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PERMUTABLE POLYNOMIALS AND RATIONAL FUNCTIONS

Garry J. Tee

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

T0(x) = 1, T1(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.

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The integer coefficients of Td are given explicitly by the following formula1 for d > 0, in Lyusternik [20, p.79]:

Td(x) =

d÷2

X

k=0

(−1)k2d−2k−1d(d − k − 1)!

k!(d − 2k)! xd−2k = 2d−1xd− 2d−3dxd−2+ 2d−6d(d − 3)xd−42d−8

3 d(d − 4)(d − 5)xd−6+ · · ·

· · · +

 (−1)d÷2 (for even d),

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

T0(x) = 1, T1(x) = x, T2(x) = 2x2− 1, T3(x) = 4x3− 3x, T4(x) = 8x4− 8x2+ 1, T5(x) = 16x5− 20x3+ 5x, T6(x) = 32x6− 48x4+ 18x2− 1, T7(x) = 64x7− 112x5+ 56x3− 7x,

T8(x) = 128x8− 256x6+ 160x4− 32x2+ 1, T9(x) = 256x9− 576x7+ 432x5− 120x3+ 9x,

T10(x) = 512x10− 1280x8+ 1120x6− 400x4+ 50x2− 1, T11(x) = 1024x11− 2816x9+ 2816x7− 1232x5+ 220x3− 11x,

T12(x) = 2048x12− 6144x10+ 6912x8− 3584x6+ 840x4− 72x2+ 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) =

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

Tj Tk(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 .

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“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) = xj, 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−1uλ and V def= λ−1vλ are permutable, since

U (V (x)) = λ−1u

λ λ−1v(λ(x))

= λ−1u(v(λ(x))) =

= λ−1v(u(λ(x))) = λ−1v

λ λ−1u(λ(x))

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

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 xj , 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)) = 2Tk2(x) − 1, (11) which is an even polynomial. Hence, with y = x2 we get the identity

Tk2(x) ≡ 12(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) = 4x2− 4x + 1, E3(x) = 16x3− 24x2+ 9x,

E4(x) = 64x4− 128x3+ 80x2− 16x + 1, E5(x) = 256x5− 640x4+ 560x3− 200x2+ 25x,

E6(x) = 1024x6− 3072x5+ 3456x4− 17923+ 420x2− 36x + 1, (13) et cetera.

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

cos2(kϑ) = Tk2(cos ϑ) = Ek(cos2ϑ), (14) and likewise

cosh2(kς) = Tk2(cosh ς) = Ek(cosh2ς) . (15) And thus

Ej Ek(cos2ϑ)

= Ej(cos2(kϑ)) = cos2(jkϑ) = Ejk(cos2ϑ). (16)

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Writing x = cos2ϑ, 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

Ej Ek(x)

≡ Ejk(x) ≡ Ekj(x) ≡ Ek Ej(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(z2) = 2Ek(z2) − 1 = T2k(z) = Tk(T2(z)) =

= Tk(2z2− 1) = Tk(λ(z2)). (19) Replacing z2 by x, we get the similarity between the chains {Ek} and {Tk}

Ek(x) ≡ λ−1

Tk λ(x)

. (20)

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

πr πs(x)

= πr xs

= xsr

= xsr =

= xrs = xrs

= πs xr

= πs πr(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 x2i+1 are permutable, and so is the reflected sequence

{x2i+1, −x2i+1} = · · · , x−1, −x−1, x, −x, x3, −x3, · · · . (27)

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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)jT2j+1(sin ϑ). (28) Denote

qj(z) def= (−1)jT2j+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 {xr} is similar (via λ(x) = −x) to the permutable sequence

{−(−x)r} = · · · , x−3, −x−2, x−1, −1, x, −x2, x3, −x4, · · · . (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 {Tj(x)/1} for j = 0, 1, 2, . . . , which is similar (via λ(x) = −x) to the permutable rational sequence {Tj(−x)/(−1)}. And since T2i+1 is an odd polynomial, there is also the reflected sequence {T2i+1(x)/1, T2i+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.

zn= (x + iy)n=

n

X

j=0

n j



ijxn−jyj= Hn(0)(x, y) + iHn(1)(x, y), (33) where the real and imaginary parts of zn are given in Lyusternik [20, Appendix 1,

§2.1] by the harmonic polynomials : Hn(0)(x, y) =

n÷2

X

k=0

(−1)k n 2k



xn−2ky2k

Hn(1)(x, y) =

(n−1)÷2

X

k=0

(−1)k

 n

2k + 1



xn−2k−1y2k+1 . (34) For all complex ρ, log(cos ρ + i sin ρ) = iρ (by Cotes’s theorem), or

cos ρ + i sin ρ = e; (35)

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

cos(nρ) + i sin(nρ) = einρ = (cos ρ + i sin ρ)n. (36)

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Therefore

cos(nρ) + i sin(nρ) = (cos ρ + i sin ρ)n =

= Hn(0)(cos ρ, sin ρ) + iHn(1)(cos ρ, sin ρ). (37) Equating real and imaginary parts (for real ρ), we get that

cos(nρ) = Hn(0)(cos ρ, sin ρ), sin(nρ) = Hn(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 cosnρ, 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)ktan2k+1ρ

n÷2

X

k=0

 n 2k



(−1)ktan2kρ

. (40)

Denote

t def= tan ρ, z def= −t2 = − tan2ρ. (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



zk, Bn(z) =

n÷2

X

k=0

 n 2k



zk, (44) and Bn(0) = 1.

