Pacific Journal of Mathematics

Transcription

1 Pacific Journal of Mathematics ELLIPTIC FUNCTIONS TO THE QUINTIC BASE HENG HUAT CHAN AND ZHI-GUO LIU Volume 226 No. July 2006

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3 PACIFIC JOURNAL OF MATHEMATICS Vol. 226, No., 2006 ELLIPTIC FUNCTIONS TO THE QUINTIC BASE HENG HUAT CHAN AND ZHI-GUO LIU We study two elliptic functions to the quintic base and find two nonlinear second order differential equations satisfied by them. We then derive two recurrence relations involving certain Eisenstein series associated with the group Ɣ 0. These recurrence relations allow us to derive infinite families of identities involving the Eisenstein series and Dedekind η-products. An imaginary transformation for one of the elliptic functions is also derived.. Introduction Let q = e 2πiτ, where Im τ > 0. The Dedekind η-function is defined by where ητ = q /2 q; q, a; q = n= aq n. n= Srinivasa Ramanujan, on a page of his τn- pn manuscript [Ramanujan 988, p. 39; Berndt and Ono 999], stated without proof that n q n q n 2 = η τ ητ, where is the Legendre symbol. This identity is of great interest historically because Ramanujan deduced from it his famous partition identity pn + q n = q ; q q; q 6, n=0 where pn is the number of partitions of n. Proofs of can be found in [Bailey 92a; 92b], and more recently in [Shen 99, p. 329; Chan 99]. MSC2000: 33E0, F, F27. Keywords: Eisenstein series, elliptic functions, Dedekind eta function. Liu was supported in part by Shanghai Priority Academic Discipline and the National Science Foundation of China. 3

4 HENG HUAT CHAN AND ZHI-GUO LIU 2 In an unpublished work, Z.-G. Liu and R. P. Lewis discovered that where 3 n= n 3 q n q 2n q 3n + q n q n nq n Aτ = + 6 q n 30 = η τ + 22 ητ = η τ ητ Aτ, nq n q n ητ ητ ητ 2. ητ The last identity relating Aτ to η-products can be found in [Berndt et al. 2000, Lemma 3.6]. If we rewrite the left-hand side of as and set n n= q n q n 2 = L k q = n= n= nq n q 2n q 3n + q n q n n k q n q 2n q 3n + q n q n, then we observe that and 2 correspond to expressions in terms of the Dedekind η-products for L k q when k = and 3 respectively. Motivated by our recent paper [Chan and Liu 2003], we propose to construct a recurrence satisfied by L 2k+ q, which would then give us expressions of L 2k+ q in terms of η products for every integer k 0. In order to achieve this, we first define V u τ = k 2u2k+ 2k +! L 2k+q. k=0 In Section 2, we will see that V u τ is a elliptic function with periods π and πτ. In Sections 3 and, we construct two differential equations associated with V u τ and Mu τ := 3 Aτ + 6 where Aτ is given by 3 and S k q = k 2u 2k k= 2k! n k q n + q 2n + q 3n + q n q n. S 2k+ q,

5 ELLIPTIC FUNCTIONS TO THE QUINTIC BASE Using these differential equations, we obtain two difference equations satisfied by S 2k+ q and L 2k+ q. These equations allow us to tabulate S 2k+ q and L 2k+ q in terms of the modular functions 6 x = ητ 6 and z = η τ ητ ητ for any k. In Section, we study the behavior of V u τ under the imaginary transformation τ /τ. As a corollary, we indicate some computations of special values of Dirichlet L-series. 2. The Jacobi theta function u τ and Vu τ Recall that the Jacobi theta functions u τ and u τ are given by u τ = 2q /8 n 0 n q nn+/2 sin2n + u, u τ = + 2 n q n2 /2 cos 2nu. From [Whittaker and Watson 927, p. 89] we know that Hence, u πτ τ + u 3 2πτ τ Similarly, = = 8 = u τ = q n/2 sin 2nu. qn q n/2 q n sin 2nu πτ + sin 2nu 3 2 πτ q n/2 cos 3nπτ sin 2nu qn q n/2 q 3n/2 + q 3n/2 q n sin 2nu = q n + q n sin 2nu. qn u + 2 πτ τ + u 2 πτ τ = q 2n + q 3n q n sin 2nu.

