PERMUTABLE POLYNOMIALS AND RATIONAL FUNCTIONS

Garry J. Tee

(Received June, 2011)

Abstract. Permutable functions have recently been applied to cryptography.

It has been known since 1951 that every sequence of permutable polynomials, which contains at least one polynomial of every positive degree, is either the sequence of simple positive powers or the sequence of Chebyshev polynomi- als of the first kind, or else it is related to those by a similarity transform by a linear function. The only known infinite sequences of permutable ratio- nal functions were the simple powers and Stirling’s functions, which express tan(nx) as rational functions of tan x: otherwise only 2 pairs of permutable rational functions have been published. Many infinite sequences of permutable rational functions are constructed on the basis of trigonometric functions and elliptic functions. Many identities connect 24 infinite sequences of permutable rational functions based on Jacobi’s 12 elliptic functions.

Contents

1 Permutable Polynomials 2 Permutable Rational Functions 3 Elliptic Functions

4 Permutable Functions αk to µk From Jacobian Elliptic Functions

5 Permutable Functions νk to ωk From Squares Of Jacobian Elliptic Functions 6 Reciprocal Parameter

7 Further Modular Transforms of Permutable Functions

8 Dualities Between the Permutable Rational Functions νk to ωk

References

1. Permutable Polynomials

The Chebyshev polynomial of the first type Td is defined by the initial values

T_{0}(x) = 1, T_{1}(x) = x, (1)

with the 3–term recurrence relation for n > 1:

Tn(x) = 2xTn−1(x) − Tn−2(x). (2) By induction on n in (2), it follows from (1) that, for all integers d ≥ 0, Td(x) is a polynomial in x of degree d with integer coefficients. Moreover, Td is an even polynomial for even d and an odd polynomial for odd d.

2010 Mathematics Subject Classification Primary 33E05, 26C05, 26C15; Secondary 20K99, 14G50.

Key words and phrases: Permutable polynomials, Chebyshev polynomials, permutable chains, permutable rational functions, Jacobi elliptic functions, real multiplication, trigonometric func- tions, rational function identities, cryptography.

The integer coefficients of Td are given explicitly by the following formula^{1} for
d > 0, in Lyusternik [20, p.79]:

T_{d}(x) =

d÷2

X

k=0

(−1)^{k}2^{d−2k−1}d(d − k − 1)!

k!(d − 2k)! x^{d−2k} =
2^{d−1}x^{d}− 2^{d−3}dx^{d−2}+ 2^{d−6}d(d − 3)x^{d−4}−2^{d−8}

3 d(d − 4)(d − 5)x^{d−6}+ · · ·

· · · +

(−1)^{d÷2} (for even d),

(−1)^{(d−1)÷2}dx (for odd d). (3)
For example,

T0(x) = 1,
T1(x) = x,
T_{2}(x) = 2x^{2}− 1,
T3(x) = 4x^{3}− 3x,
T_{4}(x) = 8x^{4}− 8x^{2}+ 1,
T5(x) = 16x^{5}− 20x^{3}+ 5x,
T6(x) = 32x^{6}− 48x^{4}+ 18x^{2}− 1,
T_{7}(x) = 64x^{7}− 112x^{5}+ 56x^{3}− 7x,

T8(x) = 128x^{8}− 256x^{6}+ 160x^{4}− 32x^{2}+ 1,
T9(x) = 256x^{9}− 576x^{7}+ 432x^{5}− 120x^{3}+ 9x,

T_{10}(x) = 512x^{10}− 1280x^{8}+ 1120x^{6}− 400x^{4}+ 50x^{2}− 1,
T11(x) = 1024x^{11}− 2816x^{9}+ 2816x^{7}− 1232x^{5}+ 220x^{3}− 11x,

T_{12}(x) = 2048x^{12}− 6144x^{10}+ 6912x^{8}− 3584x^{6}+ 840x^{4}− 72x^{2}+ 1, (4)
et cetera [20, pp. 168–169].

Write x = cos ϑ, so that T0(cos ϑ) = cos(0 × ϑ), T1(cos ϑ) = cos(1 × ϑ). By induction on n in (2), it follows from (1) that for all complex x and integer n ≥ 0, Tn(x) = Tn(cos ϑ) = 2 cos ϑ cos((n − 1)ϑ) − cos((n − 2)ϑ) = cos(nϑ). (5) Likewise,

Tn(cosh ς) = cosh(nς). (6)

Therefore, for all complex x and non–negative integers j, k, Tj Tk(x)

= Tj Tk(cos ϑ)

= Tj(cos(kϑ)) = cos(jkϑ) =

= T_{jk}(cos ϑ) = T_{jk}(x), (7)
giving the functional identity for complex x and non–negative integers j, k:

T_{j} T_{k}(x)

≡ Tjk(x). (8)

Thus, the Chebyshev polynomials, with the binary operation of function com- position, form an infinite Abelian group, with

Tj Tk(x)

≡ Tjk(x) ≡ Tkj(x) ≡ Tk Tj(x). (9)

1The symbol ÷ denotes integer division, yielding integer quotient. For integers n and d > 0, q = n ÷ d, where n = qd + r, with remainder 0 ≤ r < d .

