Raouf Abd

Chouikha

FUNCTIONS RELATED TO JACOBI THETA FUNCTIONS AND APPLICATIONS II

Keywords: innite products, theta functions, elliptic functions, q-trigonometry, ntransformation, 2010 Mathematics Subject Classication. 33E05, 11F11, 11F20, 11F27

Functions related to Jacobi Theta Functions and applications II

1. Introduction 1.1. Jacobi theta functions and the g j (v, τ ) functions.

To carry out our investigations, we need to introduce the Jacobi theta functions The Jacobi theta functions for j = 1, 2, 3, 4 are dened as ( [START_REF] Enneper | Elliptische Functionen: Theorie und Geschichte[END_REF], [START_REF] Erdelyi | Higher transcendental functions[END_REF], [START_REF] Schwarz | Formeln und Lehrsatze zum Gebrauche der elliptischen functionen[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 = e iπτ such that | q |< 1 or τ : Im(τ ) > 0. These series converge for all complex values of v whenever τ has positive imaginary part. Moreover, these series converge absolutely and uniformly on compact subsets. Theta functions are entire and periodic functions of v, [START_REF] Apostol | Modular Functions and Dirichlet Series in Number Theory[END_REF], [START_REF] Appell | Fonctions elliptiques et applications Gauthiers-Villard[END_REF], [START_REF] Erdelyi | Higher transcendental functions[END_REF].

Using the triple product identity, we can derive the innite product representations of the Jacobi theta functions [8, (19),p.357] : θ 1 (v, τ ) = 2q 0 q 1/4 sin πv n≥1 1 -2q 2n cos 2πv + q 4n , θ 2 (v, τ ) = 2q 0 q 1/4 cos πv n≥1 1 + 2q 2n cos 2πv + q 4n , θ 3 (v, τ ) = q 0 n≥1 1 + 2q 2n-1 cos 2πv + q 4n-2 , θ 4 (v, τ ) = q 0 n≥1 1 -2q 2n-1 cos 2πv + q 4n-2 , where q 0 = n≥1 [1 -q n ].

Corollary 3-3 below will give another innite product representations of the Jacobi theta functions.

We will consider for that the following functions

g 4 (v, τ ) = k≥0 sin[(k + 1 2 )πτ + πv] sin[(k + 1 2 )πτ ] , g 3 (v, τ ) = k≥0 cos[(k + 1 2 )πτ + πv] cos[(k + 1 2 )πτ ] , g 1 (v, τ ) = k≥1 sin[kπτ + πv] sin[kπτ ] , g 2 (v, τ ) = k≥1 cos[kπτ + πv] cos[kπτ ]
where v ∈ C and τ belongs to the half plane ( τ : Im(τ ) > 0). These functions are connected to theta functions (Propositions 3-1, 3-2):

θ 4 (v, τ ) = θ 4 (0, τ ) g 4 (v, τ ) g 4 (-v, τ ), θ 3 (v, τ ) = θ 3 (0, τ ) g 3 (v, τ ) g 3 (-v, τ ), θ 1 (v, τ ) = (π sin(πv) θ 1 (0, τ ) g 1 (v, τ ) g 1 (-v, τ ), θ 2 (v, τ ) = cos(πv) θ 2 (0, τ ) g 2 (v, τ ) g 2 (-v, τ ).
Furthermore, we had considered in [START_REF] Chouikha | Functions related to Jacobi Theta Functions and applications I[END_REF] another representation of the Jacobi theta functions by mean of f i (v, τ ), i = 1, 2, 3, 4. which appear to have remarkable properties. These functions are naturally connected to g i (v, τ ), i = 1, 2, 3, 4. : 1.2. The aim of the paper. The purpose of this paper is to bring out some intrinsic properties of these functions g j (v, τ ). We will see in the sequel these functions which are related between them seem play a particular role.

f 1 (v, τ ) g 1 (v, τ ) = k≥1 sin [kπτ -πv]
On the other hand, we derive the Fourier series expansions for g j (v, τ ) , as well as for log(g j (v, τ )). Some other signicant properties and equalities will be derived. We also provide a n-order transformation formulas for these functions and nd again those for the Jacobi theta functions. The last part is devoted to applications in q-trigonometry introduced by Gosper [START_REF] Erdelyi | Higher transcendental functions[END_REF]. We will rstly express sin q (v), cos q (v) as innite products : sin q (v) = (sin πv) k≥1 sin(πv + kπ τ ) sin( kπ τ -πv) (cos kπ τ ) 2

, cos q (v) = (cos πv) k≥1 cos(πv + kπ τ ) cos(πv -kπ τ ) (cos kπ τ ) 2 , for | Imv |< τ.

All these representations permits to provide some properties and identities, notably for duplication and n-transformations.

1.3. Connection with functions introduced by Eisenstein. These g j (v, τ ) functions allowing a representation of theta do not seem to have been explicitly mentioned in the literature. However, a few allusive remarks lead one to believe that this approach has been considered. Indeed, Eisenstein ([6, p.258] or [12, p.94-116]) mentioned the double product function dened as

φ(v) = v m,k 1 - v m + kτ ,
the prime symbol means that the term corresponding to m = k = 0 is to be excluded. This author established a connection between his product function φ(v) and the theta function

φ(v) = 2 πθ 1 (0, τ ) θ 1 (v, τ )
where

θ 1 (v, τ ) = δθ δv (v, τ ).
Eisenstein observed that the innite product representation of sinus function

sin(πv) = πv lim M →∞ -M ≤m≤M 1 + v m would imply ∞ m=-∞ 1 - v m + kτ = sin(kπτ -πv) sin(kπτ ) .
Therefore one gets the expression

φ(v) = v ∞ k=-∞ sin(kπτ -πv) sin(kπτ ) .
The function g 1 (v, τ ) we will consider is such that

φ(v) = sin(πv) g 1 (v, τ )g 1 (-v, τ ).
On the other hand, following Eisenstein [5, p.275-280] who consider the theta product

ξ(e iπv ) = (e iπv -e -iπv ) k≥1 (1 -q 2k e 2iπv )(1 -q 2k e -2iπv
), so that we may write

f (u, v) = k≥1 m 1 - v m + xτ + x = ξ(e iπ(v-u) ) ξ(e iπv ) .
We then obtain (Proposition 2-7 below)

g 1 (v, τ ) = k≥1 m 1 + v m + kτ = lim u→0 [-uf (u, v)].
Analogous expressions may be found for the other g j (v, τ ), j = 2, 3, 4 since they are related.

