On functions of Jacobi and Weierstrass (I)

0.26 MB·English

by  Yu. V. Brezhnev

ON FUNCTIONS OF JACOBI AND WEIERSTRASS (I)
YU.V.BREZHNEV
Abstract. We present some new results in theory of classical θ-functions of Jacobi and σ-
6
functions of Weierstrass: ordinarydifferential equations (dynamical systems) and series expan-
0
sions. Thepaperisbasicallyorganizedasastreamofnewformulasandconstitutes,inre-casted
0
form,somepartofauthor’sownhandbookforelliptic/modularfunctions.
2

n
a
J

9 1. Introduction
1
Theta-functions of Jacobi and the basis of Weierstrassian functions σ,ζ,℘,℘ occur in numerous
′
] theories and applications. Since their arising, this field was a subject of intensive studies and
A
towards the end of XIX century the state of the art in this field could be characterized about as
C
follows. Ifsomeformulaisnotofparticularinterestthenithasalreadybeenobtained. Themajority
.
h ofresultswereobtainedinworksofJacobiandWeierstrassthemselvesandtheircontemporariesand
t followers (Hermite, Kiepert, Hurwitz, Halphen, Frobenius, Fricke, and others). There is extensive
a
m bibliography, including handbooks, on the theory of elliptic/modular and related functions. Most
thorough works are four volumes by Tannery & Molk [7], two volume set by Halphen with a
[
posthumous edition of the third volume [3] and, of course, Werke by Weierstrass [8] and Jacobi

3 [6]. A large number of significant examples can be found in “A Course of Modern Analysis” by
v
Whittaker.
1
In this work we would like to present some results concerning the θ-functions of Jacobi and σ-
7
3 functionsofWeierstrass. Namely,powerseriesexpansionsofthefunctionsanddifferentialequations
1 to be obeyed by them. In a separate work we shall expound another way of construction of the
0 Jacobi–Weierstrass theory different from Jacobi’s approach of inversion of a holomorphic elliptic
6
integral and Weierstrass’s construction of elliptic functions as doubly-periodic and meromorphic
0
ones.
/
h Series expansion of elliptic/modular functions is a widely exploited apparatus which, due to
t
a analytic properties, makes it possible to get exact results. It is suffice to mention the function
m series for various θ-quotients, the famous number-theoretic q-series, and their consequences like
: “Moonshine Conjecture” and McKay–Thompsonseries.
v
Proposeddynamicalsystemsandtheir solutionshaveanindependentinterestbecausesolutions
i
X ofphysicallysignificantdifferentialequations,iftheseareexpressedthroughJacobi’sθ-ormodular
r functions, are consequences of the equations presented below. First of all we should mention the
a
theory of integrable nonlinear equations and dynamical systems. Differential properties of the
“modular part” of Jacobi’s functions are more transcendental and diverse. They gave rise to nice
applications over the last decade known as monopoles and dynamical systems of the Halphen–
Hitchin type [5], Chazy–Picard–Fuchsequations consideredin worksby Harnad, McKay, Ohyama
and others. Our interest in this part of the problem was initiated by a problem of analytic
description of moduli space of algebraic curves and differential-geometric structures on it. This
problem is completely open, so that the offered “differential technique” can be useful in those
situations where some classes of algebraic curves cover elliptic tori. We shall not touch this topic
and restrict ourselves only to “book-keeping” part of the apparatus. It is of interest in its own
right. Applications suggest themselves from the formulas.
1
2 YU.V.BREZHNEV
In sections 3–10 we shall be keeping the following rule. Unless one mentioned explicitly or if
nothing has been said about formula then the result is new1. In some cases, even though a result
is obvious, consequences can be nontrivial. In particular, this concerns the differential equations
on θ-functions with respect to their first argument. Differential identities sometimes occur in old
literature (see for example [7]), but the differential closure is needed. In most cases the derivation
of formulas is self-suggested once a given formula(s) has been displayed. For this reason we omit
proofs and restrict ourselves to comments to applications or elucidations some technical details
which are important for calculations.
Contentofsection2isstandardandpresentedheretofixnotation. Listofbibliography,modern
and classical, reduced to a minimum since, even in a shortened form, it would require mentioning
tens important works.
2. Definitions and notations
2.1. Functions of Jacobi. Fourfunctionsθ andtheirequivalentsundernotationwithchar-
1,2,3,4
acteristics are defined by the following series:
∞
−θ11 : θ1(x|τ)=−i (−1)ke(k+21)2πiτe(2k+1)πix
k=−∞
X
∞
θ10 : θ2(x|τ)= e(k+12)2πiτe(2k+1)πix
kX=−∞ .
