Annales Mathématiques Blaise Pascal

Bruno Deschamps

Ivan Suarez Atias

Groupes totaux

Ivan Marin

On the representation theory of braid groups

Georges-Henri Cottet

Jean-Matthieu Etancelin

Franck Perignon

Christophe Picard

Florian De Vuyst

Christophe Labourdette

Is GPU the future of Scientific Computing ?

Mickaël Crampon

Ludovic Marquis

Un lemme de Kazhdan-Margulis-Zassenhaus pour les géométries de Hilbert

Christian Kassel

Examples of polynomial identities distinguishing the Galois objects over finite-dimensional Hopf algebras

Stéphanie Allassonnière

Jérémie Bigot

Joan Alexis Glaunès

Florian Maire

Frédéric J.P. Richard

Statistical models for deformable templates in image and shape analysis

Stefan Czekalski

ANNALES MATHÉMATIQUES
BLAISE PASCAL
Stefan Czekalski
The new properties of the theta functions
Volume 20, no 2 (2013), p. 391-398.
<http://ambp.cedram.org/item?id=AMBP_2013__20_2_391_0>
© Annales mathématiques Blaise Pascal, 2013, tous droits réservés.
L’accès aux articles de la revue « Annales mathématiques Blaise Pas-
cal » (http://ambp.cedram.org/), implique l’accord avec les condi-
tions générales d’utilisation (http://ambp.cedram.org/legal/). Toute
utilisation commerciale ou impression systématique est constitutive
d’une infraction pénale. Toute copie ou impression de ce fichier doit
contenir la présente mention de copyright.
Publication éditée par le laboratoire de mathématiques
de l’université Blaise-Pascal, UMR 6620 du CNRS
Clermont-Ferrand — France
cedram
Article mis en ligne dans le cadre du
Centre de diffusion des revues académiques de mathématiques
http://www.cedram.org/
Annales mathématiques Blaise Pascal 20, 391-398 (2013)
The new properties of the theta functions
Stefan Czekalski
Abstract
It is shown, that the function
H(x) =
∞∑
k=−∞
e−k
2x
satisfies the relation
H(x) =
∞∑
n=0
(2π)2n
(2n)! H
(n)(x).
The theta function is defined by the following equation:
Θ0 =
∞∑
k=1
e−k
2πt t > 0. (1.1)
For computational purposes, it is convenient to introduce the function
Θ(x) = Θ0(
x
π
) =
∞∑
k=1
e−k
2x, x > 0.
Theorem 1.1. The following formula holds
∞∑
k=1
e−k
2x = 1√
π
∞∫
−∞
e−x−y
2
1− e2(−x+i
√
xy)dy. (1.2)
391
S. Czekalski
Proof. For x > 0 we have |e2(−x+i
√
xy| < 1, therefore
1
√
π
∞∫
−∞
e−x−y2
1−e2(−x+i
√
xy)dy
= 1√
π
∞∫
−∞
e−x−y
2
∞∑
k=0
e2(−x+i
√
xy)kdy
= 1√
π
∞∑
k=0
e−(2k+1)x
∞∫
−∞
e−y
2+2iy
√
xkdy
= 1√
π
∞∑
k=0
e−(2k+1)x
( ∞∫
−∞
e−y
2
cos2y
√
xk dy
+ i
∞∫
−∞
e−y
2
sin2y
√
xk dy
)
here the first integral is equal
√
πe−k
2x ; second integral - zero; therefore
we have
= 1√
π
∞∑
k=0
e−(k
2+2k+1)x√π =
∞∑
k=1
e−k
2x.