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

r2j+1(t) = t A2j+1(z) B2j+1(z) =

= t 2j + 1 + 2j+13 z + 2j+15 z2+ · · · + 2j+12 zj−1+ zj

1 + 2j+12 z + 2j+14 z2+ · · · + 2j+13 zj−1+ (2j + 1)zj , (45) where the polynomials A2j+1and B2j+1are mutually reciprocal:

zjA2j+1

 1 z



= B2j+1(z), zjB2j+1

 1 z



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

r2j(t) = t A2j(z)

B2j(z) = t 2j + 2j3z + 2j5z2+ · · · + 2j3zj−2+ 2jzj−1

1 + 2j2z + 2j4z2+ · · · + 2j2zj−1+ zj , (47)

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where both A2j and B2j are self–reciprocal polynomials:

A2j(z) = zj−1A2j

 1 z



, B2j(z) = zjB2j

 1 z



, (48)

for all z 6= 0.

For example, r0(t) = t0

1 = 0 , r1(t) = t1 1 = t , r2(t) = t 2

1 + z , r3(t) = t 3 + z 1 + 3z , r4(t) = t 4 + 4z

1 + 6z + z2 , r5(t) = t 5 + 10z + z2 1 + 10z + 5z2 , r6(t) = t 6 + 20z + 6z2

1 + 15z + 15z2+ z3 ,

(49)

et cetera .

Since cot ρ = tan 12π − ρ, we get that rn(cot ρ) = rn

 tanπ

2 − ρ

= tan 2 − nρ

. (50)

For odd n = 2j + 1, this becomes

r2j+1(cot ρ) = tan jπ +12π − (2j + 1)ρ = tan 12π − (2j + 1)ρ

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

r2j(cot ρ) = tan (jπ − 2jρ) = − tan(2jρ). (52) 2.2. Permutable rational functions rn(t) and vn(y). For the infinite sequence of odd rational functions {rn}, it follows from (42) that, for all complex t and non–negative integers j, k, with ρ = tan−1t,

rj rk(t)

= rj rk(tan ρ)

= rj(tan(kρ)) = tan(jkρ) =

= rjk(tan ρ) = rjk(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 − 4t3 1 − 6t2+ t4



= 4rk(t) − 4r3k(t)

1 − 6r2k(t) + r4k(t) . (56) Thus we get for n = 0, 1, 2, . . . the reflected sequence of permutable odd rational functions {rn(t), −rn(t)}.

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Now, with t = tan ρ and y = t2= −z, (rn(t))2 = t2 An(−t2)

Bn(−t2)

2

= vn(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 ,

v2(y) = y

 2 1 − y

2

, v3(y) = y 3 − y

1 − 3y

2 ,

v4(y) = y

 4 − 4y 1 − 6y + y2

2

, v5(y) = y 5 − 10y + y2 1 − 10y + 5y2

2 ,

v6(y) = y

 6 − 20y + 6y2 1 − 15y + 15y2− y3

2 ,

(58)

et cetera.

Hence, we get the identities

vn(tan2ρ) ≡ (rn(tan ρ))2 ≡ tan2(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 = tan2ρ,

vj vk(y)

= vj vk(tan2ρ)

= vj(tan2(kρ)) = tan2(jkρ) =

= vjk(tan2ρ) = 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

vjk(y) ≡ vj vk(y)

≡ vk vj(y). (62)

Since cot ρ = tan 12π − ρ, we get that vn(cot2 ρ) = vn



tan2π 2 − ρ

= tan2 2 − nρ

. (63)

For odd n = 2j + 1, this becomes

v2j+1(cot2 ρ) = tan2 jπ +12π − (2j + 1)ρ = tan2 12π − (2j + 1)ρ

= cot2((2j + 1)ρ), (64) and for even n = 2j, this becomes

v2j(cot2ρ) = tan2(jπ − 2jρ) = (− tan(2jρ))2 = tan2(2jρ). (65)

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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−1u(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−1uξ and ξ−1vξ 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 {fn(x) = Tn(x)/1} has the reciprocal permutable sequence {sn(x) = 1/Tn(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(cos2ϑ) = cos2(nϑ), Fn(sec2ϑ) = sec2(nϑ). (68) The odd rational function rn(t) has the reciprocal odd rational function hn(t) = 1/rn(1/t), for which

rn(tan ρ) = tan(nρ), hn(cot ρ) = cot(nρ). (69) In view of (51) and (52), this can be rewritten as the identities

h2k+1(t) ≡ r2k+1(t), h2k(t) ≡ −1

r2k(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

vn(tan2ρ) = tan2(nρ), zn(cot2ρ) = cot2(nρ). (71) Squaring the equations (70), this can be rewritten as the identities

z2k+1(y) ≡ v2k+1(y), z2k(y) ≡ 1

v2k(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 y2= 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].