∞
θ : θ (xτ)= ek2πiτe2kπix
00 3 |
kX=−∞
∞
θ : θ (xτ)= ( 1)kek2πiτe2kπix
01 4 | −
k=−∞
X
We shall use the designation: θ θ (xτ). Values of θ-functions under x = 0 are called ϑ-
k k
≡ |
constants: ϑ ϑ (τ)=θ (0τ). For handy formatting formulas we shalluse twofoldnotationfor
k k k
≡ |
θ-functions with characteristics:
∞ πi(k+α)2τ+2πi(k+α)(x+β)
θ α(xτ)=θ (xτ)= e 2 2 2 .
β | αβ |
k=−∞
(cid:2) (cid:3) X
We consider arbitrary integral characteristics and therefore the functions θ are always equal to
αβ
θ . Shifts by 1-periods lead to shifts of characteristics:
± 1,2,3,4 2
θα x+ n + mτ τ =θα+m(xτ) e–πim(x+β+2n+m4τ) , (n, m)=0, 1, 2,...
β 2 2 β+n | · ± ±
Twofold(cid:2) s(cid:3)(cid:0)hifts by 1-per(cid:12)(cid:12)io(cid:1)ds yi(cid:2)eld t(cid:3)he law of transformation of θ-function into itself:
2
θαβ(x+n+mτ τ)=θαβ(xτ) e–πi(m2τ+2mx)( 1)nα−mβ, (n, m)=0, 1, 2,....
| | · − ± ±
Value of any θ-function at any 1-period is a certain ϑ-constant with an exponential multiplier:
2
θα n + mτ τ =ϑα+m(τ) e–πim(β+2n+m4τ) , (n, m)=0, 1, 2,...
β 2 2 β+n · ± ±
(cid:2) (cid:3)(cid:0) (cid:12) (cid:1) (cid:2) (cid:3)
(cid:12)
1Thismeansthattheformulawasnotfoundintheliterature,butdoes notmeanthatitisreallynewone.
ON FUNCTIONS OF JACOBI AND WEIERSTRASS (I) 3
2.2. FunctionsofWeierstrass. Weierstrassiannotationσ(xω,ω′),℘(xω,ω′), ...andσ(x;g ,g ),
| | 2 3
℘(x;g ,g ), ... is the commonly accepted. Due to known relations of homogeneity for functions
2 3
σ, ζ, ℘, ℘′, the two parameters (ω,ω′) or (g ,g ) can be replaced by one τ = ω′ which is called
2 3 ω
modulus of elliptic curves. We shall designate that functions as follows
σ(xτ) σ(x1,τ), ζ(xτ) ζ(x1,τ) ℘(xτ) ℘(x1,τ) ℘′(xτ) ℘′(x1,τ).
| ≡ | | ≡ | | ≡ | | ≡ |
Weierstrassian invariants g , g and ∆ = g3 27g2 are functions of periods or modulus and
2 3 2 − 3
definedbyknownformulasofWeierstrass–Eisenstein. Theseseriesareentirelyunsuitedfornumeric
computations. Hurwitz, in his dissertation(1881), found a nice transition to function series. Such
series are most effective for computations:
g (τ)=20π4 1 + ∞ k3e2kπiτ , g (τ)= 7π6 1 ∞ k5e2kπiτ .
2 (240 1 e2kπiτ) 3 3 (504 − 1 e2kπiτ)
k=1 − k=1 −
X X
η-function of Weierstrass is defined by the formula η(τ) = ζ(11,τ) and the following formula is
|
used for computations:
η(τ)=2π2 1 ∞ e2kπiτ , η aτ+b =(cτ +d)2η(τ) πic(cτ +d),
24 − 1 e2kπiτ 2 cτ+d − 2
 Xk=1 −  (cid:16) (cid:17)
where the numbers (a,b,c,d(cid:0)) are integ(cid:1)er and ad bc = 1. Modular transformations are used not
  −
only in theory but also in practice since the value of modulus strongly affects the convergence of
the function series. Moving τ into a fundamental domain of modular group (the process is easily
automatized),oneobtainsthevaluesofτ withtheminimallyallowableimaginarypart (τ)= √3.
ℑ 2
At such “the most worst” point the series converge very fast.
Three functions of Weierstrass σ are defined by the following expressions:
λ
σ(xω,ω′)= eη(ω,ω′)2xω2 θλ+1 2xω ωω′ , where λ=1,2,3.