By inserting x = 1 into (1.2), we obtain
∞∑
k=1
e−k
2 = 1√
π
∞∫
−∞
e−1−y
2
1− e2(−1+iy)
dy . (1.3)
In the same way we prove the relation
∞∑
k=1
(−1)k−1e−k2x = 1√
π
∞∫
−∞
e−x−y
2
1 + e2(−x+iy
√
x)dy . (1.4)
When x = 1 we have
∞∑
k=1
(−1)k−1e−k2 = 1√
π
∞∫
−∞
e−1−y
2
1 + e2(−1+iy)
dy . (1.5)
392
The new properties of the theta functions
By virtue of (1.2) we easy find the formula
∞∑
k=1
ke−k
2x = 1√
π
∞∫
−∞
e−x−y
2 [2− e2(−x+iy
√
x)]
[1− e2(−x+iy
√
x)]2
dy . (1.6)
By taking x = 1, we obtain
∞∑
k=1
ke−k
2 = 1√
π
∞∫
−∞
e−1−y
2 [2− e2(−1+iy)]
[1− e2(−1+iy)]2
dy . (1.7)
Let us consider the following formula
∞∑
k=0
e−k
2xe−2axk = 1√
π
∞∫
−∞
e−y
2
1− e−2ax+2iy
√
x
dy a > 0. (1.8)
Let h = e−x and z = e−ax be, then
∞∑
k=0
hk
2
z2k = 1√
π
∞∫
−∞
e−y
2
1− z2e2iy
√
− lnh
dy . (1.9)
If we introduce h = e−x and z = e−ax−
1
2x in (1.8) then we obtain
∞∑
k=0
h(k−
1
2 )
2
z2k−1 = 1√
π
∞∫
−∞
e−y
2
zh−
1
4
1− z2h−1e2iy
√
− lnh
dy . (1.10)
We define the function H(x) = 2Θ(x) + 1.
Theorem 1.2. H(x) can be expanded in a series
H(x) = e−x
∞∑
n=0
(−1)n (2x)
2n
(2n)! H
(n)(x) . (1.11)
Proof. Let us consider the functions
Θ(x) =
∞∑
k=1
e−k
2x and v(x) =
∞∑
k=1
ke−k
2x.
We evaluate the sums
∞∑
n=0
(−1)n (2x)
2n
(2n)! Θ
(n)(x) =
∞∑
k=1
e−k
2x[1 + k
2(2x)2
2! +
k4(2x)4
4! + · · · ]
393
S. Czekalski
∞∑
n=0
(−1)n (2x)
2n+1
(2n+ 1)!v
(n)(x) =
∞∑
k=1
e−k
2x[k2x1! +
k3(2x)3
3! +
k5(2x)5
5! + · · · ].
The above two series are absolutely convergent, therefore we can change
the order of the terms. Addition and subtraction of both sides of equations
yields
∞∑
n=0
(−1)n (2x)
2n
(2n)! Θ
(n)(x) +
∞∑
n=0
(−1)n (2x)
2n+1
(2n+ 1)!v
(n)(x)
=
∞∑
k=1
e−k
2xe2kx = exΘ(x) + ex
∞∑
n=0
(−1)n (2x)
2n
(2n)! Θ
(n)(x) +
∞∑
n=0
(−1)n (2x)
2n+1
(2n+ 1)!v
(n)(x)
=
∞∑
k=1
e−k
2xe−2kx = exΘ(x)− 1 .
We add and subtract the equations once again. Thus we get
∞∑
n=0
(−1)n (2x)
2n
(2n)! Θ
(n)(x) = exΘ(x) + 12e
x − 12
∞∑
n=0
(−1)n (2x)
2n+1
(2n+ 1)!v
(n)(x) = 12e
x + 12 .
Introduction H(x) = 2Θ(x) + 1 and U(x) = 2v(x) yields
∞∑
n=0
(−1)n (2x)
2n
(2n)! H
(n)(x) = exH(x) (1.12)
∞∑
n=0
(−1)n (2x)
2n+1
(2n+ 1)!U
(n)(x) = ex + 1 . (1.13)