“Two polynomials, p and q, are called permutable if p(q(x)) = q(p(x)) for all x”,
in Rivlin [23, p.192]. And “A sequence of polynomials, each of positive degree, con-
taining at least one of each positive degree and such that every two polynomials in it
are permutable is called a chain. The Chebyshev polynomials T1(x), . . . , Tn(x), . . .
form a chain. So do the powers πj(x) = x^{j}, j = 1, 2, . . . , as is easily verified.” [23,
p.194]

The linear transform y = λ(x) = ax + b (with a 6= 0) has the inverse transform
x = λ^{−1}(y) = (y − b)/a. A pair of functions u(x) and v(x) are permutable, if and
only if the transformed functions U ^{def}= λ^{−1}uλ and V ^{def}= λ^{−1}vλ are permutable,
since

U (V (x)) = λ^{−1}u

λ λ^{−1}v(λ(x))

= λ^{−1}u(v(λ(x))) =

= λ^{−1}v(u(λ(x))) = λ^{−1}v

λ λ^{−1}u(λ(x))

= V (U (x)). (10)
If u(x) is a polynomial in x of degree n, then so is the transformed function λ^{−1}uλ.

Two chains are called similar if there is a linear transformation λ such that
each polynomial in one chain is similar (via λ) to the polynomial of the same
degree in the other chain. Block & Thielman [4] and Jacobsthal [14] proved the
remarkable theorem that “every chain is either similar to x^{j} , j = 1, 2, . . . or
to {Tj}, j = 1, 2, . . .”: see Rivlin [23, p.195], Lausch & N¨obauer [17, p.156] and
Borwein & Erd´elyi [5, p.34].

1.1. Squares of Chebyshev polynomials. From (8) with j = 2, we get
T2k(x) = T2(Tk(x)) = 2T_{k}^{2}(x) − 1, (11)
which is an even polynomial. Hence, with y = x^{2} we get the identity

T_{k}^{2}(x) ≡ ^{1}_{2}(T2k(x) + 1) ≡ Ek(y), (12)
where Ek(y) is a polynomial of degree k, with integer coefficients.

Rewriting x for y, we get from (4) the sequence of polynomials E0(x) = 1,

E1(x) = x,

E2(x) = 4x^{2}− 4x + 1,
E_{3}(x) = 16x^{3}− 24x^{2}+ 9x,

E4(x) = 64x^{4}− 128x^{3}+ 80x^{2}− 16x + 1,
E_{5}(x) = 256x^{5}− 640x^{4}+ 560x^{3}− 200x^{2}+ 25x,

E6(x) = 1024x^{6}− 3072x^{5}+ 3456x^{4}− 1792^{3}+ 420x^{2}− 36x + 1, (13)
et cetera.

Hence, for all complex ϑ, ς and non–negative integers j, k,

cos^{2}(kϑ) = T_{k}^{2}(cos ϑ) = Ek(cos^{2}ϑ), (14)
and likewise

cosh^{2}(kς) = T_{k}^{2}(cosh ς) = Ek(cosh^{2}ς) . (15)
And thus

E_{j} E_{k}(cos^{2}ϑ)

= E_{j}(cos^{2}(kϑ)) = cos^{2}(jkϑ) = E_{jk}(cos^{2}ϑ). (16)

Writing x = cos^{2}ϑ, we get an identity for non–negative integers j, k:

Ej Ek(x)

≡ Ejk(x). (17)

Thus, the polynomials Ek, with the binary operation of function composition, form an infinite Abelian group, with

E_{j} E_{k}(x)

≡ Ejk(x) ≡ E_{kj}(x) ≡ E_{k} E_{j}(x), (18)
and hence they form a chain of permutable polynomials for every k ≥ 0.

By the theorem of Block, Thielman and Jacobsthal, the chain {Ek} must be similar (by λ(x) = ax + b for constants a 6= 0 and b) either to the simple positive powers or to the Chebyshev polynomials. Indeed, with λ(x) = 2x − 1 we get the identities

λ(Ek(z^{2}) = 2Ek(z^{2}) − 1 = T2k(z) = Tk(T2(z)) =

= T_{k}(2z^{2}− 1) = T_{k}(λ(z^{2})). (19)
Replacing z^{2} by x, we get the similarity between the chains {Ek} and {Tk}

E_{k}(x) ≡ λ^{−1}

T_{k} λ(x)

. (20)

1.2. Permutable even and odd functions. The sequence of rational functions
πr(x) = {x^{r}} (for integer r) are all permutable, since

πr πs(x)

= πr x^{s}

= x^{s}r

= x^{sr} =

= x^{rs} = x^{r}^{s}

= π_{s} x^{r}

= π_{s} π_{r}(x). (21)
A function f is called an even function iff

f (−x) = f (x) (22)

for all x. If u and v are permutable even functions then −u and −v are permutable even functions, since

−u(−v(x)) = −u(v(x)) = −v(u(x)) = −v(−u(x)). (23) Hence, if {wr} is a permutable sequence of even functions for integer r, then the negated sequence {−wr} is permutable.