λ | · ϑ ω′
λ(cid:0)+1 ω(cid:12) (cid:1)
(cid:12)
The σ-function ofWeierstrass,asa functionof(x(cid:0), g2(cid:1), g3), satisfiesthe lineardifferentialequations
obtained by Weierstrass:
∂σ ∂σ ∂σ
x 4g 6g σ =0
∂x − 2 ∂g − 3 ∂g −
 2 3
. (1)
 ∂2σ 12g ∂σ 2g2 ∂σ + 1 g x2σ =0
∂x2 − 3 ∂g − 3 2 ∂g 12 2
2 3
These equations make itpossible to write down the recursive relation for coefficients Ck(g2,g3) of
the power series of the function σ:
σ(x;g ,g )=C x+C x3 + =x g2 x5 g3 x7+ , C =1, C =0 . (2)
2 3 0 1 3! ··· − 240 − 840 ··· 0 1
Two such recurrences are known. One is due to Halphen: (cid:0) (cid:1)
C = DC 1(k 1)(2k 1)g C , where D= 12g ∂ 2g2 ∂ . (3)
k − k−1− 6 − − 2 k−2 − 3 ∂g − 3 2 ∂g
2 3
The second was obtained by Weierstrass:
b b
∞
g m n x4m+6n+1 A = 1
σ(x;g ,g )= A 2 2g , 0,0 .
2 3 m,n 2 3 (4m+6n+1)! Am,n = 0 m<0, n<0
mX,n=0 (cid:16) (cid:17) (cid:0) (cid:1) (cid:26)
(cid:0) (cid:1)
16 1
A = (n+1)A +3(m+1)A (2m+3n 1)(4m+6n 1)A .
m,n m 2,n+1 m+1,n 1 m 1,n
3 − − − 3 − − −
4 YU.V.BREZHNEV
Recently, in connection with theory of Kleinian σ-functions, yet another recurrence was obtained
[2]. Among all the recurrences, Weierstrassian one is the least expendable. There are only multi-
plications of integers in it and its comparison efficiency rapidly grows under the growing of order
of the expansions. Weierstrass proves separately that the numbers A are integers.
m,n
2.3. Function of Dedekind. Due to coincidence the standard designations for Weierstrassian
function η(τ) and Dedekind’s one, we shall use, for the last one, the sign η(τ):
∞ ∞
η(τ)=e1π2iτ 1 e2kπiτ =e1π2iτ ( 1)ke(3k2+k)πiτ . (Ebuler (1748))
− −
Yk=1(cid:0) (cid:1) kX=−∞
b 1 dη i
Thereisadifferentialrelationbetweenthetwofunctions. Itisexpressedbytheformula = η.
η dτ π
b
3. Differentiation of functions σ,ζ,℘,℘′ b
Differentiations of Weierstrassian functions with respect to invariants are known [3, 7]. Rules
of transformations between differentiations with respect to (g ,g ) and (ω,ω′) are also known
2 3
(Frobenius–Stickelberger (1882)) [3]. Applying them, and assuming ω′ >0, one obtains
ℑ ω
∂σ = i ω′ ℘ ζ2 1 g x2 +2η′ xζ (cid:0)1 (cid:1)σ
∂ω −π − − 12 2 −
 ∂σ = i nω (cid:16)℘ ζ2 1 g x2 (cid:17)+2η x(cid:0)ζ 1 (cid:1)oσ ,
∂ω′ π − − 12 2 −
 ∂ζ i n (cid:16) 1 (cid:17) (cid:0) (cid:1)o
= ω′ ℘′+2ζ℘ g x +2η′ ζ x℘
∂ω −π − 6 2 −
 ∂ζ = i nω (cid:16)℘′+2ζ℘ 1g x(cid:17)+2η (cid:0)ζ x℘(cid:1)o ,
∂ω′ π − 6 2 −
 ∂℘ = 2in ω(cid:16)′ 2℘2+ζ℘′ 1g(cid:17) η′ (cid:0)2℘+x℘(cid:1)′o
∂ω π − 3 2 −
 ∂℘ = 2i nω (cid:16)2℘2+ζ℘′ 1g (cid:17) η (cid:0)2℘+x℘′(cid:1)o ,
∂ω′ −π − 3 2 −
∂℘′= i ω′ 6℘℘n′+(cid:16)12ζ℘2 g ζ η(cid:17)′ 6℘′(cid:0)+12x℘2(cid:1)og x
∂ω π − 2 − − 2
 ∂℘′ = i nω (cid:0)6℘℘′+12ζ℘2 g ζ(cid:1) η (cid:0)6℘′+12x℘2 g x(cid:1)o .
