (v, τ ) θ 2 (0, τ ) = cos πv k≥1 1 + 2q 2k cos 2πv + q 4k (1 + q 2k ) 2

Proof Starting with the classical expansions of the theta functions as innite product [START_REF] Whittaker | A course of Modern Analysis Cambridge[END_REF]:

θ 4 (v, τ ) = k≥0 (1 -q 2k ) 1 -2q 2k+1 cos 2πv + q 4k+2 θ 3 (v, τ ) = k≥0
(1 -q 2k ) 1 + 2q 2k+1 cos 2πv + q 4k+2 θ 1 (v, τ ) = 2q 1/4 sin πv k≥1 (1 -q 2k ) 1 -2q 2k cos 2πv + q 4k θ 2 (v, τ ) = 2q 1/4 cos πv k≥1 (1 -q 2k ) 1 + 2q 2k cos 2πv + q 4k . In particular θ 4 (0, τ ) = k≥0 (1 -q 2k )(1 -q 2k+1 ) 2 , θ 3 (0, τ ) = k≥0 (1 -q 2k )(1 + q 2k+1 ) 2 θ 2 (0, τ ) = 2q 1/4 k≥1 (1 -q 2k )(1 + q 2k ) 2 , θ 1 (0, τ ) = 2πq 1/4 k≥1 (1 -q 2k ) 3 .

Thus, since q = e iπτ Corollary 4 yields (1 + q 2k+1 ) 2 θ 1 (v, τ ) θ 1 (0, τ ) = (π sin πv) θ 1 (0, τ ) (1 -q 2k ) 2 θ 2 (v, τ ) θ 2 (0, τ ) = (cos πv) θ 2 (0, τ ) (1 + q 2k ) 2 . θ 4 (v, τ ) θ 4 (0, τ ) = k≥0 (q k+ 1 2 ) 2 q -2k-1 -2 cos 2πv + q 2k+1 (1 -q 2k+1 ) 2 = 2 k≥0 q -2k-1 -2 cos 2πv + q 2k+1 sin(k +

2

 2 Let (g n ) n a sequence of holomorphic functions dened on Ω ∈ C

θ 3 and θ 4

 4 is dened in the "strip" | Imv |< 1 2 Imτ, while θ 1 and θ 2 are dened in the "strip" | Imv |< Imτ.

  θ 3 and θ 4 is dened in the band | Imv |< 1 2 Imτ, while θ 1 and θ 2 are dened in the band | Imv |< Imτ.

1 -

 1 2q 2k+1 cos 2πv + q 4k+2

1 - 1 -q 2k+1 ) 2 θ 3 (v, τ ) θ 3 (

 11233 2q 2k+1 cos 2πv + q 4k+2 (2k+1 cos 2πv + q 4k+2

1 -

 1 2q 2k cos 2πv + q 4k

  2k cos 2πv + q 4k

  πv) -π 3 and θ 4 is dened in the "strip" | Imv |< 1 2 Imτ, while θ 1 and θ 2 is dened in the "strip" | Imv |< Imτ.

		sin 2πv
	k≥1	(sin kπτ ) 2 -(sin πv) 2

θ

  [START_REF]Chouikha Expansions of Theta Functions and Applications ArXiv[END_REF][START_REF] Copson | Theory of Functions of a Complex Variable[END_REF][START_REF] Erdelyi | Higher transcendental functions Vol. I-II[END_REF][START_REF] Lang | Elliptic functions Addison-Wesley[END_REF][START_REF] Lawden | Elliptic functions and applications[END_REF][START_REF] Magnus | Formulas and Theorems for the Special Functions of Mathematical Physics Bd[END_REF][START_REF] Whittaker | A course of Modern Analysis Cambridge[END_REF][10][11] Consider the following trigonometric equalities

	sin(2t + 2x) sin(2t -2) = 4 sin(t + x) cos(t + x) sin(t -x) cos(t -x),
	sin(4t + 4x) sin(4t -4x) = 4 sin(2t + 2x) cos(2t + 2x) sin(2t -2x) cos(2t -2x)
	= 16 sin(t + x) sin(t -x) cos(t + x) cos(t -x) cos(2t + 2x) cos(2t -2x).

  Repetition of this calculation (details omitted) for θ 3 (v, τ ), θ 2 (v, τ ), θ 1 (v, τ ) yields the other expressions.

		1 2 )πτ	2	= 4	k≥0	-cos (2k + 1)πτ + cos 2πv sin (k + 1 2 )πτ 2
	=	k≥0	(sin (k + 1 2 )πτ ) 2 -(sin πv) 2 sin (k + 1 2 )πτ

2

On the other hands, since θ 4 (v + 1 2 , τ ) = θ 3 (v, τ ), and

we may derive Let q = e iπτ and v ≤ Im τ 2 the innite products satisfy the following relations

In particular for v = 0 we have

With respect the quasi periodicity we have

which permits to derive Corollary 3-7 Let q = e iπτ and v ≤ Im τ 2 we have the relations

Another expression of theta functions as innite products Corollary 3-10 Let q = e iπτ , | q |< 1. Then the followings identities hold

However, By the same way we may prove Since θ 4 (0, 2τ ) = θ 3 (0, τ )θ 4 (0, τ ) = θ 1 (0, τ )θ 2 (0, τ ) 2θ 1 (0, 2τ )