6 6 HENG HUAT CHAN AND ZHI-GUO LIU This implies that 2 V u τ = u πτ τ + u 3 2πτ τ u + 2 πτ τ u 2πτ τ, where V u τ is given by. By using the transformation formulas we find that u τ = i + u 2 πτ τ, u πτ τ = u τ 2i, 2 2 u + πτ τ + u πτ τ = u πτ τ + u 3 2πτ τ = q n + q n sin 2nu, qn 2 3 u + 2πτ τ + u 2πτ τ = u + 2 πτ τ + u 2πτ τ = Combining formulas 2 through 2 3, we conclude that 2 V u τ = u+πτ τ + u πτ τ q 2n + q 3n q n sin 2nu. u+2πτ τ u 2πτ τ. 3. Differential equations satisfied by Vu τ and Mu τ Before we proceed with the derivation of a differential equation satisfied by V u τ, we first define the classical Eisenstein series to be used in the sequel. For Im τ > 0, the Eisenstein series E 2 τ is given by 3 E 2 τ = 2 For integer k 2, we define 3 2 E 2k τ = + where ζs = nq n q n, q = e2πiτ. 2πi 2k n 2k q n 2k! ζ2k q n,, Re s >. ns

7 ELLIPTIC FUNCTIONS TO THE QUINTIC BASE 7 We now construct a differential equation satisfied by V u τ. Define f v:= f v τ= v+u+πτ τ v+u πτ τ v u+2πτ τ v u 2πτ τ v τ, where u < π τ. Our choice of f v τ is clearly motivated by 2. Using the transformation formulas t + π τ = t τ and t + πτ τ = q /2 e 2it t τ from [Whittaker and Watson 927, p. 63], we verify that f v is an elliptic function with periods π and πτ. It has a pole of order at v = 0. Now, set Fv = v f v and φv = F v Fv. By elementary calculations, we find that res f v; v = 0 = 6 F0 φ φ0φ 0 + φ 0, where res f v; v = p denotes the residue of f at the pole p. Since the sum of residues of an elliptic function over a period parallelogram is zero and F0 = 0, we conclude that 3 3 φ φ0φ 0 + φ 0 = 0. Next, note that 3 φv = v u τ + v + u + πτ τ + v + u πτ τ + v u + 2πτ τ + v u 2πτ τ = 3 E 2τv + v + u + πτ τ + v + u πτ τ where we have used the expansion 3 + v u + 2πτ τ + v u 2πτ τ + Ov 3, t τ = cot t + q n sin 2nt qn from [Whittaker and Watson 927, p. 89] and the representation given in 3. From 3, we find that φ0 = V u τ.

8 8 HENG HUAT CHAN AND ZHI-GUO LIU Differentiating 3 with respect to v and setting v = 0, we find that 3 6 φ 0 = 3 E 2τ + u + πτ τ + u πτ τ + u + 2πτ τ + u 2πτ τ. Differentiating both sides of 2 2 and 2 3, we conclude from 3 6 that Mu τ = φ 0, where Mu τ is given by. Finally differentiating 3 twice and setting v = 0, we conclude that φ 0 = V u τ, by 2. Substituting these expressions for φ0, φ 0 and φ 0 into 3 3, we conclude: Theorem 3.. V u τ + 3Mu τv u τ + V 3 u τ = 0. Corollary 3.2. For integer k, k L 2k+ = AL 2k + 6 l= 2k!L 2k l + S 2l+ 2l! 2k l! k l=0 k l 2 m=0 where L k := L k q and S k := S k q. 2k!L 2l+ L 2m+ L 2k l m 2+ 2l +! 2m +! 2k l m 2 +!, The recurrence in the corollary involves both L k and S k. In order to tabulate L k, we need another difference equation expressing S k in terms of L s and S t for 3 s, t < k. To achieve this, we need to obtain a second differential equation satisfied by M := Mu τ and V := V u τ. Using the method in the proof of Theorem 3. with f replaced by gv = 2v τ 8 v τ v + u + πτ τ v + u πτ τ v u + 2πτ τ v u 2πτ τ, we find that V 6 + V M + 60MV 2 V + M E 6τ + 6V V + 20V 2 V 3 + 0V 2 + M 2 V E τ + M 2 M + V 2 = 0, where E τ and E 6 τ are given by 3 2 and i denotes i-fold partial differentiation with respect to u.

