aa r X i v : . [ m a t h . C V ] S e p THE ELLIPTIC FUNCTION dn OF SHEN
P.L. ROBINSON
Abstract.
We analyze the elliptic function dn introduced by Li-Chien Shen, contributingto the Ramanujan theory of elliptic functions in signature four. Introduction
In [4] Li-Chien Shen introduced and studied a signature-four counterpart dn to the classicalJacobian elliptic function dn. He showed that the function dn is elliptic, expressing it in termsof its coperiodic Weierstrass function and in terms of associated Jacobian functions; he alsolocated its poles and gave expressions for its fundamental periods. The formulation throughoutwas based on theta functions, and included much more information than this summary suggests.Our aim in the present paper is twofold. On the one hand, we amplify several aspects ofthe account presented in [4] and view matters from a different perspective, completely avoidingtheta functions. On the other hand, we reinforce the position of dn in Ramanujan’s theory ofelliptic functions to alternative bases, in part by showing how dn enables the establishment ofhypergeometric identities relating signature four both to itself and to the classical signature. Preliminary results
Our primary purpose in this opening section is to construct the elliptic function dn andpresent some of its most basic properties. We do not aim at anything approaching completeness:for further detail, see the original account [4] in which dn made its first appearance; see alsothe partial reworking [3] in line with the present paper.We begin by fixing a modulus κ ∈ ( , ) and defining f ( T ) = ∫ T F ( , ; ; κ sin t ) d t where F = F is the hypergeometric function with the indicated parameters. Here, f isanalytic and invertible in a sufficiently small open neighbourhood of 0; its analytic inverse φ isthen defined in a suitably small open disc D about 0 and an auxiliary analytic function ψ isdefined in D by ψ ( ) = ψ = κ sin φ .Now dn is initially the analytic function defined in D by the ruledn = cos ψ. A companion analytic function sn is defined in D by the rulesn = sin φ. Note that in D there holds the counterpartdn = − sin ψ = − κ sin φ = − κ sn to the identity dn + κ sn = D is largely immaterial: it turns out that dn as defined above isin fact the restriction to D of an elliptic function, this elliptic function being dn proper.It is now convenient to introduce the complementary modulus λ = √ − κ ∈ ( , ) . Theorem 1.
The function dn satisfies dn ( ) = and the differential equation ( dn ′ ) = ( − dn )( dn − λ ) . Proof. As φ is inverse to f and as f ′ ○ φ = F ( , ; ; κ sin φ ) = F ( , ; ; sin ψ ) it follows from the standard hypergeometric evaluation F ( , ; ; sin ψ ) = cos ψ cos ψ that φ ′ = cos ψ cos ψ . Additionally, sin ψ = κ sin φ implies ψ ′ = κ cos φ cos ψ while dn = cos ψ implies dn ′ = − sin ψ ψ ′ . As sin ψ = ψ cos ψ it follows thatdn ′ = − ( sin 12 ψ )( κ cos φ ) . As cos ψ = − ψ and κ − sin ψ = cos ψ − λ it follows that ( dn ′ ) = ( − dn )( dn − λ ) . (cid:3) At present, dn is defined locally around 0. The following result allows us to globalize. Theorem 2.
The function dn satisfies ( − dn )( + ℘) = κ where ℘ is the Weierstrass elliptic function with invariants g = − κ = λ + g = − κ = λ − . Proof.
The function p = − + κ − dn has a pole at 0 and direct calculation shows that it satisfies ( p ′ ) = p − ( λ + ) p − ( λ − ) . This initial value problem characterizes the Weierstrass function with the stated invariants. (cid:3)
Briefly, we may say that dn is the elliptic function given bydn = − κ /( + ℘) . Were greater clarity desirable, we could initially define d = cos ψ on D : from d ( ) = ( d ′ ) = ( − d )( d − λ ) it would then follow that d is the restriction to D of this elliptic function. HE ELLIPTIC FUNCTION dn OF SHEN 3
The invariants of ℘ being real and its discriminant∆ = g − g = κ λ being positive, the period lattice of ℘ is rectangular. Let 2 ω = K and 2 ω ′ = K ′ be fundamentalperiods with K > K ′ >
0. It is a familiar fact that ℘ has purely real values on the edgesof the half-period rectangle Q with vertices0 , K, K + i K ′ , i K ′ ;also that the values of ℘ decrease strictly from +∞ to −∞ upon counterclockwise passage aroundthe perimeter of Q starting from and returning to 0. The (midpoint) values of ℘ at the non-zerovertices of Q are the zeros of the cubic 4 w − g w − g in w : explicitly, ℘( K ) = + λ, ℘( K + i K ′ ) = − λ, ℘( i K ′ ) = − . Theorem 3.