∂ω′ −π − 2 − − 2
Setting here(ω =1, ω′ =τ)non(cid:0)e obtains the dynamic(cid:1)al sys(cid:0)tem with a paramete(cid:1)rox:
∂σ = i ℘ ζ2+2η(xζ 1) 1 g x2 σ
∂τ π − − − 12 2
 n o
 ∂∂∂℘τζ == 2πii n℘2′℘+2+2η℘ζ′(+ζ 2℘x(ηζ)−x2ηη)℘− 161g2gxo .
∂τ − π − − − 3 2
 ∂∂℘τ′ = −6πi nn℘′(℘−η)+(cid:16)2℘2− 61g2(cid:17)(ζ −xoη)o
ON FUNCTIONS OF JACOBI AND WEIERSTRASS (I) 5
The function σ(xτ) satisfies one differential equation
|
∂2σ 2xη ∂σ πi∂σ + 2η+ 1 g x2 σ =0,
∂x2 − ∂x − ∂τ 12 2
(cid:16) (cid:17)
which is an analogue of heat equation for the function θ(xτ). All the differential equations in
|
variables x,ω,ω′,τ contain no the quantity g . Depending on representation (g ,g ), (ω,ω′) or τ,
3 2 3
it is either algebraicintegralof these equationsg =4℘3 g ℘ ℘′2 or determines a fix algebraic
3 − 2 −
relation between the variables. Each of the functions σ,ζ,℘,℘′(xτ) satisfies ordinary differential
|
equation of 4-th order with variable coefficients g (τ), η(τ) in the variable τ. These equations
2,3
have defied simplifications. By virtue of linear connection between function σ and θ, the last one
also satisfies ordinary differential equations.
4. Power series for σ-functions of Weierstrass
Weierstrassian recurrence has the following graphical interpretation:
H
n H
H(m 2,n+1)
.......................................H.....................................t`(−..m..H....−.....H1.....................`t.,...n..H..)[email protected].(........................m.dq..@,..n..()m@..................`t+@1,@n−@@1)
0 m
The explicitly grouped σ-series (2) has the form ([n] denotes an integral part of the number)
∞ k/2 x2k+1
σ(x;g2,g3)= ( 22k−5νA3ν−k,k−2ν ·g23ν−kg3k−2ν)(2k+1)! . (4)
Xk=0 ν=X[k/3]
Denote e ℘(ω ω,ω′). Then the functions σ satisfy the equations of Halphen:
λ ≡ λ| λ
∂σ ∂σ ∂σ
x λ 2e λ 4g λ = 0
∂x − λ ∂e − 2 ∂g
 λ 2
.
 ∂2σλ 4e2 2g ∂σλ 12 4e3 g e ∂σλ + e + 1 g x2 σ = 0
∂x2 − λ − 3 2 ∂e − λ − 2 λ ∂g λ 12 2 λ
λ 2
Herefromone obtain(cid:16)s analogue o(cid:17)f the recur(cid:0)rence (4): (cid:1) (cid:16) (cid:17)
∞
k/2 x2k
σλ(x;eλ,g2)= ( 2−νBk−2ν,ν ·eλk−2νg2ν)(2k)!,
Xk=0 Xν=0
B = 24(n+1)B +(4m 12n 5)B
m,n m 3,n+1 m 1,n
− − − − −
4(m+1)B 1(m+2n 1)(2m+4n 3)B .
m+1,n 1 m,n 1
−3 − − 3 − − −
All the coefficients B are integrals,and B =1 and B =0 under (m,n)<0. Weierstrass
m,n 0,0 m,n
wrote out recurrences for the functions Sλ = e21eλx2σλ in representation eλ, ελ = 3eλ2 − 41g2 .
(cid:0) (cid:1)
6 YU.V.BREZHNEV
One can build a universal series for the functions σ,σ . All the functions satisfy the differential
λ
equations
∂Ξ ∂Ξ ∂Ξ
x 2e 4g (1 ε)Ξ = 0
∂x − λ ∂e − 2 ∂g − −
 λ 2
, (5)
 ∂2Ξ 4e2 2g ∂Ξ 12 4e3 g e ∂Ξ + εe + 1 g x2 Ξ = 0
∂x2 − λ − 3 2 ∂e − λ − 2 λ ∂g λ 12 2
λ 2
where case Ξ = σλ c(cid:16)orresponds(cid:17)to ε = 1, a(cid:0)nd Ξ = σ c(cid:1)orrespon(cid:16)ds to ε = 0 an(cid:17)d arbitrary eλ. The
quantity ε is a parity of θ-characteristic. The universal series and integer recurrence acquire the
following form:
∞
k/2 x2k+1 ε
Ξ(x;e ,g )= 2 νB(ε) ek 2νgν − ,
λ 2 ( − k−2ν,ν · λ− 2)(2k+1 ε)!