Functions defined by an infinite product

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

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

These functions are naturally related. Replacing v by v + 1, v + 1 2 and τ by

τ + 1, τ + 1 2 we then obtain Proposition 2-1
The above function g j (v, τ ) veries the following properties for τ > 0, Imv < 1 2 τ F or j = 1, 2, 3, 4 g j (0, τ ) = 1, g j (v, τ + 2) = g j (v, τ ),

g 4 (v, τ + 1) = g 3 (v, τ ), g 3 (v, τ + 1) = g 4 (v, τ ), g i (v, τ +1) = g i (v, τ ) f or i = 1, 2, g 1 (v, τ + 1 2 ) = g 2 (v, τ ), g 2 (v, τ + 1 2 ) = g 1 (v, τ ). Proposition 2-2
The following identities hold for

Imv < 1 2 τ g 4 (v + 1 2 , τ ) = k≥0 cot[(k + 1 2 )πτ ] g 3 (v, τ ), g 4 (v + τ 2 , τ ) = g 1 (v, τ ) g 4 ( τ 2 , τ ), g 1 (v + 1 2 , τ ) = k≥0 cot(kπτ ) g 2 (v, τ ), g 3 (v + τ 2 , τ ) = g 2 (v, τ ) g 3 ( τ 2 , τ ), g 4 (v + τ, τ ) = g 4 (v, τ ) sin(πv + πτ 2 ) , g 1 (v + τ, τ ) = g 1 (v, τ ) sin(πv + πτ ) .
Proof Indeed, concerning the last identity we may write

k≥1 sin[(k + 1)πτ + πv] sin[kπτ ] = 1 sin(πv + πτ ) k≥1 sin[kπτ + πv] sin[kπτ ] .
Thus

g 1 (v + τ, τ ) sin(πv + πτ ) = g 1 (v, τ ).
For any integer n the functions g j (v, τ ) may also be expressed

g 4 (v, τ ) = 0≤k≤n-1 sin[(k + 1 2 )πτ + πv] g 4 (v + nτ, τ ) g 1 (v, τ ) = 1≤k≤n-1 sin[kπτ + πv] g 1 (v + nτ, τ ).
We may provide also a duplication formula and n-order transformation Proposition 2-4

The following identities hold

g 4 (2v, 2τ ) = g 4 (v, τ ) g 3 (v, τ ) = g 4 (v, τ ) g 4 ( π 4 -v, τ ) g 3 ( π 4 -v, τ ), g 1 (2v, 2τ ) = g 1 (v, τ ) g 2 (v, τ ) = g 1 (v, τ ) g 1 ( π 4 -v, τ ) g 2 ( π 4 -v, τ ).
For any integer n we get,

g 1 (v + nτ, τ ) = g 1 (v, τ ) n m=1 sin[mπτ + πv] , g 2 (v + nτ, τ ) = g 2 (v, τ ) n m=1 cos[mπτ + πv] , g 4 (v + nτ, τ ) = g 4 (v, τ ) n m=1 sin[( mπ 2 )τ + πv] , g 3 (v + nτ, τ ) = g 3 (v, τ ) n m=1 cos[( mπ 2 )τ + πv]
.

Proof

By Proposition 2-2 we have g 4 (v + τ, τ ) = g4(v,τ ) sin(πv+ πτ 2 ) , then

g 4 (v + 2τ, τ ) = g 4 (v + τ, τ ) sin(πv + 3πτ 2 ) = g 4 (v, τ ) sin[ πτ 2 + πv] sin[( 3π 2 )τ + πv] , g 1 (v + 2τ, τ ) = g 1 (v + τ, τ ) sin(πv + 3πτ ) = g 1 (v, τ ) sin[πτ + πv] sin[(3π)τ + πv] .
Moreover,

g 1 (v + 3τ, τ ) = g 1 (v + 2τ, τ ) sin(πv + 4πτ ) = g 1 (v + τ ) sin[2πτ + πv] sin[(4π)τ + πv] ,
thus

g 1 (v + 3τ, τ ) = g 1 (v, τ ) 4 m=1 sin[mπτ + πv] , g 4 (v + 3τ, τ ) = g 4 (v, τ ) 4 m=1 sin[ mπ 2 )τ + πv]
.

The last assertion of the Proposition may be proved by induction.

2.1. n-order transformations. These functions have many other interesting properties. For example, we can express in a simple way n-order transformation (for an integer). In the sense that we can highlight a relationship between g j (nv, nτ ) and g j (v + m n , τ ).

Theorem 2-5

Let n be any integer then the following identities hold

g 1 (nv, nτ ) = 2 n-1 k≥1 (sin kπτ ) n sin nkπτ 0≤m≤n-1 g 1 (v + mπ n , τ ), g 4 (nv, nτ ) = 2 n-1 k≥0 (sin(k + 1 2 )πτ ) n sin n(k + 1 2 )πτ 0≤m≤n-1 g 4 (v + mπ n , τ ).
When n is an even integer then

g 1 (nv, nτ ) = (-1) n 2 2 n-1 k≥1 (cos kπτ ) n sin nkπτ 0≤m≤n-1 g 2 (v + mπ n , τ ), g 4 (nv, nτ ) = 2 n-1 k≥0 (cos(k + 1 2 )πτ ) n sin nkπτ 0≤m≤n-1 g 3 (v + mπ n , τ ).
When n is odd then the following hold :

g 1 (nv, nτ ) = 2 n-1 k≥1 (sin kπτ ) n sin nkπτ 0≤m≤n-1 g 1 (v + mπ n , τ ), g 2 (nv, nτ ) = (-1) n-1 2 2 n-1 k≥1 (cos kπτ ) n cos nkπτ 0≤m≤n-1 g 2 (v + mπ n , τ ), g 3 (nv, nτ ) = (-1)
n-1

2 2 n-1 k≥0 (cos(k + 1 2 )πτ ) n cos n(k + 1 2 )πτ 0≤m≤n-1 g 4 (v + mπ n , τ ), g 4 (nv, nτ ) = 2 n-1 k≥0 (sin(k + 1 2 )πτ ) n sin n(k + 1 2 )πτ 0≤m≤n-1 g 4 (v + mπ n , τ ).
Proof of Theorem 2-5

Starting from the classical trigonometric product formulas valid for n integer

sin(nz) = 2 n-1 sin(z) sin(z+ π n ) sin(z+ 2π n ).... sin(z+ (n -1)π n ) = 2 n-1 n-1 m=0 sin(z+ mπ n ).
When n is even then it yields

sin(nz) = (-1) n/2 2 n-1 cos(z) cos(z + π n ) cos(z + 2π n ).... cos(z + (n -1)π n ) = (-1) n 2 2 n-1 n-1 m=0 cos(z + mπ n ).
When n is odd we then obtain

cos(nz) = (-1) n-1/2 2 n-1 cos(z) cos(z + π n ) cos(z + 2π n ).... cos(z + (n -1)π n ) = (-1) n-1 2 2 n-1 n-1 m=0 cos(z + mπ n ).
Therefore we may deduce for any integer k

sin[nkπτ + nπv] sin(nkπτ ) = 2 n-1 sin(nkπτ ) n-1 m=0 sin[kπτ + πv + mπ n ] , sin[n(k + 1 2 )πτ + nπv] sin(n(k + 1 2 )πτ ) = 2 n-1 sin(n(k + 1 2 )πτ ) n-1 m=0 sin[(k + 1 2 )πτ + πv + mπ n ] , cos[nkπτ + nπv] cos(nkπτ ) = 2 n-1 cos(nkπτ ) n-1 m=0 cos[kπτ + πv + mπ n ] , cos[n(k + 1 2 )πτ + nπv] cos(n(k + 1 2 )πτ ) = 2 n-1 cos(n(k + 1 2 )πτ ) n-1 m=0 cos[(k + 1 2 )πτ + πv + mπ n ] .
Then it follows for any integer n

g 1 (nv, nτ ) = k≥1 sin[nkπτ + nπv] sin[nkπτ ] = k≥1 2 n-1 sin(nkπτ ) n-1 m=0 sin[kπτ + πv + mπ n ] = k≥1 2 n-1 [sin(kπτ )] n sin(nkπτ ) n-1 m=0 sin[kπτ + πv + mπ n ] sin(kπτ ) ,
which means

g 1 (nv, nτ ) = k≥1 2 n-1 [sin(kπτ )] n sin(nkπτ ) 0≤m≤n-1 g 1 (v + mπ n , τ ),
and