The elliptic function dn has K and K ′ as fundamental periods. It has adouble pole at i K ′ ; also, dn ( K ) = λ and dn ( K + i K ′ ) = − λ. Proof.
Theorem 2 shows that the elliptic functions dn and ℘ are coperiodic. Moreover, thelattice midpoint i K ′ is a double zero of + ℘ and hence a double pole of dn . The values ofdn at K and K + i K ′ follow from those of ℘ . (cid:3) Here, note that the values of dn decrease strictly from +∞ to −∞ upon counterclockwisepassage around the perimeter of Q starting from and returning to i K ′ .The Weierstrass function ℘ has an associated triple sn , cn , dn of classical Jacobian ellipticfunctions. With the semistandard notation e = ℘( K ) , e = ℘( K + i K ′ ) , e = ℘( i K ′ ) for the midpoint values in decreasing order, these Jacobian functions have modulus k ∈ ( , ) given by k = e − e e − e and the Jacobian modular sine function sn is related to ℘ by ℘( z ) = e + e − e sn [ z √ e − e ] . Theorem 4.
The elliptic function dn satisfies dn ( z ) = − ( − λ ) sn [ z √ + λ ] where sn = sn ( ● , k ) is the Jacobian sine function with modulus k given by k = − λ + λ . Proof.
From Theorem 2, noting that e − e = ( + λ )/ e − e = ( − λ )/ (cid:3) We close this section by returning to the initial definition of dn . Note that the formula for f ( T ) is valid whenever T is real, thereby defining a function f ∶ R → R whose periodic derivativeis strictly positive. The inverse φ of f is therefore defined on the whole real line: if u ∈ R then u = ∫ φ ( u ) F ( , ; ; κ sin t ) d t. More is true. From the proof of Theorem 1 we see that ( φ ′ ) = ψ + cos ψ = + dn . Here, dn has poles at ± i K ′ , while taking the value − ∈ ( −∞ , − λ ) at a point w on the upperedge of the rectangle Q and at the conjugate point w . It follows (by ‘square-rooting’) that φ ′ P.L. ROBINSON extends holomorphically to the open band { z ∈ C ∶ ∣ Im z ∣ < K ′ } and then (by integrating from φ ( ) =
0) that φ itself extends holomorphically to the same band. Similarly, as the zeros of1 − dn (at points congruent to 0 modulo periods) are double, the identitydn + κ sn = extends holomorphically to the very same band, throughout which the relationsn = sin ○ φ continues to hold. Fundamental periods
Here, we derive explicit formulae for the fundamental periods 2 K and 2i K ′ of the ellipticfunction dn . In fact, we derive more than one formula for each of these periods and take morethan one approach to do so. Our formulae express these periods in terms of the modulus κ andcomplementary modulus λ = √ − κ .In the interests of clarity, we shall replace dn by dn κ as a symbol for the elliptic func-tion hitherto constructed with κ as modulus, making similar replacements when referring tofundamental periods and half-period rectangles. The complementary modulus λ engenders acorresponding elliptic function dn λ with fundamental periods 2 K λ and 2i K ′ λ .Recall from Theorem 1 that dn κ satisfies the differential equation ( dn ′ κ ) = ( − dn κ )( dn κ − λ ) . Recall also the behaviour of dn κ on the rectangle Q κ : it decreases strictly from dn κ ( ) = κ ( K κ ) = λ along the lower edge [ , K κ ] and decreases strictly from dn κ ( K κ ) = λ todn κ ( K κ + i K ′ κ ) = − λ along the right edge [ K κ , K κ + i K ′ κ ] . It follows thatdn ′ κ = − √ ( − dn κ )( dn κ − λ ) along [ , K κ ] and therefore that ∫ K κ = − ∫ dn κ ( K κ ) dn κ ( ) d x √ ( − x )( x − λ ) or K κ = ∫ λ d x √ ( − x )( x − λ ) . In order to calculate K ′ κ we integrate up the right edge of Q κ as follows: the function d definedby d ( y ) = dn κ ( K κ + i y ) satisfies − ( d ′ ) = ( − d )( d − λ ) and decreases along [ , K ′ κ ] so that on this intervald ′ = − √ ( − d )( λ − d ) and therefore K ′ κ = ∫ λ − λ d y √ ( − y )( λ − y ) . In preparation for the following theorem, specify the complementary acute angles α and β by requiring that κ = cos α and λ = cos β . Further, when γ is an acute angle write I ( γ ) = ∫ γ cos t √ cos t − cos γ d t. HE ELLIPTIC FUNCTION dn OF SHEN 5
Theorem 5.
The fundamental periods K κ and K ′ κ of dn κ are given by K κ = I ( β ) and K ′ κ = √ I ( α ) . Proof.