Xk=0 Xν=0 −
(6)
B(ε) = 24(n+1)B(ε) +(4m 12n 4 ε)B(ε)
m,n m−3,n+1 − − − m−1,n−
4(m+1)B(ε) 1(m+2n 1)(2m+4n 1 2ε)B(ε) .
−3 m+1,n−1− 3 − − − m,n−1
Transition between pairs (g ,g ), (e ,g ) and (e ,e ) is one-to-one therefore the universal recur-
2 3 λ 2 λ µ
rence can be written for any of these representations. In the last case it is symmetrical. The
representation (6) is a five-term one, in contrast to four-term recurrence of Weierstrass A , but
m,n
the representation (g ,g ) is not possible for the functions σ .
2 3 λ
5. Power series for θ-functions of Jacobi
JacobimadeanattempttoobtainsuchseriesasearlyasbeforeappearanceofhisFundamentaNova
[6, I: 259–260]. These seriesare the serieswith polynomialcoefficients inη(τ) and ϑ(τ)-constants.
This fact follows from the formulas
θ1(x|τ)=πη3(τ)·e−2η(τ)x2σ(2x|τ), θλ(x|τ)=ϑλ(τ)·e−2η(τ)x2σλ−1(2x|τ).
ϑ-constants make it possible to rewrite Weierstrassian representations g ,g ,e into ϑ-constant
b 2 3 λ
ones. Formulasfor branch-pointse throughthe ϑ-constants arewell known. In turn, ϑ-constants
k
areconnectedvia Jacobiidentity ϑ4 =ϑ4+ϑ4. All this enables oneto convertthe representations
3 2 4
choosing arbitrary pair. We shall speak (α,β)-representation, if formulas are written through the
constants (ϑ ,ϑ ) under (α,β)=(0,0). Operator (3) has Weierstrassianrepresentation
α0 0β 6
D= 12 4e3 g e ∂ 4e2 2g ∂ =
− λ − 2 λ ∂g − λ − 3 2 ∂e
2 λ
(cid:0) (cid:1) (cid:16) (cid:17)
b = 4 2e2+2e e e2 ∂ + 4 2e2+2e e e2 ∂ ,
3 µ µ λ− λ ∂e 3 λ λ µ− µ ∂e
λ µ
(cid:0) (cid:1) (cid:0) (cid:1)
and also ϑ-constant one. For example (ϑ ,ϑ )-representation
2 4
π2 ∂ π2 ∂
D= ϑ8+2ϑ4ϑ4 ϑ8+2ϑ4ϑ4 . (7)
3 2 2 4 ∂ϑ4 − 3 4 2 4 ∂ϑ4
2 4
(cid:8) (cid:9) (cid:8) (cid:9)
Let us denote hαi=( b1)α. General (α,β)-representation of Halphen’s operator has the form
−
π2 ∂ π2 ∂
D= hαiϑ8 +2hβiϑ4 ϑ4 hβiϑ8 +2hαiϑ4 ϑ4 .
3 0β α0 0β ∂ϑ4 − 3 α0 α0 0β ∂ϑ4
0β α0
(cid:0) (cid:1) (cid:0) (cid:1)
b
ON FUNCTIONS OF JACOBI AND WEIERSTRASS (I) 7
Let ε is defined by a parity of characteristic of the function θ (xτ):
αβ |
ε= hαβi+1, ε=0 under θαβ =±θ1 .
2 (ε=1 under θαβ =±θ2,3,4)
The equation (5) for the function Ξ=(σ,σ ) has the form
λ
∂2Ξ +DΞ+ εe (ϑ)+ π4 ϑ8+ϑ4ϑ4+ϑ8 x2 Ξ=0,
∂x2 λ 122 2 2 4 4
n o
D (cid:2) (cid:3)
where is determined by fobrmula (7).
5.1. Function θ(xτ). Expansion of function
b 1 |
θ(xτ)= ∞ C (τ) x2k+1 =2πη3 x 2η x3+ 2η2 π4 ϑ8+ϑ4ϑ4+ϑ8 x5+ (8)
1 | k · − · − 180 2 2 4 4 · ···
Xk=0 n (cid:16) (cid:0) (cid:1)(cid:17) o
has the following analytic representbation:
∞
(4πi)k dkη3
θ(xτ)= 2π x2k+1 =
1 | (2k+1)! dτk ·
k=0
X b (9)
∞
=2πη3 ( 2)k k (−6)−νπ2ν ηk−ν ν(ϑ) x2k+1 ,
− ( (k−ν)!(2ν+1)!· N )
Xk=0 Xν=0
where polynomials (ϑ), dbepending on combinations of ϑ-constants, have the form
ν
N
ν Gν−s,s·ϑ44sϑ24(ν−s)
Nν(ϑ)=Xs=0 ((−−11))ssGGsν,−νs−,ss··ϑϑ4343ssϑϑ2444((νν−−ss)).