A function f is called an odd function iff

f (−x) = −f (x) (24)

for all x. If u and v are permutable odd functions then −u and v are permutable odd functions, since

−u(v(x)) = −v(u(x)) = v(−u(x)), (25)

and similarly −v and u permute. If u and v are permutable odd functions then −u and −v are permutable odd functions, since

−u(−v(x)) = −(−u(v(x))) = u(v(x)) =

= v(u(x)) = −(−v(u(x))) = −v(−u(x)). (26)
Hence, if wrare permutable odd functions for integer r, then the reflected sequence
{wr, −wr} is permutable. For odd r = 2i + 1 the odd rational functions x^{2i+1} are
permutable, and so is the reflected sequence

{x^{2i+1}, −x^{2i+1}} = · · · , x^{−1}, −x^{−1}, x, −x, x^{3}, −x^{3}, · · · . (27)

And the odd Chebyshev polynomials T2j+1 are permutable, and hence the re- flected sequence of polynomials {T2j+1, −T2j+1} is permutable. For all ϑ, we have the identity in Lyusternik [20, p.79]:

sin((2j + 1)ϑ) = (−1)^{j}T2j+1(sin ϑ). (28)
Denote

qj(z) ^{def}= (−1)^{j}T2j+1(z), (29)
so that qj(sin ϑ) = sin((2j + 1)ϑ). Thus the sequence {qj} is a subsequence of
the reflected permutable sequence {T2j+1, −T2j+1}. Indeed, for all non-negative
integers j and k,

qj(qk(sin ϑ)) = qj(sin((2k + 1)ϑ) = sin((2j + 1)(2k + 1)ϑ) =

= sin((4jk + 2j + 2k + 1)ϑ) = q2jk+j+k(sin ϑ) = qk(qj(sin ϑ)), (30)
The linear transform with λ(x) = −x converts u(x) to −u(−x). Hence, if u is an
odd function then it is unchanged, but if u is an even function then it is converted
to −u. Thus, the permutable sequence of rational functions {x^{r}} is similar (via
λ(x) = −x) to the permutable sequence

{−(−x)^{r}} = · · · , x^{−3}, −x^{−2}, x^{−1}, −1, x, −x^{2}, x^{3}, −x^{4}, · · · . (31)
And the permutable chain of Chebyshev polynomials {Tj} is similar (via λ(x) =

−x) to the permutable chain

{−Tj(−x)} = −1, x, −T2(x), T3(x), −T4(x), T5(x), · · · . (32) 2. Permutable Rational Functions

A simple example of an infinite sequence of permutable rational functions is given
by {T_{j}(x)/1} for j = 0, 1, 2, . . . , which is similar (via λ(x) = −x) to the permutable
rational sequence {T_{j}(−x)/(−1)}. And since T_{2i+1} is an odd polynomial, there is
also the reflected sequence {T_{2i+1}(x)/1, T_{2i+1}(x)/(−1)} of permutable rational
functions. Another simple example is given by {Ej(x)/1}.

2.1. Tangents of multiple angles. For complex z = x + iy and positive integer n.

z^{n}= (x + iy)^{n}=

n

X

j=0

n j

i^{j}x^{n−j}y^{j}= H_{n}^{(0)}(x, y) + iH_{n}^{(1)}(x, y), (33)
where the real and imaginary parts of z^{n} are given in Lyusternik [20, Appendix 1,

§2.1] by the harmonic polynomials :
H_{n}^{(0)}(x, y) =

n÷2

X

k=0

(−1)^{k} n
2k

x^{n−2k}y^{2k}

H_{n}^{(1)}(x, y) =

(n−1)÷2

X

k=0

(−1)^{k}

n

2k + 1

x^{n−2k−1}y^{2k+1} . (34)
For all complex ρ, log(cos ρ + i sin ρ) = iρ (by Cotes’s theorem), or

cos ρ + i sin ρ = e^{iρ}; (35)

and hence (De Moivre’s Theorem) for integer n ≥ 0

cos(nρ) + i sin(nρ) = e^{inρ} = (cos ρ + i sin ρ)^{n}. (36)

Therefore

cos(nρ) + i sin(nρ) = (cos ρ + i sin ρ)^{n} =

= H_{n}^{(0)}(cos ρ, sin ρ) + iH_{n}^{(1)}(cos ρ, sin ρ). (37)
Equating real and imaginary parts (for real ρ), we get that

cos(nρ) = H_{n}^{(0)}(cos ρ, sin ρ), sin(nρ) = H_{n}^{(1)}(cos ρ, sin ρ); (38)
and hence

tan(nρ) = sin(nρ)

cos(nρ) = Hn^{(1)}(cos ρ, sin ρ)
Hn^{(0)}(cos ρ, sin ρ)

. (39)