g 4 (nv, nτ ) = k≥0 2 n-1 [sin((k + 1 2 )πτ )] n sin(n(k + 1 2 )πτ ) 0≤m≤n-1 g 4 (v + mπ n , τ ).
On the other hand, when n is even we have

sin[nkπτ + nπv] sin[nkπτ ] = 2 n-1 sin(nkπτ ) n-1 m=0 cos[kπτ + πv + mπ n ] ,
which means

g 1 (nv, nτ ) = k≥1 2 n-1 [cos(kπτ )] n sin(nkπτ ) 0≤m≤n-1 g 2 (v + mπ n , τ ).
By the same way we deduce for n even

g 4 (nv, nτ ) = k≥0 n-1 m=0 2 n-1 [cos((k + 1 2 )πτ )] n sin(n(k + 1 2 )πτ ) 1≤m≤n g 4 (v + mπ n , τ ).
Turn now to the case n odd, we easily prove by the same method

g 2 (nv, nτ ) = k≥1 2 n-1 [cos(kπτ )] n cos(nkπτ ) 0≤m≤n-1 g 2 (v + mπ n , τ ), g 3 (nv, nτ ) = k≥0 2 n-1 [cos((k + 1 2 )πτ )] n cos(n(k + 1 2 )πτ ) 0≤m≤n-1 g 3 (v + mπ n , τ ).
Taking the logarithmic derivative of functions g j (v, τ ) it yields Corollary 2-6

Let n be any integer then the following hold

g 1 g 1 (nv, nτ ) = π k≥1 cot(kπτ + πv) = 0≤m≤n-1 g 1 g 1 (v + mπ n , τ ), g 4 g 4 (nv, nτ ) = π k≥1 cot((k + 1 2 )πτ + πv) = 0≤m≤n-1 g 4 g 4 (v + mπ n , τ ).
For any odd integer n :

g j g j (nv, nτ ) = 0≤m≤n-1 g j g j (v + mπ n , τ ), j = 1, 2, 3, 4.
For any even integer n :

g 1 g 1 (nv, nτ ) = 0≤m≤n-1 g 2 g 2 (v + mπ n , τ ), g 4 g 4 (nv, nτ ) = 0≤m≤n-1 g 3 g 3 (v + mπ n , τ ).
Notice that

g 2 g 2 (nv, nτ ) = -π k≥1 tan(kπτ + πv), g 3 g 3 (nv, nτ ) = -π k≥1 tan((k + 1 2 )πτ + πv).
Moreover, these functions may also be expressed as double innite products Proposition 2-7

The functions g j (v, τ ) may be expressed

g 1 (v, τ ) = k≥1 m 1 + v m + kτ , g 2 (v, τ ) = k≥1 m 1 + v m + kτ + k 2 , g 4 (v, τ ) g 4 ( τ 2 , τ ) = k≥1 m 1 + v -τ 2 m + kτ , g 3 (v, τ ) g 3 ( τ 2 , τ ) = k≥1 m 1 + v -τ 2 m + kτ + k 2 . Proof of Proposition 2-7
Indeed, following Eisenstein [6, p.213] starting from product formula for sin πx

sin πx = πx m 1 + x m ,
(the prime means the integer m = 0) it follows for any integer n

sin[kπτ + nπv] = (kπτ + πv) m 1 + kτ + v m ,
as well as

sin[kπτ ] = (kπτ ) m 1 + kτ m .
Then,

sin[kπτ + nπv] sin[kπτ ] = (kτ + v) (kτ ) m 1 + kτ +v m 1 + kτ m = (kτ + v) (kτ ) m m + kτ + v m + kτ = m 1 + v m + kτ .
We then deduce g 1 (v, τ ) = k≥1 m 1 + v m+kτ . The others expressions for g 2 , g 3 , g 4 are derived from Proposition 2-2.

Remarks 2-8

To go further and explore other properties of functions g j we can take inspiration from the method developed by Eisenstein [6, p.275] which consists in expressing g j as the limit of a function of 2 variables f (u, v). This method has been used successfully to reveal the connection between his product function φ(v) and the Jacobi function θ 1 . More explicitly, following 1-3 consider again

f (u, v) = k≥1 m 1 - v m + xτ + x .
We observe that

m 1 - v m + kτ + x = sin[kπτ + nπv -πu] sin[kπτ + πv] ,
thus it could be written

f (u, v) = k≥1 sin[kπτ + nπv -πu] sin[kπτ + πv] .
Following Eisenstein [5,p.275-280] who consider the theta product ξ(e iπv ) = (e iπv -e -iπv ) k≥1

(1 -q 2k e 2iπv )(1 -q 2k e -2iπv ), so that we may write

f (u, v) = k≥1 m 1 - v m + xτ + x = ξ(e iπ(v-u) ) ξ(e iπv ) .
Notice that since by Proposition 2-7

g 1 (v, τ ) = k≥1 m 1 + v m + kτ ,
we then obtain the identity

f (u, v) = g 1 (v -u) g 1 (v) ,
and

g 1 (v, τ ) = lim u→0 [-uf (u, v)].

Connection of g j with the 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) or by the triple product identity, it is possible to obtain innite products representing this function. We refer to [START_REF] Chouikha | Functions related to Jacobi Theta Functions and applications I[END_REF][START_REF]Chouikha Expansions of Theta Functions and Applications ArXiv[END_REF] in order to nd more ample information as well as other aspects and developments as innite products concerning these theta functions.

3.1. Connection with θ 4 (v, τ ). A natural relation with θ 4 (v, τ ) is given by Proposition 3-1

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

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

g 4 (v, τ ) = k≥0 sin[(k + 1 2 )πτ + πv] sin[(k + 1 2 )πτ ] 1 θ 4 ∂θ 4 ∂v (v, τ ) = 1 g 4 ∂g 4 ∂v (v, τ )+ 1 g 4 ∂g 4 ∂v (-v, τ ) = k≥0 cot[(k+ 1 2 )πτ +πv]+cot[(k+ 1 2 )πτ -πv]. θ 4 is dened in the band | Imv |< 1 2 τ.
Proof of Proposition 3-1

Let q = e iπτ then we may write

g 4 (v, τ ) = k≥0 sin[(k + 1 2 )πτ + πv] sin[(k + 1 2 )πτ ] = k≥0 e (k+ 1 2 )πτ +iπv -e -(k+ 1 2 )πτ -iπv e iπ(k+ 1 2 )τ -e -iπ(k+ 1 2 )τ = k≥0 q k e iπv -q -k e -iπv q k -q -k = k≥0
q 2k e iπv -e -iπv q 2k -1 .