Integer recurrence Gm,n (G0,0 =1) looks as follows: 
G =4(n 2m 1)G 4(m 2n 1)G
m,n m,n 1 m 1,n
− − − − − − − −
.
2(m+n 1)(2m+2n 1) G +G +G
m 2,n m 1,n 1 m,n 2
− − − − − − −
The recurrenceGm,n is antisymmetric: Gm,n =( 1(cid:0))m+nGn,m. Onemaywrite outr(cid:1)epresentation
−
of the type θ (xτ) = C gmgnηpxk through the recurrence of Weierstrass A , but G
1 | mnp 2 3 m,n m,n
is more effective than A since polynomials are already grouped together in ϑ-constants.
m,n
P
Odd derivatives θ(2k+1)(0τ) = ϑ(2k+1)(τ), i.e. expressions in front of x2k+1 in (9), generate
1 | 1
polynomials (8) in (η,ϑ) which are exactly integrable k times in τ.
5.2. Functions θ (xτ). Expansions of functions θα = θ of the form
2,3,4 | β ± 2,3,4
θα(xτ)= ∞ C(α,β)(τ) x2k =ϑα ϑα 2η+ π(cid:2)2(cid:3) hβiϑα–14 hαiϑ 0 4 x2+ (10)
β | k · β − β 6 0 − β–1 ···
(cid:2) (cid:3) Xk=0 (cid:2) (cid:3) (cid:2) (cid:3)n (cid:16) (cid:2) (cid:3) (cid:2) (cid:3) (cid:17)o
have the following analytic representation:
∞
(4πi)k dkϑα
θα(xτ)= β x2k . (11)
β | (2k)! dτk ·
(cid:2) (cid:3)
(cid:2) (cid:3) Xk=0
The equation (5) on functions Ξ=(σ,σ ) in (α,β)-representation has the form
λ
∂2Ξ +DΞ+ e (ϑ)+ π4 ϑ8 +hα+βiϑ4 ϑ4 +ϑ8 x2 Ξ=0,
∂x2 γδ 122 α0 α0 0β 0β
n (cid:0) (cid:1) o
b
8 YU.V.BREZHNEV
where e (ϑ) correspond (independently of representation (α,β)) to chosen function σ or σ :
γδ λ
e (ϑ)= π2 hδiϑγ–14 hγiϑ 0 4 , e11 ≡0, e10 ≡e1 . (12)
γδ 12 0 − δ–1 e e , e e
(cid:16) (cid:2) (cid:3) (cid:2) (cid:3) (cid:17) (cid:26) 00 ≡ 2 01 ≡ 3 (cid:27)
Representation is called symmetrical if (γ,δ)=(α,β).
In symmetrical representation the series (11) has the following grouped form:
∞
θαβ (x|τ)=ϑαβ (τ) (−2)k( k ((k−−6)ν−)ν!(π22νν)!·ηk−νNν(α,β)(ϑ))x2k (13)
(cid:2) (cid:3) (cid:2) (cid:3) Xk=0 Xν=0
with the universal integer recurrence (G(α,β) =1):
0,0
ν
(α,β)(ϑ) = G(α,β) ϑ 0 4sϑα–14(ν−s) ,
Nν ν−s,s· β–1 0
Xs=0 (cid:2) (cid:3) (cid:2) (cid:3)
G(α,β) = hαi(4n 8m 3)G(α,β) hβi(4m 8n 3)G(α,β)
m,n − − m,n−1− − − m−1,n−
2(m+n 1)(2m+2n 3) G(α,β) +hα+βiG(α,β) +G(α,β) .