Dividing numerator and denominator in (39) by cos^{n}ρ, we get in Lyusternik [20,
p.79] tan(nρ) as Stirling’s rational function of tan ρ, with integer coefficients:

tan(nρ) = Hn^{(1)}(1, tan ρ)
Hn^{(0)}(1, tan ρ)

=

(n−1)÷2

X

k=0

n

2k + 1

(−1)^{k}tan^{2k+1}ρ

n÷2

X

k=0

n 2k

(−1)^{k}tan^{2k}ρ

. (40)

Denote

t ^{def}= tan ρ, z ^{def}= −t^{2} = − tan^{2}ρ. (41)
Then, it follows from (40) that

tan(nρ) = rn(tan ρ), (42)

where rn(t) is Stirling’s odd rational function with positive integer coefficients rn(t) = t An(z)

Bn(z) , (43)

(except that A0(t) = 0), with : An(z) =

(n−1)÷2

X

k=0

n

2k + 1

z^{k}, Bn(z) =

n÷2

X

k=0

n 2k

z^{k}, (44)
and Bn(0) = 1.

In more detail, for odd n = 2j + 1:

r2j+1(t) = t A2j+1(z)
B_{2j+1}(z) =

= t 2j + 1 + ^{2j+1}_{3} z + ^{2j+1}_{5} z^{2}+ · · · + ^{2j+1}_{2} z^{j−1}+ z^{j}

1 + ^{2j+1}_{2} z + ^{2j+1}_{4} z^{2}+ · · · + ^{2j+1}_{3} z^{j−1}+ (2j + 1)z^{j} , (45)
where the polynomials A2j+1and B2j+1are mutually reciprocal:

z^{j}A2j+1

1 z

= B2j+1(z), z^{j}B2j+1

1 z

= A2j+1(z) (46) for all z 6= 0. For even n = 2j:

r_{2j}(t) = t A_{2j}(z)

B2j(z) = t 2j + ^{2j}_{3}z + ^{2j}_{5}z^{2}+ · · · + ^{2j}_{3}z^{j−2}+ 2jz^{j−1}

1 + ^{2j}_{2}z + ^{2j}_{4}z^{2}+ · · · + ^{2j}_{2}z^{j−1}+ z^{j} , (47)

where both A2j and B2j are self–reciprocal polynomials:

A2j(z) = z^{j−1}A2j

1 z

, B2j(z) = z^{j}B2j

1 z

, (48)

for all z 6= 0.

For example,
r_{0}(t) = t0

1 = 0 , r_{1}(t) = t1
1 = t ,
r2(t) = t 2

1 + z , r3(t) = t 3 + z
1 + 3z ,
r_{4}(t) = t 4 + 4z

1 + 6z + z^{2} , r_{5}(t) = t 5 + 10z + z^{2}
1 + 10z + 5z^{2} ,
r6(t) = t 6 + 20z + 6z^{2}

1 + 15z + 15z^{2}+ z^{3} ,

(49)

et cetera .

Since cot ρ = tan ^{1}_{2}π − ρ, we get that
rn(cot ρ) = rn

tanπ

2 − ρ

= tannπ 2 − nρ

. (50)

For odd n = 2j + 1, this becomes

r2j+1(cot ρ) = tan jπ +^{1}_{2}π − (2j + 1)ρ =
tan ^{1}_{2}π − (2j + 1)ρ

= cot((2j + 1)ρ), (51) and for even n = 2j, this becomes

r_{2j}(cot ρ) = tan (jπ − 2jρ) = − tan(2jρ). (52)
2.2. Permutable rational functions r_{n}(t) and v_{n}(y). For the infinite sequence
of odd rational functions {r_{n}}, it follows from (42) that, for all complex t and
non–negative integers j, k, with ρ = tan^{−1}t,

rj rk(t)

= rj rk(tan ρ)

= rj(tan(kρ)) = tan(jkρ) =

= r_{jk}(tan ρ) = r_{jk}(t), (53)
giving the functional identity for complex t and non–negative integers j, k:

rj rk(t)

≡ rjk(t). (54)

Thus, the odd rational functions rn, with the binary operation of function com- position, form an infinite Abelian group as in Borwein & Erd´elyi [5, p.34], with

rjk(t) ≡ rj rk(t)

≡ rk rj(t). (55)

For example, for all complex t and integer k ≥ 0, r4k(t) = rk

4t − 4t^{3}
1 − 6t^{2}+ t^{4}

= 4rk(t) − 4r^{3}_{k}(t)

1 − 6r^{2}_{k}(t) + r^{4}_{k}(t) . (56)
Thus we get for n = 0, 1, 2, . . . the reflected sequence of permutable odd rational
functions {r_{n}(t), −r_{n}(t)}.