Therefore

g 4 (v, τ ) g 4 (-v, τ ) = k≥0 q 2 k+1 e iπ v -e -iπ v q 2 k+1 e -iπ v -e iπ v (q 2 k+1 -1) 2 = k≥0 q 4 k+2 -q 2 k+1 e 2 iπ v -e -2 iπ v q 2 k+1 -1 (q 2 k+1 + 1) 2 .
On the other hand, by [8, (16) p.369] the innite products representing the theta functions

θ 4 (v, τ ) = k≥1 (q 2 k -1) k≥1 [1 -2q 2k-1 cos 2πv + q 4k-2 , θ 4 (0, τ ) = k≥1 (q 2 k -1) k≥1 [1 -2q 2k-1 + q 4k-2 = (q 2 k -1) 3 .
We thus derive expression of theta as innite product

θ 4 (v, τ ) θ 4 (0, τ ) = k≥1 1 -2q 2k-1 cos 2πv + q 4k-2 (q 2 k -1) 2 = k≥0 sin[(k + 1 2 )πτ + πv] sin[(k + 1 2 )πτ -πv] (sin[(k + 1 2 )πτ ]) 2 = g 4 (v, τ ) g 4 (-v, τ ).
3.2. Connection with the other theta functions. Consider again the functions g i (v, τ ), i = 1, 2, 3.

g 3 (v, τ ) = k≥0 cos[(k + 1 2 )πτ + πv] cos[(k + 1 2 )πτ ] , g 1 (v, τ ) = k≥1 sin[kπτ + πv] sin[kπτ ] , g 2 (v, τ ) = k≥1 cos[kπτ + πv] cos[kπτ ] .
Recall that they are related

g 3 (v, τ ) = g 4 (v, τ +1), g 4 (v, τ ) = g 3 (v, τ +1), g 2 (v, τ + 1 2 ) = g 1 (v, τ ), g 1 (v, τ + 1 2 ) = g 2 (v, τ ).
By the same manner as we prove Proposition 3-1 and using expansions of theta functions [8, (16)

p.369] θ 1 (v, τ ) = 2 n≥1 [1 -q n ]q 1/4 sin πv n≥1 1 -2q 2n cos 2πv + q 4n , θ 2 (v, τ ) = 2 n≥1 [1 -q n ]q 1/4 cos πv n≥1 1 + 2q 2n cos 2πv + q 4n , θ 3 (v, τ ) = n≥1 [1 -q n ] n≥1 1 + 2q 2n-1 cos 2πv + q 4n-2 ,
one derives the following Proposition 3-2

The other theta functions may be expressed

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

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

We then deduce expressions for Jacobi theta functions as innite products 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 sin[(k + 1 2 )πτ + πv] sin[(k + 1 2 )πτ -πv] (sin[(k + 1 2 )πτ ]) 2 (1)
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
(2) This type of expansion was already known to the literature. We refer for that for example to the book of H.A. Schwarz [13 p.36] in which he exposed a complete theory of the elliptic sigma functions as well as various trigonometric expansions but in a dierent context than we use for theta functions. As we know that these sigma and theta functions are related, by an elementary calculation we may nd again expansions of Corollary 3-3. Note however that our approach is dierent. It seems also Eisenstein [6, to have been aware of this type of expressions. To be more precise, he used a method of summation (according to his notation) for s ≥ 1

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 (3)
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 (
(s, x) = ∞ m,k=-∞ 1 (x + m + kτ ) s , (s * , 0) = ∞ m,k=-∞ 1 (x + m + kτ ) s .
These sums are called later by A. Weil as Eisenstein series. Eisenstein observed that by periodicity and after term-by-term dierentiation

(s, x + 1) = (s, x), d dx (s, x) = -s(s + 1, x). Since ∞ m=-∞ 1 (y+m) = π cot πy we then deduce ∞ m=-∞ 1 (x + m + kτ ) 2 = -π d dx cot(π(x + kτ )) = π 2 [sin(π(x + kτ ))] 2 .
Thus, (2, x + τ ) = (2, x) and

(2, x) = ∞ m,k=-∞ 1 (x + m + kτ ) 2 = k π 2 [sin(π(x + kτ ))] 2
is a double periodic function. Note in addition that Eisenstein also pointed out all the important properties of his function (2, x) -(2 * , 0), called later ℘(x) of Weierstrass.

We may derive from the preceding For any integer n we have

(-1) n e -iπn(2v+τ ) = n m=1 sin[mπτ + πv] sin[-mπτ -πv], e -iπn(2v+τ ) = n m=1 cos[mπτ + πv] cos[-mπτ -πv], e -iπn(2v+τ ) = n m=1 cos[ mπ 2 τ + πv] cos[- mπ 2 τ -πv], (-1) n e -iπn(2v+τ ) = n m=1 sin[ mπ 2 τ + πv] sin[- mπ 2 τ -πv].
Proof Recall by Proposition 2-4

g 1 (v + nτ, τ ) = g 1 (v, τ ) n m=1 sin[mπτ + πv] , g 2 (v + nτ, τ ) = g 2 (v, τ ) n m=1 cos[mπτ + πv] , g 4 (v + nτ, τ ) = g 4 (v, τ ) n m=1 sin[( mπ 2 )τ + πv] , g 3 (v + nτ, τ ) = g 3 (v, τ ) n m=1 cos[( mπ 2 )τ + πv] . Since θ 4 (v + τ, τ ) θ 4 (v, τ ) = (-1)e -iπ(2v+τ ) , θ 3 (v + τ, τ ) θ 3 (v, τ ) = e -iπ(2v+τ ) , θ 1 (v + τ, τ ) θ 1 (v, τ ) = (-1)e -iπ(2v+τ ) , θ 2 (v + τ, τ ) θ 2 (v, τ ) = e -iπ(2v+τ ) ,
By induction we obtain for any integer

θ 4 (v + nτ, τ ) θ 4 (v, τ ) = (-1) n e -iπn(2v+τ ) , θ 3 (v + nτ, τ ) θ 3 (v, τ ) = e -iπn(2v+τ ) , θ 1 (v + nτ, τ ) θ 1 (v, τ ) = (-1) n e -iπn(2v+τ ) , θ 2 (v + nτ, τ ) θ 2 (v, τ ) = e -iπn(2v+τ ) .
Moreover, since

θ j (v + nτ, τ ) θ j (v, τ ) = g j (v + nτ, τ )g j (-v -nτ, τ ) g j (v, τ )g j (-v, τ ) ,
We then deduce

θ 1 (v + τ, τ ) θ 1 (v, τ ) = (-1) n e -iπn(2v+τ ) = 1 n m=1 sin[mπτ + πv] sin[-mπτ -πv] , θ 2 (v + τ, τ ) θ 2 (v, τ ) = e -iπn(2v+τ ) = 1 n m=1 cos[mπτ + πv] cos[-mπτ -πv] , θ 3 (v + nτ, τ ) θ 3 (v, τ ) = e -iπn(2v+τ ) = 1 n m=1 cos[ mπ 2 τ + πv] cos[-mπ 2 τ -πv] , θ 4 (v + nτ, τ ) θ 3 (v, τ ) = (-1) n e -iπn(2v+τ ) = 1 n m=1 sin[ mπ 2 τ + πv] sin[-mπ 2 τ -πv]
.

Recall that the transformation theory of elliptic functions or theta functions deals with the relations between these functions belonging to dierent pairs of primitive periods. Observe [9, p.252] that any transformation of order n > 1 may be represented as a product of transformations of rst order and of transformations of higher order with matrix

M = 1 0 0 n
Moreover, any transformation τ = nτ can be separated into a product when n has prime factors. Therefore, we only study the case of transformation when n is a prime and limit our study for this type of matrix. We refer to an n-order modular equation as a relation between θ(τ ) and θ(nτ ).

Consequently from Proposition 2-5 and Corollary 3-3 we may derive formulas of multiplication by an integer n for theta functions which are well known for the odd case since Jacobi (see [7, p.374]) who asserted

θ j (nv, nτ ) = s≥1 (1 -q 2sn (1 -q 2s ) n 0≤m≤(n-1) θ j (v + m n , τ ).
We will nd again this result, but the proof we give here is dierent and will have various consequences and applications, in particular in q-trigonometry. This will be discussed in part 5.