− − − m−2,n m−1,n−1 m,n−2
Symmetry of the recurrence G(α,β) with respect t(cid:0)o permutation of indices is determined(cid:1) by the
m,n
properties:
G(α,β) =( 1)(m+n)(α+β+1)G(α,β) ,
n,m − m,n
G(β,α) =( 1)(m+n)(α+β) G(α,β) .
m,n − m,n
This means that there are only two recurrence G(α) under α= 1,0 and β =0:
m,n { }
G(mα,)n = hαi(4n−8m−3)G(mα,)n−1−(4m−8n−3)G(mα−) 1,n−
2(m+n 1)(2m+2n 3) G(α) +hαiG(α) +G(α)
− − − m−2,n m−1,n−1 m,n−2
with property G(α) =( 1)(m+n)(α+1)G(α) : (cid:0) (cid:1)
n,m − m,n
G(0) =( 1)(m+n)G(0) , G(1) =G(1) .
n,m − m,n n,m m,n
Developments (9) and (13) differ from eachother only ina multiplier and shape ofthe recurrence.
They can be unified into one (by introducing the parity ε) but the quantity hαi is left and (m,n)-
entries of matrices G(β,α) differ only in a sign.
Even derivatives θ(2k)(0τ) = ϑ(2k)(τ), i.e. expressions in front of x2k in (13), generate polyno-
αβ | αβ
mials (10) in (η,ϑ) which are integrable k times in τ. Their integrability is a consequence of one
dynamical system considered in sect.7.
Makinguseoftheseexpansionsonecanbuildotheronesaboutpointsx= 1, τ . Described
{±2 ±2}
series will be, up to obvious modifications, changed into one another.
6. Dynamical systems on θ-functions
6.1. Differential equations in x. The five functions
∂θ
θ1,2,3,4 and θ1′ ≡ ∂x1
ON FUNCTIONS OF JACOBI AND WEIERSTRASS (I) 9
satisfy closed ordinary autonomous differential equations:
∂θ2 = θ1′ θ πϑ2 θ3θ4 ∂θ1′ = θ1′2 π2ϑ2ϑ2 θ22 4η+ π2 ϑ4+ϑ4 θ
∂x θ 2− 2· θ ∂x θ − 3 4·θ − 3 3 4 · 1
 1 1 1 1
 ∂∂θx3 = θθ11′ θ3−πϑ23·θθ2θ14 ∂∂θx1 =θ1′ . n (cid:0) (cid:1)o
∂θ θ θ θ
Thesee∂qxu4a=tioθn11′sθa4re−eqπuϑiv24a·leθ2n1t3torelationsbetweenWeierstrassianfunctions(σ,ζ,℘,℘′)(xτ). Gen-
|
eral form of differentiation of the functions θ is as follows (ϑ =0):
1,2,3,4 1
∂θ θ θ θ 8k 28 10k 28
k = 1′ θ πϑ2 ν µ , where ν = − , µ= − , k =(1,2,3,4).
∂x θ k− k· θ 3k 10 3k 8
1 1 − −
It immediately follows that the following identities are obeyed 2
θ θ θ θ
n′ m′ =πϑ2 1 k sign(n−m), where (k =n, n=m, k=m, k,n,m=2,3,4),
θ − θ k·θ θ 6 6 6
n m n m
and also well-known polynomial identities:
θ4+θ4 =θ4+θ4, ϑ2θ2 =ϑ2θ2+ϑ2θ2, sign(n−m) ϑ2θ2 =ϑ2 θ2 ϑ2 θ2 . (14)
1 3 2 4 3 3 2 2 4 4 · k 1 m n− n m
6.2. Differential equations in τ. The functions θ ,θ satisfy closed non-autonomous ordi-
1,2,3,4 1′
nary differential equations in τ:
∂θ1 = −i θ1′2 + + πiϑ2ϑ2 θ22 + iη+ πi ϑ4+ϑ4 θ
∂τ 4π θ 4 3 4·θ π 12 3 4 · 1
 1 1 n (cid:0) (cid:1)o
 ∂∂∂∂θθττ32 == 44−−ππii(cid:26)θθ1′θ12θ211′θ−3+πϑ2i22ϑ·23θθ·31θθθ242θ(cid:27)42θθθ121′2−+π4π4iiϑϑ2223ϑϑ2324··θθθθ4122122θ3++nnππiiηη++ 11ππ22ii(cid:0)(cid:0)ϑϑ4343++ϑϑ4444(cid:1)(cid:1)oo··θθ23 . (15)
General f∂∂o∂∂rθθττm14′ ==of4−4−τππi-idθθθθi1′112ff′3122e+θre4n3+t(cid:26)ia2πi4tiϑiϑo2423n·θϑo224θf·3tθθhθθ212212e1′+−fuπniπ4cηitϑ+io22nϑ1πs2i24(cid:0)θ·ϑ1θθ,4331222,+3θ,44ϑ+i44s(cid:1)na(cid:27)sπiθηf1′o−+lloπw1π222is(cid:0)i:ϑϑ4322+ϑ23ϑϑ4424(cid:1)·oθ·2θθθ4132θ4
∂θk = −i θ1′2 θ + i ϑ2 θ θ θ1′ + πi ϑ2ϑ2 θ2 ϑ2ϑ2ϑ2 θν2 + θµ2 θ +
∂τ 4π θ2 k 2 k· ν µθ2 4 3 4· 2− k ν µ· ϑ2 ϑ2 k
1 1 (cid:26) (cid:18) ν µ(cid:19)(cid:27)
+ i η+ πi ϑ4+ϑ4 θ , where ν = 8k−28, µ= 10k−28, k =(1,2,3,4).