Now, with t = tan ρ and y = t^{2}= −z,
(r_{n}(t))^{2} = t^{2} An(−t^{2})

Bn(−t^{2})

^{2}

= v_{n}(y) ^{def}= y An(−y)
Bn(−y)

^{2}

. (57)

Thus, from (49) we get that

v0(y) = y 0 1

^{2}

= 0 , v1(y) = y 1

1

^{2}

= y ,

v_{2}(y) = y

2 1 − y

^{2}

, v_{3}(y) = y 3 − y

1 − 3y

^{2}
,

v4(y) = y

4 − 4y
1 − 6y + y^{2}

2

, v5(y) = y 5 − 10y + y^{2}
1 − 10y + 5y^{2}

^{2}
,

v6(y) = y

6 − 20y + 6y^{2}
1 − 15y + 15y^{2}− y^{3}

^{2}
,

(58)

et cetera.

Hence, we get the identities

v_{n}(tan^{2}ρ) ≡ (r_{n}(tan ρ))^{2} ≡ tan^{2}(nρ). (59)
For the infinite sequence of rational functions {vn}, it follows from (59) that, for
all complex y and non–negative integers j, k, with y = tan^{2}ρ,

vj vk(y)

= vj vk(tan^{2}ρ)

= vj(tan^{2}(kρ)) = tan^{2}(jkρ) =

= vjk(tan^{2}ρ) = vjk(y), (60)
giving the functional identity for complex y and non–negative integers j, k:

vj vk(y)

≡ vjk(y). (61)

Thus, the rational functions vn, with the binary operation of function composi- tion, form an infinite Abelian group, with

v_{jk}(y) ≡ v_{j} v_{k}(y)

≡ v_{k} v_{j}(y). (62)

Since cot ρ = tan ^{1}_{2}π − ρ, we get that
vn(cot^{2} ρ) = vn

tan^{2}π
2 − ρ

= tan^{2}nπ
2 − nρ

. (63)

For odd n = 2j + 1, this becomes

v_{2j+1}(cot^{2} ρ) = tan^{2} jπ +^{1}_{2}π − (2j + 1)ρ =
tan^{2} ^{1}_{2}π − (2j + 1)ρ

= cot^{2}((2j + 1)ρ), (64)
and for even n = 2j, this becomes

v_{2j}(cot^{2}ρ) = tan^{2}(jπ − 2jρ) = (− tan(2jρ))^{2} = tan^{2}(2jρ). (65)

2.3. Reciprocal rational functions. The bilinear transform y = ξ(x) = (ax +
b)/(cx + d), where ad − bc 6= 0, has the inverse bilinear transform x = ξ^{−1}(y) =
(dy − b)/(a − yc). A function u(x) is rational if and only if the transformed function
ξ^{−1}u(ξ(x)) is also a rational function of x. Similarly to (10), functions u(x) and
v(x) are permutable if and only if the transformed functions ξ^{−1}uξ and ξ^{−1}vξ are
permutable.

In the important special case where ξ(x) = 1/x, any function v(x) gets converted to another function 1/v(1/x), which we call the reciprocal of the function v. The reciprocal of an even function is even, and the reciprocal of an odd function is odd.

Any pair of functions v(x) and w(x) are permutable, if and only if their reciprocal functions f (x) = 1/v(1/x) and g(x) = 1/w(1/x) are permutable.

The permutable rational infinite sequence {f_{n}(x) = T_{n}(x)/1} has the reciprocal
permutable sequence {s_{n}(x) = 1/T_{n}(1/x)}, for which

fn(cos ϑ) = cos(nϑ), sn(sec ϑ) = sec(nϑ). (66) The sequence {sn(x)} is similar (via λ(x) = −x) to the permutable rational se- quence {−sn(−x)} = −s0(x), s1(x), −s2(x), s3(x), · · · . The rational function s2i+1

is an odd function, giving the permutable reflected rational sequence {s2i+1, −s2i+1}.

The permutable odd rational sequence {qj(x) = T2j+1(x)/(−1)^{j}} has the recip-
rocal permutable sequence {uj(x) = (−1)^{j}/T2j+1(1/x)}, and in view of (28),

qj(sin ϑ) = sin((2j + 1)ϑ), uj(cosec ϑ) = cosec((2j + 1)ϑ). (67) The odd rational function uj has the reflected permutable sequence {uj, −uj}.

The permutable rational infinite sequence {En(x)/1} has the reciprocal per- mutable rational sequence {Fn(x) = 1/En(1/x)}, with

En(cos^{2}ϑ) = cos^{2}(nϑ), Fn(sec^{2}ϑ) = sec^{2}(nϑ). (68)
The odd rational function rn(t) has the reciprocal odd rational function hn(t) =
1/rn(1/t), for which

r_{n}(tan ρ) = tan(nρ), h_{n}(cot ρ) = cot(nρ). (69)
In view of (51) and (52), this can be rewritten as the identities

h_{2k+1}(t) ≡ r_{2k+1}(t), h_{2k}(t) ≡ −1

r_{2k}(t) . (70)

Since {rn} is an odd permutable sequence, we get the odd permutable reflected reciprocal sequence {hn, −hn}.