9 ELLIPTIC FUNCTIONS TO THE QUINTIC BASE 9 It turns out that there is a differential equation simpler than the one just given: 3 7 M E 6τ M2 + 3 M 3 A V 2 + 3V 2 V V 2 = 0. We will devote the next section to the proof this equation. In the meantime, here is a result that follows immediately from 3 7: Corollary 3.3. For k 2, k 2k 2!L 2l+ L 2k l + S 2k+ = AS 2k + 6L L 2k + 6 2l! 2k l +! k l= l= 2k 2! S 2k l + S 2l+ + L 2l+ L 2k l + 2l! 2k l! k 3 2 l=0 k l 3 m=0 2k 2!L 2l+ L 2m+ S 2k l m 2+ 2l +! 2m +! 2k l m 2!. Alternate applications of Corollaries 3.2 and 3.3 yield tables for L 2k+ and S 2k+ in terms of z and x given by 6. Here are first few terms derived from these recurrences: L q = zx, S 3 q = z 2 x + 3x 2, L 3 q = zx Aτ, S q = z 2 Aτx + 3x 2, L q = z 3 x + 0x + 33x 2, S 7 q = z x + 3x + 32x x 3. We first note that for k,. Sketch of the proof of 3 7 S 2k+ q = n 2k+ q n q n n 2k+ q n q n = k+ 2k +! ζ2k + 2E 2k+2 τ E 2k+2 τ 2π 2k+2, where E 2k τ is given by 3 2. Ramanujan [Berndt et al. 2000, Theorems 3., 3.2] offered the identities expressing E τ, E τ, E 6 τ and E 6 τ in terms of x, z and A, and since E 2k τ can be expressed in terms of E τ and E 6 τ by [Serre 973, Corollary 2, p. 89], we can compute S 2k+ q in terms of x, z, and Aτ for any positive integer k, and in particular for k 9. Now, using Corollary 3.2, we compute L 2k+ q in terms of x, z and A for 0 k 8 using the identity A 2 τ = z x + 2x 2,

10 60 HENG HUAT CHAN AND ZHI-GUO LIU which follows from 3 and the definitions of x and z in 6. These yield a total of 8 identities, which gives L 2s+ and S 2t+ in terms of x, z and Aτ for 0 s 8 and t 9. The coefficients of u 2k+ in V u τ and u 2k in Mu τ are constant multiples of L 2k+ and S 2k+ respectively, by and. Hence, we may write 2 8 V u τ = F k x, z, Aτu 2k+ + Ou 9, k=0 Mu τ = 9 3 Aτ + G k x, z, Aτu 2k + Ou 20, k= where F k and G k are certain polynomials in x, z and Aτ. Identity 3 7 is discovered by first assuming that there is a relation between M 2 + α E 6 τ,m 2,M + α 2 A V 2, V 2 V, and V 2, namely, M 2 + α E 6 τ + β M 2 + β 2 M + α 2 A V 2 + β 3 V 2 V + β V 2 = 0. We then use the representations 2 to establish relations among β i s by equating the coefficients of u k. The determination of β i s gives 3 7 immediately. In order to show that 3 7 is valid, we note that M and V are both elliptic functions with the total number of poles being 8 and respectively; see 2 and 3 6. Therefore the left-hand side hu of 3 7 is an elliptic function with at most 6 poles. We know that an elliptic function has the same number of poles and zeros in a period parallelogram. Hence, hu must have at most 6 zeros in a period parallelogram. This last step can be established by using the representations 2 and showing that hu = Ou 7. Let. Behavior of Vu τ under the imaginary transformation τ τ Uu τ = u + π τ + u π τ u + 2π τ u 2π τ. Theorem.. The function Uu τ and V u τ satisfy the transformation formula u V τ = τuu τ. τ Proof. The function u τ satisfies the relation [Chan and Liu 2003,.2] u τ = 2iu τ π + τ u τ.