Theorem 3-6

Let n be any integer then : (i) -For j = 1, 4 the following identities hold

θ 1 (nv, nτ ) πθ 1 (0, nτ ) = 2 2n-2 sin nπv (sin πv) n [πθ 1 (0, τ )] n k≥1
[sin(kπτ )] n sin(nkπτ )

2 0≤m≤n-1 θ 1 (v + m n , τ ), θ 4 (nv, nτ ) θ 4 (0, nτ ) = 2 2n-2 1 [θ 4 (0, τ )] n k≥0 [sin((k + 1 2 )πτ )] n sin(n(k + 1 2 )πτ ) 2 0≤m≤n-1 θ 4 (v + m n , τ ).
(ii) -When n is an even integer then

θ 1 (nv, nτ ) πθ 1 (0, nτ ) = 2 2n-2 sin nπv (cos πv) n [θ 2 (0, τ )] n k≥1 [sin(kπτ )] n sin(nkπτ ) 2 0≤m≤n-1 θ 2 (v + m n , τ ), θ 4 (nv, nτ ) θ 4 (0, nτ ) = 2 2n-2 1 [θ 3 (0, τ )] n k≥0 [sin((k + 1 2 )πτ )] n sin(n(k + 1 2 )πτ ) 2 0≤m≤n-1 θ 3 (v + m n , τ ).
(iii) -When n is odd then the following hold

θ 1 (nv, nτ ) πθ 1 (0, nτ ) = 2 2n-2 sin nπv (sin πv) n [πθ 1 (0, τ )] n k≥1 [sin(kπτ )] n sin(nkπτ ) 2 0≤m≤n-1 θ 1 (v + m n , τ ), θ 4 (nv, nτ ) θ 4 (0, nτ ) = 2 2n-2 1 [θ 4 (0, τ )] n k≥0 [sin((k + 1 2 )πτ )] n sin(n(k + 1 2 )πτ ) 2 0≤m≤n-1 θ 4 (v + m n , τ ),
and

θ 2 (nv, nτ ) θ 2 (0, nτ ) = 2 2n-2 cos nπv (cos πv) n [θ 2 (0, τ )] n k≥1 [cos(kπτ )] n cos(nkπτ ) 2 0≤m≤n-1 θ 2 (v + m n , τ ), θ 3 (nv, nτ ) θ 3 (0, nτ ) = 2 2n-2 1 [θ 3 (0, τ )] n k≥0 [cos((k + 1 2 )πτ )] n cos(n(k + 1 2 )πτ ) 2 0≤m≤n-1 θ 3 (v + m n , τ ).
Proof of Theorem 3-6

By Theorem 2-5,

g 1 (nv, nτ ) = 2 n-1 k≥1 (sin kπτ ) n sin nkπτ 0≤m≤n-1 g 1 (v + mπ n , τ ), g 4 (nv, nτ ) = 2 n-1 k≥0 (sin(k + 1 2 )πτ ) n sin n(k + 1 2 )πτ 0≤m≤n-1 g 4 (v + mπ n , τ ).
When n is an even integer then

g 1 (nv, nτ ) = (-1) n 2 2 n-1 k≥1 (cos kπτ ) n sin nkπτ 0≤m≤n-1 g 2 (v + mπ n , τ ), g 4 (nv, nτ ) = 2 n-1 k≥0 (cos(k + 1 2 )πτ ) n sin nkπτ 0≤m≤n-1 g 3 (v + mπ n , τ ).
When n is odd then the following hold :

g 1 (nv, nτ ) = 2 n-1 k≥1 (sin kπτ ) n sin nkπτ 0≤m≤n-1 g 1 (v + mπ n , τ ), g 2 (nv, nτ ) = (-1)
n-1

2 2 n-1 k≥1 (cos kπτ ) n cos nkπτ 0≤m≤n-1 g 2 (v + mπ n , τ ), g 3 (nv, nτ ) = (-1)
n-1

2 2 n-1 k≥0 (cos(k + 1 2 )πτ ) n cos n(k + 1 2 )πτ 0≤m≤n-1 g 4 (v + mπ n , τ ), g 4 (nv, nτ ) = 2 n-1 k≥0 (sin(k + 1 2 )πτ ) n sin n(k + 1 2 )πτ 0≤m≤n-1 g 4 (v + mπ n , τ ).
Consequently,

θ 1 (nv, nτ ) πθ 1 (0, nτ ) = sin(nπv) [πθ 1 (0, τ )] n 0≤m≤n-1 θ 1 (v + m n , τ ) sin(v + m n )
. n even :

θ 1 (nv, nτ ) πθ 1 (0, nτ ) = sin(nπv) [θ 2 (0, τ )] n 0≤m≤n-1 θ 2 (v + m n , τ ) cos(v + m n ) , θ 4 (nv, nτ ) θ 4 (0, nτ ) = 1 [θ 3 (0, τ )] n 0≤m≤n-1 θ 3 (v + m n , τ ).
n odd :

θ j (nv, nτ ) θ j (0, nτ ) = 1 [θ j (0, τ )] n 0≤m≤n-1 θ j (v + m n , τ ),
for j = 3, 4, and

θ 2 (nv, nτ ) πθ 2 (0, nτ ) = cos(nπv) [θ 2 (0, τ )] n 0≤m≤n-1 θ 2 (v + m n , τ ) cos(v + m n )
.

The results follow since when n even one has

sin(nz) = (-1) n/2 2 n-1 cos(z) cos(z + π n ) cos(z + 2π n ).... cos(z + (n -1)π n ) = (-1) n 2 2 n-1 n-1 m=0 cos(z + mπ n ).
For n odd one has

cos(nz) = (-1) n-1/2 2 n-1 cos(z) cos(z + π n ) cos(z + 2π n ).... cos(z + (n -1)π n ) = (-1) n-1 2 2 n-1 n-1 m=0 cos(z + mπ n ).
Remark 3-7

Recall that for the odd case of n the formula of [7, p.249]. For

j = 1, 2, 3, 4 θ j (nv, nτ ) η n (τ ) η(nτ ) = θ j (v, τ ) 1≤m≤ n-1 2 θ j (v + m n , τ )θ j (v - m n , τ ).
In [6, p.272] there is another analogous formula always for n odd

θ j (v, τ n ) η n (τ ) η( τ n ) q 1-n 2 24n = θ j (v, τ ) 1≤m≤ n-1 2 θ j (v + mτ n , τ )θ j (v - mτ n , τ ).
For any integer n Enneper proved [6, p.248]

θ j (nv, nτ ) = s≥1 (1 -q 2sn ) (1 -q 2s ) n 0≤m≤n-1 θ j (v + m n , τ ), j = 1, 2, 3, 4.
Another alternative expression originally due to Jacobi :

θ j (nv, nτ ) = (q 2n ; q 2n ) ∞ (q 2 ; q 2 ) n ∞ 0≤m≤n-1 θ j (v + m n , τ ), j = 1, 2, 3, 4.

Connection with f j (v, τ ).