The rational function vn(y) has the reciprocal rational function zn(y) = 1/vn(1/y), for which

v_{n}(tan^{2}ρ) = tan^{2}(nρ), z_{n}(cot^{2}ρ) = cot^{2}(nρ). (71)
Squaring the equations (70), this can be rewritten as the identities

z2k+1(y) ≡ v2k+1(y), z2k(y) ≡ 1

v_{2k}(y) . (72)

3. Elliptic Functions

An integral of the form R R(x, y) dx, where R(x, y) is a rational function of x
and y, and y^{2}= P (x) where P is a polynomial of degree 3 or 4, is called an elliptic
integral, as in Louis Melville Milne–Thomson [21, §17.1].

3.1. Legendre elliptic integrals. We shall use Milne–Thomson’s notation [21]

for Legendre’s elliptic integrals and Jacobian elliptic functions.

Adrien–Marie Legendre’s Incomplete Elliptic Integral of the First Kind, with amplitude ϕ and parameter m, is defined [21, §17.2.7] as

F (ϕ | m) ^{def}=
Z sin ϕ

0

dt

p(1 − t^{2})(1 − mt^{2}) =
Z ϕ

0

dx
p1 − m sin^{2}x

. (73)
(Earlier authors often used the modulus k, where m = k^{2}.) F (ϕ | m) is often
abbreviated to F (ϕ), when the parameter m is to be understood.

If the parameter m is real, then it can be reduced [21, §17.4.15 & §17.4.17] to the case with 0 ≤ m ≤ 1.

Legendre’s Complete Elliptic Integral of the First Kind [21, §17.3.1] is
K(m) ^{def}= F (^{1}_{2}π | m) =

Z 1 0

dt

p(1 − t^{2})(1 − mt^{2}) =

= Z π/2

0

dx
p1 − m sin^{2}x

. (74)

As m % 1, then K(m) % ∞.

Legendre’s Complementary Complete Elliptic Integral of the First Kind is defined [21, §17.3.5] as

K^{0}(m) ^{def}= K(1 − m). (75)

K(m) and K^{0}(m) are often abbreviated to K and K^{0}, when the parameter m is
to be understood.

The elliptic integral u = F (ϕ | m) is single-valued for integration along the real interval [0, sin ϕ], but it has infinitely many values in the case of complex integration.

3.2. Jacobian elliptic functions. In 1827, Niels Henrik Abel inverted elliptic integrals to get elliptic functions [1, p.264], and he shewed that elliptic functions are doubly–periodic single–valued functions. Then Carl Gustav Jacob Jacobi developed greatly the theory of elliptic functions.

As functions of the complex variable u, the Jacobian elliptic functions sn u, cn u, dn u et cetera are doubly–periodic single–valued functions of u.

3.2.1. Jacobian elliptic function sn u. The inverse function of the Legendre elliptic
function F is ϕ = F^{−1}(u), and the Jacobian elliptic function sn(u | m) ^{def}= sin ϕ
is often abbreviated to sn u, with the parameter m implied. The function sn is
single–valued for all complex parameters [21, §16.1.3], with

u = Z snu

0

dt

p(1 − t^{2})(1 − mt^{2}) , (76)
and sn u is an odd single–valued function of u.

For real u, the function sn has real period 4K(m) and range [−1, 1], with sn(0) = 0, sn(K) = 1, sn(2K) = 0, sn(3K) = −1 and sn(4K) = 0 [21, §16.2]. Let τ = sn u, so that

sn^{−1}τ = u =
Z τ

0

dt

p(1 − t^{2})(1 − mt^{2}) = F sin^{−1}τ | m . (77)

On the real interval −K ≤ u ≤ K the function sn increases monotonically from −1
to 1, and so for real τ ∈ [−1, 1] the function sn^{−1}τ has a single value in the real
interval [−K, K].

In addition to the real period 4K, sn also has the imaginary period i 2K^{0}.
3.2.2. Jacobian elliptic function cn u. The function cn is defined by

cn(u | m) ^{def}= cos ϕ. (78)

It is often abbreviated to cn u, with the parameter m implied. Thus, cn u = p

1 − sn^{2}u , (79)

and cn u is an even single–valued function of u. The branch of the square root function in (79) is determined by (78).

For real u, the function cn has real period 4K(m) and range [−1, 1], with cn(0) =
1, cn(K) = 0, cn(2K) = −1, cn(3K) = 0 and cn(4K) = 1 [21, §16.2]. On the real
interval 0 ≤ u ≤ 2K the function cn decreases monotonically from 1 to −1, and so
for real r ∈ [−1, 1] the function cn^{−1}r has a single value in the real interval [0, 2K].

In addition to the real period 4K, cn has the complex period 2K + i 2K^{0}.
3.2.3. Jacobian elliptic function dn u. The Jacobian elliptic function dn is defined
by

dn(u | m) ^{def}= p

1 − m sn^{2}u . (80)

It is often abbreviated to dn u, with the parameter m implied. Thus, dn u is an even single–valued function of u.

For real u, the function dn has real period 2K(m) and range [√

1 − m, 1], with dn(0) = 1, dn(K) = √

1 − m and dn(2K) = 1 [21, §16.2]. On the real interval 0 ≤ u ≤ K the function dn decreases monotonically from 1 to√

1 − m, and so for real r ∈ [√

1 − m, 1] the function dn^{−1}r has a single value in the real interval [0, K].