The following argument is extracted from [4] Theorem 3.3. In the integral formula K κ = ∫ λ d x √ ( − x )( x − λ ) substitute x = cos t and (as agreed) λ = cos β : there follows K κ = ∫ β − sin t √ ( − cos t )( cos t − cos β ) d t whence by trigonometric dimidiation and cancellation K κ = ∫ β cos t √ cos t − cos β d t. This establishes the first of the advertised formulae; the second is a little more demanding. Inthe integral formula K ′ κ = ∫ λ − λ d y √ ( − y )( λ − y ) substitute first y = cos θ where β ⩽ θ ⩽ π − β and then t = θ − π where − α = β − π ⩽ t ⩽ π − β = α. The first substitution leads to K ′ κ = ∫ π − ββ cos θ √ cos β − cos θ d θ essentially as above, whereupon the second substitution gives K ′ κ = √ ∫ α − α ( cos t − sin t )√ cos t − cos α d t because cos α + cos β = = cos t + cos θ ; finally, trigonometric parities justify K ′ κ = √ ∫ α cos t √ cos t − cos α d t. (cid:3) An interesting conclusion to draw here is the symmetric pair of relations K ′ κ = √ K λ and K ′ λ = √ K κ involving the complementary moduli κ and λ . These relations have implications for the size andshape of the half-period rectangles Q κ and Q λ associated to the corresponding elliptic functionsdn κ and dn λ . Thus K κ K ′ κ = K λ K ′ λ (so that the half-period rectangles have equal area) and K ′ κ K κ K ′ λ K λ = . An alternative evaluation of K κ and i K ′ κ proceeds by recognizing that the formulae derivedleading up to Theorem 5 express these ‘half-periods’ as elliptic integrals. For this evaluation,we quote the following results from Section 43 of the classic text [2] by Greenhill, with minornotational adjustments. P.L. ROBINSON
Let the cubic T = ( t − a )( t − b )( t − c ) have real roots a > b > c. Then ∫ ab d t √ − T = √ a − c K ( a − ba − c ) and ∫ bc d t √ T = √ a − c K ( b − ca − c ) where if 0 < k < K ( k ) stands for the complete elliptic integral given by K ( k ) = ∫ π d θ √ − k sin θ = πF ( , ; 1; k ) . Theorem 6.
The fundamental periods K κ and K ′ κ of dn κ are given by K κ = √ + λ K ( − λ + λ ) and K ′ κ = √ + λ K ( λ + λ ) . Proof.
Here, we take the cubic T given by T = ( t − )( t − λ )( t + λ ) and apply the foregoing results quoted from [2] with a = b = λ and c = − λ so that a − c = + λ, a − ba − c = − λ + λ and b − ca − c = λ + λ . (cid:3) Otherwise said, K κ = π √ + λ F ( ,
12 ; 1; 1 − λ + λ ) and K ′ κ = π √ + λ F ( ,
12 ; 1; 2 λ + λ ) . The results of Theorem 6 can equivalently be approached from a Weierstrassian direction.Let dn κ have coperiodic Weierstrass function ℘ as in Theorem 2. As noted prior to Theorem3, the midpoint values of ℘ (named according to one of the standard conventions) are e = + λ, e = − λ, e = − . At this stage, we merely quote from Section 51 of [2]: the half-period ω = K κ of ℘ is given by ω = K ( k )√ e − e where e − e = + λ k = e − e e − e = − λ + λ HE ELLIPTIC FUNCTION dn OF SHEN 7 whence we recover the previous formula for K κ . Similarly, ω ′ = i K ( k ′ )√ e − e where now k ′ = e − e e − e = λ + λ and we recover the previous formula for K ′ κ .The same results can be approached from a Jacobian elliptic direction. Recall from Theorem4 that dn κ ( z ) = − ( − λ ) sn (√ + λ z ) where sn = sn ( ● , k ) is the Jacobian sine function with modulus k given by k = − λ + λ . Now, from dn κ ( K κ ) = λ it follows that sn (√ + λ K κ ) =
1; moreover, if 0 < x < K κ then1 > dn κ ( x ) > λ so that 0 < sn (√ + λ x ) <
1. It follows easily that √ + λ K κ is the least positive x such that sn ( x ) =
1: that is, √ + λ K κ = K ( − λ + λ ) or again K κ = √ + λ K ( − λ + λ ) . Similarly up the imaginary axis: dn κ encounters its first (double) pole at i K ′ κ while sn encountersits first (simple) pole at i K ′ ( − λ + λ ) = i K ( λ + λ ) so that again K ′ κ = √ + λ K ( λ + λ ) . Certain values of the modulus κ perhaps deserve special mention. The self-complementaryvalue κ = /√ κ = dn λ (and of the coperiodic Weierstrass function) is given by the ratio K ′ κ / K κ = √
2; as expected, the corresponding Jacobian modulus k is √ − k = − λ + λ = √ − √ + = (√ − ) . Next, let κ = √ / λ = /
3. In this case, Theorem 2 shows that the coperiodicWeierstrass function has g = k = /√ − λ = λ . Instead, let κ = / λ = √ / K ′ κ / K κ = k = √ − λ + λ = ¿ÁÁÀ − √ + √ = − √ = tan π . P.L. ROBINSON
Hypergeometric identities
Thus far, we have avoided what may be the most natural formula for the fundamentalperiod 2 K κ of dn κ . We proceed to rectify this omission, after which we extract from thisformula and those of the previous section a number of hypergeometric function identities. Noneof these identities is new; however, our approach to them presents new aspects and opens upthe possibility of further identities.For the following paragraph, we temporarily return to the earlier notation dn in place ofdn κ and K in place of K κ .Recall from the close of our ‘Preliminary Results’ that both sn and φ extend to functionsthat are holomorphic in the open band { z ∈ C ∶ ∣ Im z ∣ < K ′ } where they continue to satisfy thedefining relation sn = sin ○ φ. Recall also the identitydn + κ sn = and sn in the discussion following Theorem 6, thatsn ( K ) = φ ( K ) = π . Lastly, recall that if u ∈ R then u = ∫ φ ( u ) F ( , ; ; κ sin t ) d t. Let us now reinstate the more informative notation dn κ and K κ . Theorem 7.