Recall the functions introduced in [START_REF] Chouikha | Functions related to Jacobi Theta Functions and applications I[END_REF] allowing us to represent the Jacobi theta functions :

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). These functions are connected to g j (v, τ )

Proposition 3-8

The following identities hold

f 1 (v, τ ) g 1 (v, τ ) = k≥1 sin [kπτ -πv] 2 sin [kπτ +πv] 2 , f 4 (v, τ ) g 4 (v, τ ) = k≥0 sin [(k+ 1 2 )πτ -πv] 2 sin [(k+ 1 2 )πτ +πv] 2 , f 2 (v, τ ) g 2 (v, τ ) = k≥1 sin [kπ(τ +1)-πv] 2 sin [kπ(τ +1)+πv] 2 , f 3 (v, τ ) g 3 (v, τ ) = k≥0 sin [(k+ 1 2 )π(τ +1)-πv] 2 sin [(k+ 1 2 )π(τ +1)+πv] 2 . f j (v, τ ) g j (v, τ ) = g j (-v, τ ) f j (-v, τ ) , j = 1, 2, 3, 4.
Proof of Proposition 3-8

Indeed, one has

f 4 (v, τ ) g 4 (v, τ ) = k≥0 1 - sin πv sin[(k+ 1 2 )πτ ] k≥0 sin[(k+ 1 2 )πτ +πv] sin[(k+ 1 2 )πτ +πv] = k≥0 sin[(k + 1 2 )πτ ] -(sin πv) k≥0 sin[(k + 1 2 )πτ ] = k≥0 sin [(k+ 1 2 )πτ -πv] 2 sin [(k+ 1 2 )πτ +πv] 2 .
We prove in exactly the same way for other quotients fj (v,τ )

By Corollary 3-5 we may also other identities as

f 1 (v + nτ, τ ) f 1 (v, τ ) = n k=1 sin [kπτ +πv] 2 sin [kπτ -πv] 2 sin[kπτ + πv] . Remark 3-9
In [START_REF] Chouikha | Functions related to Jacobi Theta Functions and applications I[END_REF] we have highlighted the impact of the quotient fj (v,τ ) fj (v+1,τ )

allowing to establish modular relations. These quotients may also be written

fj (v,τ ) fj (v+1,τ ) = gj (v,τ )
gj (v+1,τ ) .

Infinite products for sin p and cos p

In this part we are interested in the relations between q-trigonometry and innite product, as a consequence of the previous study on the developments of the Jacobi theta functions. Gosper [START_REF] Gosper | Experiments and discoveries in q-trigonometry[END_REF] introduced q-analogues to sin and cos

sin q (v) = q ( v-1 2 ) 2 k≥1 (1 -q 2n-2v )(1 -q 2n+2v-2 ) (1 -q 2n-1 ) 2 = (q 2-2v ; q 2 ) ∞ (q 2v ; q 2 ) ∞ (q; q 2 ) 2 ∞ q (v-1 2 ) 2 , cos q (v) = q ( v-1 2 ) 2 k≥1 (1 -q 2n-2v-1 )(1 -q 2n+2v-1 ) (1 -q 2n-1 ) 2 = (q 1-2v ; q 2 ) ∞ (q 1+2v ; q 2 ) ∞ (q; q 2 ) 2 ∞ q v 2 ,
where (a; q) = k≥0 (1 -aq) k and gave relations between them and theta function

sin q (v) = θ 1 (v, τ ) θ 1 ( 1 2 , τ ) , cos q (v) = θ 2 (v, τ ) θ 2 (0, τ ) ,
where τ = -1 τ . It is proved that lim q→1 sin q (v) = sin(v), lim q→1 cos q (v) = cos(v).

4.1.

Allusive remark on q-trigonometry in Enneper's works. Notice that these q-analogues to trigonometric functions was already known to the ancients mathematicians. Indeed, Enneper after developing a ne analysis introduced [7, p.114] the quotients

θ 2 (x, q) θ 2 (0, q) = θ 2 (x, q 4 ) θ 2 (0, q 4 ) F (x, q 4 ), θ 1 (x, q) θ 2 (0, q) = θ 1 (x, q 4 ) θ 2 (0, q 4 ) F 1 (x, q 4 )
where

F (x, q 4 ) = θ 3 (x, q 4 ) θ 3 (0, q 4 ) (θ 2 (x, q 4 )) 2 + (θ 3 (x, q 4 )) 2 (θ 2 (0, q 4 )) 2 + (θ 3 (0, q 4 )) 2 , F 1 (x, q 4 ) = θ 4 (x, q 4 ) θ 3 (0, q 4 ) (θ 1 (x, q 4 )) 2 + (θ 4 (x, q 4 )) 2 (θ 2 (0, q 4 )) 2 + (θ 3 (0, q 4 )) 2 .
Moreover, in each of these equations, put q 4 , q 16 , q 24 ... instead of q, multiply the results obtained in this way with each other. That gives

θ 1 (x, q) θ 2 (0, q) = θ 1 (x, q 16 ) θ 2 (0, q 16 ) F 1 (x, q 4 )F 1 (x, q 16 ) = ...., θ 2 (x, q) θ 2 (0, q) = θ 2 (x, q 16
) θ 2 (0, q 16 ) F (x, q 4 )F (x, q 16 ) = .....

If one goes over to the limit, it follows θ 2 (x, q) θ 2 (0, q) = lim l→∞ θ 2 (x, q l ) θ 2 (0, q l ) F (x, q 4 )F (x, q 16 )F (x, q 24 )...., θ 1 (x, q) θ 2 (0, q) = lim l→∞ θ 1 (x, q l ) θ 2 (0, q l ) F (x, q 4 )F (x, q 16 )F (x, q 24 )....

for l unlimited integer. Since q < 1 , therefore lim →∞ q l = 0 and thus lim l→∞ θ 2 (x, q l ) θ 2 (0, q l ) = cos(x).

Changing x by x + π 2 then lim l→∞ θ 1 (x, q l ) θ 2 (0, q l ) = sin(x).

These remarks of Enneper undoubtedly deserve to be developed and exploited, since cos q and sin q appear in fact as limit product of a combination of theta functions.

Replace for example τ by τ = -1/τ then q = e -iπ τ . We thus obtain by using modern notations where q = e iπτ

θ 2 (x, τ ) θ 2 (0, τ ) = θ 2 (x, 4τ ) θ 2 (0, 4τ ) F (x, q 4 ), θ 1 (x, τ ) θ 2 (0, τ ) = θ 1 (x, 4τ ) θ 2 (0, 4τ ) F 1 (x, q 4 ).
So, cos q (x) = cos q 1/4 F (x, q 4 ), sin q (x) = sin q 1/4 F 1 (x, q 4 ).

Or equivalently, cos q 4 (x) = cos q (x)F (x, q ), sin q 4 (x) = sin q (x)F 1 (x, q ). Theorem 4-1

sin q (v) = (sin πv) k≥1 (tan kπτ ) 2   1 - sin πv sin kπ τ 2   = (sin πv) k≥1 sin(πv + kπ τ ) sin( kπ τ -πv) (cos kπ τ ) 2 , cos q (v) = (cos πv) k≥1   1 - sin πv cos kπ τ 2   = (cos πv) k≥1 cos(πv + kπ τ ) cos(πv -kπ τ ) (cos kπ τ ) 2
.

sin q (v) and cos q (v) are dened in the band | Imv |< τ.