In addition to the real period 2K, dn also has the imaginary period i 4K^{0}.
3.2.4. The 12 Jacobi elliptic functions. In J. W. L. Glaisher’s systematic notation,
there are 9 other Jacobi elliptic functions [21, §16.3]:

cd u ^{def}= cn u

dn u dc u ^{def}= dn u

cn u ns u ^{def}= 1
sn u
sd u ^{def}= sn u

dn u nc u ^{def}= 1

cn u ds u ^{def}= dn u
sn u
nd u ^{def}= 1

dn u sc u ^{def}= sn u

cn u cs u ^{def}= cn u
sn u .

(81)

The 6 functions sn u, sc u, sd u, ns u, cs u, ds u are odd functions of u, and the other 6 functions are even functions of u.

Milne–Thomson gave clear graphs [21, Figures 16.1, 16.2, 16.3] of the 12 Jacobi elliptic functions for real u (and m = 0.5). Eugene Jahnke and Fritz Emde gave very effective graphs [15, pp. 92–93] of sn u, cn u and dn u for complex u (and m = 0.64).

3.3. Weierstraß elliptic function ℘. Karl Wilhelm Theodor Weierstraß devel- oped his version of elliptic functions on the basis of his elliptic function ℘, with complex (or real) parameters g2 and g3, which is defined by the differential equa- tion as in Southard [24, §18.1.6]

℘^{02}(z) = 4℘^{3}(z) − g_{2}℘(z) − g_{3}. (82)
Hence, Weierstraß’s function ℘ is the inverse of an elliptic integral, with

z = Z ∞

℘(z)

dw

p4w^{3}− g_{2}w − g_{3} (83)

for complex z, as in Jahnke & Emde [15, p.91].

Many mathematicians (including Eric Temple Bell [3, p.207]) regard Weierstraß’s function ℘ as being much more difficult than the Jacobi elliptic functions. But some mathematicians seem to have responded to that difficulty as a challenge, and they have chosen to use Weierstraß’s version of elliptic functions.

For the parameters m = 0 and m = 1, the Jacobi elliptic functions reduce respectively [21, §16.6] to trigonometric and hyperbolic functions:

sn(u|0) = sin u, cn(u|0) = cos u, dn(u|0) = 1,

sn(u|1) = tanh u, cn(u|1) = sech u, dn(u|1) = sech u, (84) et cetera.

For the Weierstraß function, as u → 0,

℘(u) = 1

u^{2} + O(u^{2}). (85)

The simplest form of ℘(u) occurs as the limit when one period is i∞. Denote the other period as 2ω (with nonzero real part). Then [15, p.105]

℘(u) = −1 3

π 2ω

^{2}
+

π 2ω sinπu

2ω

2

. (86)

In 1956, Harold and Bertha Swiles Jeffreys mildly remarked that “Had Abel been
alive he would probably have remarked that this property corresponds to taking
the fundamental trigonometric function as cosec^{2}x −^{1}_{3} ” [16, p.688].

3.4. Addition formulæ for elliptic functions. For all complex α and β, the Jacobi elliptic functions have the rational addition formulæ: [21, §16.17],

sn(α + β) = sn α · cn β · dn β + sn β · cn α · dn α
1 − m sn^{2}α · sn^{2}β ,
cn(α + β) = cn α · cn β − sn α · dn α · sn β · dn β

1 − m sn^{2}α · sn^{2}β ,
dn(α + β) = dn α · dn β − m sn α · cn α · sn β · cn β

1 − m sn^{2}α · sn^{2}β . (87)
But the function ℘ has the irrational addition formula in Southard [24, §18.4.1]:

℘(α + β) = −℘(α) − ℘(β) + +

" p

4℘(α)^{3}− g2℘(α) − g_{3}−p4℘(β)^{3}− g2℘(β) − g_{3}
2(℘(α) − ℘(β))

#^{2}

(88)

for ℘(α) 6= ℘(β).

3.5. Elliptic functions of multiple argument. For brevity, we shall write s, c and d for sn u, cn u and dn u.

Put α = β = u in (87), and then it follows that sn 2u = 2scd

1 − ms^{4} , cn 2u = 1 − 2s^{2}+ ms^{4}

1 − ms^{4} , dn 2u =1 − 2ms^{2}+ ms^{4}

1 − ms^{4} . (89)
For m = 0 these reduce to

sin 2u = 2 sin u cos u, cos 2u = 1 − 2 sin^{2}u, 1 = 1 ; (90)
and for m = 1 these reduce to

tanh 2u = 2 tanh u

1 + tanh^{2}u (91)

and

cosh 2u = 2 cosh^{2}u − 1 (twice). (92)
For the Weierstraß function, taking the limit of (88) as β → α, we get the rational
formula

℘(2α) = −2℘(α) + (3℘^{2}(α) − ^{1}_{4}g_{2})^{2}
4℘^{3}(α) − g2℘(α) − g3

= 16℘^{4}(α) + 8g_{2}℘^{2}(α) + 32g_{3}℘(α) + g_{2}^{2}
16 4℘^{3}(α) − g2℘(α) − g3

, (93)

in Southard [24, §18.4.5]. Define

H(x) ^{def}= 16x^{4}+ 8g2x^{2}+ 32g3x + g^{2}_{2}

16 4x^{3}− g_{2}x − g_{3} , (94)
so that H(℘(α)) = ℘(2α).