The fundamental half-period K κ of dn κ is given by K κ = π F ( , ; 1; κ ) . Proof.
It is a familiar fact that if a and b are arbitrary then ∫ π F ( a, b ; ; κ sin t ) d t = πF ( a, b ; 1; κ ) as may be verified by termwise integration of the hypergeometric series. Let a = and b = ;then summon the recollections that prefaced this Theorem. (cid:3) We are now ready to extract hypergeometric identities.Firstly, direct comparison of Theorem 7 with the first formula after Theorem 6 at once yieldsthe hypergeometric identity F ( ,
34 ; 1; 1 − λ ) = √ + λ F ( ,
12 ; 1; 1 − λ + λ ) when it is recalled that κ = − λ . This identity is proved as Theorem 9.2 in [1]. If λ is replacedby ( − x )/( + x ) - equivalently, if x is replaced by ( − λ )/( + λ ) - then this identity becomesthe quadratic transform listed under Corollary 4.2 in [4].Secondly, direct comparison of Theorem 7 with the second formula after Theorem 6 at onceyields the hypergeometric identity F ( ,
34 ; 1; λ ) = √ + λ F ( ,
12 ; 1; 2 λ + λ ) when K ′ κ = √ K λ is recalled from Theorem 5. This identity is recorded on page 260 ofRamanujan’s second notebook; it is proved as Theorem 9.1 in [1].Also recorded on page 260 of Ramanujan’s second notebook is the following transformationlaw. It appears as Theorem 9.4 in [1]; there, it is proved by analytic continuation from smallvalues of the variable. It also appears as Lemma 6.2 in [4]; there, the proof involves thetafunctions and Landen transformations. HE ELLIPTIC FUNCTION dn OF SHEN 9
Theorem 8. If < x < then √ + x F ( ,
34 ; 1; x ) = F ( ,
34 ; 1; 1 − ( − x + x ) ) . Proof.
Let x and y be related by the condition2 x + x = − y + y and consider the equation F ( ,
12 ; 1; 2 x + x ) = F ( ,
12 ; 1; 1 − y + y ) . On the left side, F ( ,
12 ; 1; 2 x + x ) = √ + xF ( ,
34 ; 1; x ) by virtue of the second hypergeometric identity displayed just prior to the present Theorem.On the right side, F ( ,
12 ; 1; 1 − y + y ) = √ + y F ( ,
34 ; 1; 1 − y ) by virtue of the first hypergeometric identity displayed prior to this Theorem. Finally, thecondition relating x and y is equivalent both to y = − x + x and to 1 + y = + x + x so substitutions and cancellation conclude the proof. (cid:3) Incidentally (and perhaps contrary to appearances) the conditions relating x and y in thisproof are actually symmetric in x and y : they are equivalent to x + y + xy = R EFERENCES [1] B.C. Berndt, S. Bhargava, and F.G. Garvan,
Ramanujan’s theories of elliptic functions toalternative bases , Transactions of the American Mathematical Society (1995) 4163-4244.[2] A.G. Greenhill,
The Applications of Elliptic Functions , Macmillan and Company (1892);Dover Publications (1959).[3] P.L. Robinson,
Elliptic functions from F ( , ; ; ● ) , arXiv 1908.01687 (2019).[4] Li-Chien Shen, On a theory of elliptic functions based on the incomplete integral of thehypergeometric function F ( , ; ; z ) , Ramanujan Journal (2014) 209-225. Department of Mathematics, University of Florida, Gainesville FL 32611 USA
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