Proof of Theorem 4-1

Notice that since θ 1 (v + 1 2 , τ ) = θ 2 (v, τ ) and by Corollary 3-5 one gets

θ 1 (v + 1 2 , τ ) (π cos πv) θ 1 (0, τ ) = θ 2 (v, τ ) (π cos πv) θ 1 (0, τ ) = θ 2 (0, τ ) πθ 1 (0, τ ) k≥1 cos[kπτ + πv] cos[kπτ -πv] (cos[kπτ ]) 2 = k≥1 cos[kπτ + πv] cos[kπτ -πv] (sin[kπτ ]) 2 .
We thus deduce

θ 2 (0, τ ) = π θ 1 (0, τ ) k≥1 (cot kπτ ) 2 .
So, we nd again by another way the innite product as a q-series

π sin q (0) = πθ 1 (0, τ ) θ 2 (0, τ ) = k≥1 (tan kπτ ) 2 = π n≥0 (-1) n (2n + 1)q ( n+1 2 ) 2 n≥0 q ( n+1 2 ) 2 = k≥1 1 -q 2k 1 + q 2k 2 .
On the other hand,

θ 1 (v + 1 2 , τ ) θ 2 (0, τ ) = θ 2 (v, τ ) θ 2 (0, τ ) = θ 1 (v + 1 2 , τ ) π θ 1 (0, τ ) k≥1 (cot kπτ ) 2 = cos πv k≥1 1 - sin πv cos kπτ 2 ,
and

θ 2 (v, τ ) = (cos πv) θ 2 (0, τ ) k≥1 1 - sin πv cos kπτ 2 = (cos πv)π θ 1 (0, τ ) k≥1 (cot kπτ ) 2 k≥1 1 - sin πv cos kπτ 2 .
Thus,

θ 2 (v, τ ) θ 2 (0, τ ) = (cos πv) k≥1 1 - sin πv cos kπτ 2 = (cos πv) k≥1 (sin πv) 2 -(cos kπτ ) 2 (sin kπτ ) 2 , θ 1 (v, τ ) θ 2 (0, τ ) = (sin πv) k≥1 (tan kπτ ) 2 1 - sin πv sin kπτ = (sin πv) k≥1 (tan kπτ ) 2 (cos πv) 2 -(cos kπτ ) 2 (sin kπτ ) 2 .
Finally, we have

sin q (v) = (sin πv) k≥1 (tan kπτ ) 2   1 - sin πv sin kπ τ 2   = (sin πv) k≥1 (tan kπτ ) 2 (cos πv) 2 -cos kπ τ 2 (sin kπ τ ) 2 , cos q (v) = (cos πv) k≥1   1 - sin πv cos kπ τ 2   = (cos πv) k≥1 (sin πv) 2 -cos kπ τ 2 (sin kπ τ ) 2
.

Remark sin q (v), cos q (v) are related to the functions g j (v, τ ) dened above in Part 2 where τ = -1

τ sin q (v) = (sin πv) g 1 (v, τ ) g 1 (-v, τ ) g 1 (1/2, τ ) g 1 (-1/2, τ ) , cos q (v) = (cos πv) g 2 (v, τ ) g 2 (-v, τ ).
Taking the logarithmic derivative of functions g j (v, τ ), j = 1, 2 (see above Corollary 2-6)

g 1 (v, τ ) = k≥1 sin[kπτ + πv] sin[kπτ ] , g 2 (v, τ ) = k≥1 cos[kπτ + πv] cos[kπτ ] ,
we derive the following Corollary 4-2

Let q = e iπτ , and τ = -1 τ . Then the logarithmic derivative of sin q (v), cos q (v) are

cos q cos q (v) = 1 cos q ∂ cos q ∂v (v) = -(tan πv) + k≥1 tan[kπτ + πv] + tan[kπτ -πv]
= -(tan πv) + k≥1 sin kπτ cos kπτ -sin πv + sin kπτ cos kπτ + sin πv ,

sin q sin q (v) = 1 sin q ∂ sin q ∂v (v) = (cot πv) - k≥1 cot[kπτ + πv] + cot[kπτ -πv],
= (cot πv) + k≥1 sin kπτ cos kπτ -cos πv + sin kπτ cos kπτ + cos πv , sin q and cos q are dened in the band | Imv |< τ.

Theorem 4-3

Let q = e iπτ , and τ = -1 τ . Then the following identities hold

(i) sin q (2v) = 8 k≥1 cos( kπ τ ) (cos( kπ 2τ )) 2 2 sin q 2 (v) cos q 2 (v), (ii) tan q (2v) = sin q (2v) cos q (2v) = tan(2v) k≥1 tan(2πv + kπ τ ) tan( kπ τ -2πv). (iii) tan q 2 (v) = tan(v) k≥1 tan(πv + kπ 2τ ) tan( kπ 2τ -πv).
(iv) tan q (2v)

tan q 2 (v) = k≥1 4 (cos( kπ 2τ )) 4 (cos( kπ τ )) 2 θ 2 2 (v, τ 2 ) θ 2 (2v, τ ) = 4 (cos q 2 (v)) 2 cos q (2v) k≥1 (cos kπ 2τ ) 4 (cos kπ τ ) 2 .
Proof of Theorem 4-3

Notice that after developing and reducing the following expressions

1 - (cos πv)
Observe now from these trigonometric identities we can deduce sin q (2v) = (sin π2v)

k≥1 sin(2πv + kπ τ ) sin( kπ τ -2πv) (cos kπ τ ) 2 , sin q 2 (v) = (sin πv) k≥1 sin(πv + kπ 2τ ) sin( kπ 2τ -πv) (cos kπ 2τ ) 2 , cos q 2 (v) = (cos πv) k≥1 cos(πv + kπ 2τ ) cos(πv -kπ 2τ ) (cos kπ 2τ ) 2 . Therefore sin q 2 (v) cos q 2 (v) = (sin πv)(cos πv) k≥1 sin(πv + kπ 2τ ) sin( kπ 2τ -πv) (cos kπ 2τ ) 2 cos(πv + kπ 2τ ) cos(πv -kπ 2τ ) (cos kπ 2τ ) 2 = 1 2 (sin π2v) k≥1 sin(2πv + kπ τ ) sin( kπ τ -2πv) 4(cos kπ 2τ ) 4 = 1 8 k≥1 (cos( kπ 2τ )) 2 (cos( kπ τ )) 2 sin q (2v).
We then nd again by another way the Gosper's identity

sin q (2v) = Π q Π q 2 sin q 2 (v) cos q 2 (v) = θ 2 1 ( π 2 , τ 2 ) θ 2 1 ( π 4 , τ 2 ) sin q 2 (v) cos q 2 (v),
where

Π q = q 1 4 (q 2 ; q 2 ) 2 ∞ (q; q 2 ) 2 ∞ , Π q Π q 2 = 8 k≥1 cos( kπ τ ) (cos( kπ 2τ )) 2 2 .
To prove (iv) it suces to develop and reduce

tan π(2v + k τ ) tan π(-2v + k τ ) tan π(v + k 2τ ) tan π(-v + k 2τ ) = sin π( 2v + k τ ) sin π( -2v + kπ τ ) cos π(v + k 2τ ) cos π(-v + k 2τ ) cos π( 2v + k τ ) cos π( -2v + k τ ) sin π(v + k 2τ ) sin π(-v + kπ 2τ ) = 4 cos π(v + k 2τ ) 2 cos π(-v + k 2τ ) 2 cos π(2 v + k τ ) cos π(-2 v + k τ ) . Therefore k≥1 4 cos π(v + k 2τ ) 2 cos π(-v + k 2τ ) 2 cos π(2 v + k τ ) cos π(-2 v + k τ ) = 4 (cos q 2 (v)) 2 cos q (2v) cos(2πv) (cos(πv)) 2 k≥1 (cos kπ 2τ ) 4 (cos kπ τ ) 2 .
On the other hand this could be re-written

4 k≥1 cos π(v + k 2τ ) 2 cos π(-v + k 2τ ) 2 cos π(2 v + k τ ) cos π(-2 v + k τ ) = 4 k≥1 cos kπ 2τ 2 -1 + (cos (πv)) 2 2 cos kπ τ 2 -4 (cos (πv)) 2 + 4 (cos (πv)) 4 = 4 k≥1 cos kπ 2τ 2 -(sin (πv)) 2 2 cos kπ τ 2 -(sin (2π v)) 2 = k≥1 4 (cos( kπ 2τ )) 4 (cos( kπ τ )) 2 cos(2πv) (cos(πv)) 2 θ 2 2 (v, τ 2 ) θ 2 (2v, τ )
.