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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 − t2)(1 − mt2) = Z ϕ

0

dx p1 − m sin2x

. (73) (Earlier authors often used the modulus k, where m = k2.) 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 (12π | m) =

Z 1 0

dt

p(1 − t2)(1 − mt2) =

= Z π/2

0

dx p1 − m sin2x

. (74)

As m % 1, then K(m) % ∞.

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

K0(m) def= K(1 − m). (75)

K(m) and K0(m) are often abbreviated to K and K0, 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 − t2)(1 − mt2) , (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−1τ = u = Z τ

0

dt

p(1 − t2)(1 − mt2) = F sin−1τ | m . (77)

(11)

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−1τ has a single value in the real interval [−K, K].

In addition to the real period 4K, sn also has the imaginary period i 2K0. 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 − sn2u , (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−1r has a single value in the real interval [0, 2K].

In addition to the real period 4K, cn has the complex period 2K + i 2K0. 3.2.3. Jacobian elliptic function dn u. The Jacobian elliptic function dn is defined by

dn(u | m) def= p

1 − m sn2u . (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−1r has a single value in the real interval [0, K].

In addition to the real period 2K, dn also has the imaginary period i 4K0. 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).

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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) − g2℘(z) − g3. (82) Hence, Weierstraß’s function ℘ is the inverse of an elliptic integral, with

z = Z

℘(z)

dw

p4w3− g2w − g3 (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

u2 + O(u2). (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

2 +

π sinπu



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 cosec2x −13 ” [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 sn2α · sn2β , cn(α + β) = cn α · cn β − sn α · dn α · sn β · dn β

1 − m sn2α · sn2β , dn(α + β) = dn α · dn β − m sn α · cn α · sn β · cn β

1 − m sn2α · sn2β . (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) − g3p4℘(β)3− g2℘(β) − g3 2(℘(α) − ℘(β))

#2

(88)

(13)

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 − ms4 , cn 2u = 1 − 2s2+ ms4

1 − ms4 , dn 2u =1 − 2ms2+ ms4

1 − ms4 . (89) For m = 0 these reduce to

sin 2u = 2 sin u cos u, cos 2u = 1 − 2 sin2u, 1 = 1 ; (90) and for m = 1 these reduce to

tanh 2u = 2 tanh u

1 + tanh2u (91)

and

cosh 2u = 2 cosh2u − 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) − 14g2)2 4℘3(α) − g2℘(α) − g3

= 16℘4(α) + 8g22(α) + 32g3℘(α) + g22 16 4℘3(α) − g2℘(α) − g3

 , (93)

in Southard [24, §18.4.5]. Define

H(x) def= 16x4+ 8g2x2+ 32g3x + g22

16 4x3− g2x − g3 , (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,

℘ 2jα

= 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 − 6ms4+ 4m[1 + m]s6− 3m2s8, sn 3u = s 3 − 4[1 + m]s2+ 6ms4− m2s8 /D, cn 3u = c 1 − 4s2+ 6ms4− 4m2s6+ m2s8 /D,

dn 3u = d 1 − 4ms2+ 6ms4− 4ms6+ m2s8 /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 = sn2u, as in Cayley [12, pp. 78–87].

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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 An(z)

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

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

Dn(z) . (98) Each of An, Bn, Cn, Dn is a polynomial of degree n2÷ 2 — except that A2r has degree 12n2− 2 = 2r2− 2. In each polynomial, the coefficient of each power of z is a reciprocal polynomial in m of degree less than or equal to n2÷ 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 = s2 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 function2and the reciprocal rational functions sn perform real multiplication of the secant function; the polynomials En perform real mul- tiplication of the square of the cosine function (14)3 and the reciprocal rational functions Fn perform real multiplication of the square of the secant function (68).

The rational functions rn perform real multiplication of the tangent function and the reciprocal rational functions hn 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

Qµ,ν def= 2µK(m) + i 2νK0(m). (99) Then sn Qµ,ν= 0 for all integers µ and ν, and conversely every zero of sn is of that form Qµ,ν.

Consider an odd positive integer n = 2r + 1. Then sn(Qµ,ν/n) has n2 distinct values for all integers µ and ν, and those are all given by −r ≤ µ ≤ r and −r ≤ ν ≤ r. Those n2 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.

References

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