Define the function Wj by W0(y) ^{def}= y and for j > 0 define Wj(y) ^{def}=
H(Wj−1(y)); so that Wj(y) is a rational function of y with coefficients which are
polynomials in g2 and g3, with integer coefficients.

Then, induction on (93) shews that for integer j ≥ 0,

℘ 2^{j}α

= Wj(℘(α)), (95)

But if n is not a power of 2, then the irrational formulæ for ℘(nα) are very complicated.

Put α = 2u and β = u in (87) and we get rational functions, with denominator and numerators expressed as polynomials in s = sn u and m;

D = 1 − 6ms^{4}+ 4m[1 + m]s^{6}− 3m^{2}s^{8},
sn 3u = s 3 − 4[1 + m]s^{2}+ 6ms^{4}− m^{2}s^{8} /D,
cn 3u = c 1 − 4s^{2}+ 6ms^{4}− 4m^{2}s^{6}+ m^{2}s^{8} /D,

dn 3u = d 1 − 4ms^{2}+ 6ms^{4}− 4ms^{6}+ m^{2}s^{8} /D. (96)
Hereafter, n is a positive integer.

Those formulæ for addition of Jacobian elliptic functions can be iterated, to get
real formulæ for sn(nu) et cetera, in terms of rational functions of z = sn^{2}u, as in
Cayley [12, pp. 78–87].

For odd n = 2r + 1, sn(nu) = s An(z)

Dn(z) , cn(nu) = c Bn(z)

Dn(z) , dn(nu) = d Cn(z)

Dn(z) ; (97) and for even n = 2r,

sn(nu) = s c d A_{n}(z)

Dn(z) , cn(nu) = B_{n}(z)

Dn(z) , dn(nu) = C_{n}(z)

Dn(z) . (98)
Each of A_{n}, B_{n}, C_{n}, D_{n} is a polynomial of degree n^{2}÷ 2 — except that A_{2r} has
degree ^{1}_{2}n^{2}− 2 = 2r^{2}− 2. In each polynomial, the coefficient of each power of z
is a reciprocal polynomial in m of degree less than or equal to n^{2}÷ 4, with integer
coefficients, as in Bell [3, p.208].

N.B. Arthur Cayley advised that, when constructing the elliptic functions of
nu = ju + ku, then j = k should be used for even n and j = k ± 1 should be
used for odd n — otherwise the constructed numerators and denominators will
have common multinomial factors which need to be divided out to get the rational
functions in lowest terms [12, p.79]. The appropriate degrees of z = s^{2} in the
numerators and denominators are given in the previous paragraph. Without that
information, it would be necessary to divide numerator and denominator by their
gcd, in the form of a polynomial in z whose coefficients are polynomials in m with
integer coefficients.

3.6. Transcendental formulæ for real multiplication of elliptic functions.

Abel inverted some elliptic integrals and invented elliptic functions φ, f and F which correspond to sn, cn and dn; and he constructed an explicit formula for φ((2r + 1)u) as a rational function (in factored form) of φ(u), and similarly for f and F [1, pp.321–323]. That is called Real Multiplication of Elliptic Functions.

Since (5) Tn(cos ϑ) ≡ cos(nϑ), the Chebyshev polynomials Tn perform real mul-
tiplication of the cosine function^{2}and the reciprocal rational functions s_{n} perform
real multiplication of the secant function; the polynomials E_{n} perform real mul-
tiplication of the square of the cosine function (14)^{3} and the reciprocal rational
functions F_{n} perform real multiplication of the square of the secant function (68).

The rational functions r_{n} perform real multiplication of the tangent function and
the reciprocal rational functions h_{n} perform real multiplication of the cotangent
function (69); the rational functions vnperform real multiplication of the square of
the tangent function and the reciprocal rational functions znperform real multipli-
cation of the square of the cotangent function (71) .

For integers µ and ν, denote

Q_{µ,ν} ^{def}= 2µK(m) + i 2νK^{0}(m). (99)
Then sn Q_{µ,ν}= 0 for all integers µ and ν, and conversely every zero of sn is of that
form Q_{µ,ν}.

Consider an odd positive integer n = 2r + 1. Then sn(Qµ,ν/n) has n^{2} distinct
values for all integers µ and ν, and those are all given by −r ≤ µ ≤ r and −r ≤
ν ≤ r. Those n^{2} values include sn(Q0,0/n) = 0; so that if we exclude that zero

2And of the hyperbolic cosine function.

3And of the square of the hyperbolic cosine function.