394
The new properties of the theta functions
The following relations for H(x) and U(x) hold
∞∑
n=0
(−1)n (2x)
2n+1
(2n+ 1)!H
(n+1)(x) = exH(x) (1.14)
∞∑
n=0
(−1)n (2x)
2n
(2n)! U
(n)(x) = exU(x) + ex. (1.15)
Proof of these relations is identical as (1.12) and (1.13).
We give now a generalization of (1.12). Let us consider the following
extension of Θ(x)
Θ1(x) =
∞∑
k=1
e−k
2xe−ka x > 0, a ∈ R.
Then the more general version of (1.12) is equal
chaH1(x)− sha = e−x
∞∑
n=0
(−1)n (2x)
2n
(2n)! H
(n)
1 (x) (1.16)
where H1(x) = 2Θ1(x) + 1.
The proof of (1.16) is patterned after this of the Theorem 1.2.
By virtue of the relation
∞∑
k=−∞
e−k
2xe−ka =
∞∑
k=1
e−k
2xe−ka +
∞∑
k=1
e−k
2xeka + 1
it is easy to verify that, Ψ(x) =
∞∑
k=−∞
e−k
2xe−ka satisfies (1.16) with the
member sha removed.
Theorem 1.3. The formula (1.12) can be written in the form
H(x) =
∞∑
n=0
(2π)2n
(2n)! H
(n)(x) . (1.17)
To prove theorem 0.3 we need the lemma
Lemma 1.4.
[x−
1
2H(x)](n) = (−1)n (2n)!22n
n∑
k=0
22k
(n− k)!
x−n−k−
1
2
(2k)!
dk
d( 1x)k
H( 1
x
). (1.18)
395
S. Czekalski
Proof. Proof of Lemma 1.4 (by induction) The case n = 1 is obvious. We
suppose, that (1.18) is true for any n and differentiate it.
[x−
1
2H(x)](n+1)
= (−1)n+1 (2n)!22n
n∑
k=0
22k
(n− k)!
(n+ k + 12)x
−n−k− 32
(2k)!
dk
d( 1x)k
H( 1
x
)
+ (−1)n+1 (2n)!22n
n∑
k=0
22k
(n− k)!
x−n−k−
5
2
(2k)!
dk+1
d( 1x)k+1
H( 1
x
)
= (−1)n+1 (2n)!22n [
2n+ 1
2n! x
−n− 32H( 1
x
)
+
n∑
k=1
22k
(n− k)!
(n+ k + 12)x
−n−k− 32
(2k)!
dk
d( 1x)k
H( 1
x
)]
+ (−1)n+1 (2n)!22n
n∑
k=0
22k−2
(n+ 1− k)!
x−n−k−
3
2
(2k − 2)!
dk
d( 1x)k
H( 1
x
)
+ (−1)n+1x−2n−
5
2
dn+1
d( 1x)n+1
H( 1
x
)
and consequently
[x−
1
2H(x)](n+1) = (−1)n+1 (2n+ 2)!22n+2
x−n−
3
2
(n+ 1)!H(
1
x
)
+
n∑
k=1
22k
(n+ 1− k)!
x−n−k−
3
2
(2k)!
dk
d( 1x)k
H( 1
x
)
+ (−1)n+1x−2n−
5
2
dn+1
d( 1x)n+1
H( 1
x
)
therefore
[x−
1
2H(x)](n+1) =
(−1)n+1x−n−
3
2
(2n+ 2)!
22n+2
n+1∑
k=0
22k
(n+ 1− k)!
x−k
(2k)!
dk
d( 1x)k
H( 1
x
).

396
The new properties of the theta functions
Proof. Proof of the Theorem 1.3 The known functional relation
H(π2x) = π−
1
2x−
1
2H( 1
x
)
we apply to (1.12) and obtain
H(π2x) = π−
1
2 e−π
2x
∞∑
n=0
(−1)n (2x)
2n(π2)n
(2n)! [x
− 12H( 1
x
)](n)
We use now (1.18) and have
H(π2x) = π−
1
2x−
1
2 e−π
2x
∞∑
n=0
n∑
k=0
(2π)2k
(2k)!
(π2x)n−k
(n− k)!
dk
d( 1x)k
H( 1
x
)
and consequently
H( 1
x
) = e−π2x(
∞∑
n=0
(π2x)n
n! H(
1
x
) +
∞∑
n=1
(π2x)n−1
(n− 1)!
(2π)2
2!
d
d( 1x)
H( 1
x
)
+
∞∑
n=2
(π2x)n−1
(n− 2)!
(2π)4
4!
d2
d( 1x)2
H( 1
x
) + · · · )
therefore
H( 1
x
) =
∞∑
n=0
(2π)2n
(2n)!
dn
d( 1x)n
H( 1
x
). 
If we replace 1x by x we obtain (1.17).
The following formula for Θ1(x) holds
∞∑
k=1
e−k
2xe−ka =
∞∑
n=0
(−1)n
n! x
n( 1
ea − 1)
(2n)
a > 0. (1.19)
Proof. We transform the right-hand side of (1.19) and have
∞∑
n=0
(−1)n
n! x
n(
∞∑
k=1
e−ka)(2n) =
∞∑
k=1
e−ka
∞∑
n=0
(−1)n
n! (k
2x)n
=
∞∑
k=1
e−k
2xe−ka.
397
S. Czekalski
Using the formula (1.19) with a = 2x, we find following expression for the
theta function [1, 2]
Θ(x) = e−x[1 +
∞∑
n=0
(−1)n
n! (
x
4 )
n( 1
e2x − 1)
(2n)]. (1.20)

References
[1] R. Bellman – A Brief Introduction to Theta Functions, Hall, Rine-
hart and Winston, New York, 1961.
[2] A. Krazer – Lehrbuch der Theta - Funktionen, Chelsea, New York,
1971.
Stefan Czekalski
Ul. Marszalkowska 1 m 80
00-624 Warszawa
Poland
398


The new properties of the theta functions

Ibrahim Assem

Christophe Reutenauer

Mutating seeds: types $\mathbb{A}$ and $\widetilde{\mathbb{A}}$.

P. Marios Petropoulos

Pierre Vanhove

Gravity, strings, modular and quasimodular forms

Min Ho Lee

Quasimodular forms and quasimodular polynomials

Not a member yet