Theorem 4-4

Let q = e iπτ , and τ = -1 τ . Then the following hold

(tan q 2 (v)) 2 = Π q 2 tan q (2v) Π q + Π q 2 tan q (2v) = 1- k≥1 cot( kπ τ ) cot( kπ 2τ ) 2 Π q 4Π q 2 cos kπ τ 2 -(sin (2 v)) 2 cos kπ 2τ 2 -(sin (v)) 2 2 , cos q (2v) (cos q 2 (v)) 2 = k≥1 cot( kπ τ ) cot( kπ 2τ ) 2 Π q Π q 2 cos 2kπ τ + cos (4 v) 2 cos kπ τ + (cos (2v)) 2 ,
where

Π q = 1 4 θ 2 1 ( π 2 , τ 2 ) Π q 2 = 1 4 θ 2 1 ( π 4 , τ 2
).

Proof of Theorem 4-4

Starting from the Gosper's identities

sin q (2v) sin q 2 (v) = Π q Π q 2 cos q 2 (v), cos q (2v) = cos q 2 (v) -sin q 2 (v),
we get

tan q (2v) tan q 2 (v) = sin q (2v) cos q 2 (v) cos q (2v) sin q 2 (v) = Π q Π q 2 (cos q 2 (v)) 2 cos q (2v) = Π q Π q 2 1 1 -(tan q 2 (v)) 2 .
On the other side one has

tan q (2v) tan q 2 (v) = 4 k≥1 cot( kπ τ ) cot( kπ 2τ ) 2 cos kπ 2τ 2 -(sin (v)) 2 2 cos kπ τ 2 -(sin (2 v)) 2 . cos q (2v) (cos q 2 (v)) 2 = 1 -(tan q 2 (v)) 2 = Π q 4Π q 2 k≥1 cot( kπ τ ) cot( kπ 2τ ) 2 cos kπ τ 2 -(sin (2 v)) 2 cos kπ 2τ 2 -(sin (v)) 2 2
.

By Enneper's terminology ( [START_REF] Enneper | Elliptische Functionen: Theorie und Geschichte[END_REF] p.114)

F (x, q) = θ 3 (x, q) θ 3 (0, q) (θ 2 (x, q)) 2 + (θ 3 (x, q)) 2 (θ 2 (0, q)) 2 + (θ 3 (0, q)) 2 , F 1 (x, q) = θ 4 (x, q) θ 3 (0, q) (θ 1 (x, q)) 2 + (θ 4 (x, q)) 2 (θ 2 (0, q)) 2 + (θ 3 (0, q)) 2 , then F 1 (x, q) F (x, q) = θ 4 (x, q) θ 3 (x, q) (θ 2 (x, q)) 2 + (θ 3 (x, q)) 2 (θ 1 (x, q)) 2 + (θ 4 (x, q)) 2 .
We deduce from Corollary 4-3

tan q 4 (x) = tan(v) F 1 (v, q) F (v, q) = tan(v) k≥1 tan(πv + kπ 4τ
) tan( kπ 4τ -πv).

We thus obtain the following theta identity Corollary 4-5

Let q = e iπτ , and τ = -1 τ . The following equalities hold Let q = e iπτ , and τ = -1 τ . The following innite products hold

k≥1 cos[π(v + k 2τ )] cos[π(-v + k 2τ )] cos[π( v + k τ )] cos[π(-v + k τ )] = k≥0 cos[π(v+ (k + 1 2 ) τ )] cos[π(-v+ (k + 1 2 ) τ )], k≥1 sin[π(v + k 2τ )] sin[π(-v + k 2τ )] sin[π( v + k τ )] sin[π(-v + k τ )] = θ 4 (0, τ ) θ 3 (0, τ ) k≥0 sin[π(v+ (k + 1 2 ) τ )] sin[π(-v+ (k + 1 2 ) τ )], tan(πv) k≥1 tan[π(v + k 2τ )] tan[π(-v + k 2τ )] tan[π( v + k τ )] tan[π(-v + k τ )] = θ 4 (0, τ ) θ 3 (0, τ ) k≥1 tan[π(v + (k + 1 2 ) τ )] tan[π(-v + (k + 1 2 ) τ )].
Proof Indeed, (following Gosper notation) it suces to notice that ccs q (v) = cos q 2 (v) cos q (v) = k≥1 cos π(v + k 2τ ) cos π(-v + k 2τ ) cos π( v + k τ ) cos π(-v + k τ )

, ssn q (v) = sin q 2 (v) sin q(v) = k≥1 sin π(v + k 2τ ) sin π(-v + k 2τ ) sin π( v + k τ ) sin π(-v + k τ )

,

These functions can also be re-written ccs q (v) = θ 3 (v, τ ) θ 3 (0, τ ) , ssn q (v) = θ 4 (v, τ ) θ 3 (0, τ ) .

Therefore, we deduce the identities by Corollary 3-3 since 4.2. n-order transformations and q-trigonometry.

θ 3 (v,
Notice that Gosper [9, p.92] stated the following identity for any integer n q n 2 -1 12

(q, q 2 ) 2 ∞ (q n , q 2n ) 2n

∞ sin q nπv = n-1 k=0 sin q n (v + k n ).
In [START_REF] Bachraoui | On a theta product of Jacobi and its applications to q-gamma products[END_REF] the authors point out that this formula is in fact equivalent to the Jacobi identity :

θ j (nv, nτ ) = s≥1

(1 -q 2sn (1 -q 2s ) n 0≤m≤(n-1)

θ j (v + m n , τ ).
Another consequence of the innite product expansions deduced from Theorem 3-6 is the n-order multiplication formulas for sin q (v), cos q (v) where in addition we distinguish the two cases n odd, n even Theorem 4-7

Let n be any integer then :

(i) -The following identity holds sin q (nv) = πθ 1 (0, nτ ) θ 2 (0, nτ )

cos n q (v + m n ).
By Corollary 2-6 and Theorem 4-2 we deduce the logarithmic derivative of sin q (nv), cos q (nv). See also [START_REF] Bachraoui | On a theta product of Jacobi and its applications to q-gamma products[END_REF]Theorem 1] Corollary 4-8

Let n be any odd integer then n cos q cos q (nv) = n-1 m=0 cos q n cos q n (v + mπ n ),

n sin q sin q (nv) = -n cos q cos q ( π 2 -nv) = n-1 m=0 sin q n sin q n (v + mπ n ).
Indeed, this corollary is a consequence of Theorem 4-2 and trigonometric identities

n tan nv = n-1 m=0 tan(v + mπ n ), n cot nv = n-1 m=0 cot(v + mπ n ).
Notice that lim q→1 cos q cos q (v) = tan(v), lim q→1 sin q sin q (v) = cot(v).

2 sin [kπτ +πv] 2 , f 4 (v, τ ) g 4 (

 2244

θ 4 ((θ 2

 42 v, τ ) θ 3 (v, τ ) (θ 2 (v, τ )) 2 + (θ 3 (v, τ )) 2 (θ 1 (v, τ )) 2 + (θ 4 (v, τ )) (v, τ )) 2 + (θ 3 (v, τ )) 2 (θ 1 (v, τ )) 2 + (θ 4 (v, τ )) 2 = k≥1 tan(πv + kπ 4τ ) tan( kπ 4τ -πv) tan(πv + kπ τ ) tan( kπ τ -πv).Corollary 4-6

.

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

	Remark 3-4