π 12 3 4 · k 3k−10 3k−8
n (cid:0) (cid:1)o
2ThesefundamentalrelationsappearimplicitlyinworksbyJacobibuthavenotgotintocomprehensivehandbook
forellipticfunctionscompiledbySchwarzonthebasisofWeierstrass’slectures.
10 YU.V.BREZHNEV
7. Differential equations on ϑ,η-constants
Differential closure is provided by the constants ϑ and η:
2,3,4
1 dϑ2 = i η+ πi ϑ4+ϑ4 , 1 dϑ4 = i η πi ϑ4+ϑ4 ,
ϑ dτ π 12 3 4 ϑ dτ π − 12 2 3
2 4 (16)
1 dϑ3 = i η+ πi(cid:0)ϑ4 ϑ4(cid:1), dη = i 2η2 (cid:0)π4 ϑ8+(cid:1) ϑ8+ϑ8 .
ϑ dτ π 12 2− 4 dτ π − 122 2 3 4
3
(cid:0) (cid:1) n (cid:0) (cid:1)o
Well known differential equations on logarithms of ratios ϑ : ϑ : ϑ and also known dynamical
2 3 4
system of Halphen and its numerous varieties are widely encountered in the modern literature.
These are generated by the system of equations
dg2 = i 8g η 12g , dg3 = i 12g η 2g2 , dη = i 2η2 1g ,
dτ π 2 − 3 dτ π 3 − 3 2 dτ π − 6 2
(cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17)
which, in implicit form, was written out by Weierstrass and Halphen used it [3, I: p.331] in order
to get his famous symmetrical version of the system. Ramanujan obtained its equivalent for some
number-theoretic q-series. See for example [9, 1] and additional information in this work. It is
§
lesser known, to all appearances, that another version of the system was written out by Jacobi
in connection with power series of θ-functions. These results were published by Borchardt on the
basis of papers kept after Jacobi [6, II: 383–398].
Jacobi obtained a dynamical system on four functions (A,B,a,b) which are, in our notation,
rational functions of the ϑ,η-constants
4 η 1 ϑ4 ϑ4 ϑ2 ϑ2ϑ2
A=ϑ2, B = + 2− 4 , a=4 1 2 2 , b=2 2 4
3 π2 ϑ2 3 ϑ2 − ϑ2 ϑ4
3 3 (cid:18) 3(cid:19) 3
and showedthat the dynamical system can be useful for obtaining power series of θ-functions. He
also presented two (canonical) transformations of the variables (A,B,a,b) retaining the shape of
the equations. Jacobi noticed that the developments become simple and have a recursive form
when extracted an exponential multiplier e−21ABx2. This is, in fact, Halphen’s recurrence (3) for
theσ-functionofWeierstrass. Transitionbetweenvariablesinmentionedsystemsisnotone-to-one
but always algebraic. This is because all of them are consequences of the equations (16) since
the variables, appeared in these systems, are rationally expressible through the ϑ,η-constants. A
related dynamical system arose in a work of Jacobi earlier, when he derived his known equation
on ϑ-constants [6, II: p.176]
ϑ2ϑ 15ϑϑ ϑ +30ϑ3 2+32 ϑϑ 3ϑ2 3 = π2ϑ10 ϑϑ 3ϑ2 2.
τττ τ ττ τ ττ τ ττ τ
− − − −
(cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1)
Logarithmic derivatives of ϑ-constants, in turn, satisfy compact differential equation
d
f 2f2 f f2 +16f3f +4f2 f 6f2 =0, f = lnϑ (τ)
τ − τττ − ττ ττ τ τ − dτ 2,3,4
(cid:0) (cid:1) (cid:0) (cid:1)
and Dedekind’s function Λ=lnη(τ) satisfies the differential equation of the 3-rd order
Λ 12Λ bΛ +16Λ3 2+32 Λ 2Λ2 3 = 4 π6η24.
τττ ττ τ τ ττ τ
− − 27
(cid:8) (cid:9) (cid:8) (cid:9)
b