11 Replacing u by u + π and u + 2π u τ + π τ τ u τ + 2π τ τ ELLIPTIC FUNCTIONS TO THE QUINTIC BASE 6 = 2iu in, we find that π + 2i + τ u + π τ and u + 2π τ. = 2iu π + i + τ Replacing u by u in these two equations and using the fact that u τ is an odd function of u see 3, we find that u τ π τ = 2iu τ π 2i + τ u π τ and u τ 2π τ τ = 2iu π i + τ u 2π τ. From these four equations we deduce Theorem. immediately. Let χ n n = and define 2 E 2r, χ q = + r 2π 2r χ 2r! L2r, χ n n2r q n q n, where Lr, χ = χ n n r. Our function Uu τ can be expressed in terms of E 2r, χ q: Theorem.2. Uu τ = 2 u 2r 2r L2r, χ π E 2r, χ q. r Proof. Using 3, we find that Uu τ = Su + S n u q n, where Su = cot u + π + cot u π cot u + 2π cot u 2π S n u = sin 2n u + π + sin 2n u π sin 2n u + 2π = χ n sin 2nu. q n, sin 2n u 2π Here the last equality follows by elementary calculations using trigonometric identities.

12 62 HENG HUAT CHAN AND ZHI-GUO LIU From the expansion cot u = m 3 Su = χ k cot u + kπ k= = m = r 0 Next, we find that and m k= k= χ k u r π u mπ + u+m π χ k u+kπ/ mπ + u +kπ/ m π r+ m k r+ = m m k= m k= k= χ k m k r+ + χ m k m k r+ = m χ k m +k r+ = Lr +, χ. Hence, we conclude from 3 that Su = 2 u 2r+ 2r+2 L2r + 2, χ π r 0, we then find that r m +k r+ χ n n r+ = Lr +, χ. and this completes the proof of Theorem.2. Now, using Theorem.2 and equating the coefficients of powers of u in Theorem., we derive a transformation formula for L 2k+ and E 2k+, χ : Theorem.3. L2k + 2, χ E 2k+2, χ e 2πiτ = k+ 2k +! 2π 2k+2 τ 2k 2 L 2k+ e 2πi/τ. This result shows that there is a one-to-one correspondence between identities satisfied by E 2k+2, χ and L 2k+ if we know the corresponding behavior of all the functions in the identities under the transformation τ /τ. We illustrate our observation in the case k = 0. In this case, L2, χ E 2, χ e 2πiτ 2π 2 = τ 2 L e 2πi/τ. From and the transformation formula η = iτητ, τ

13 ELLIPTIC FUNCTIONS TO THE QUINTIC BASE 63 we find that L e 2πi/τ = τ 2 ze 2πiτ, which upon substitution into yields L2, χ E 2, χ e 2πiτ = π 2 2 ze2πiτ. Since the q-expansions of E 2, χ and zq start with, we conclude from that and L2, χ = π E 2, χ q = zq. The latter is another identity of Ramanujan [Ramanujan 988, p. 39]. The equivalence of 6 and was illustrated in [Chan 996]. However, in that work, the special value L2, χ needed to be computed. Here, we obtain the value L2, χ as a consequence of the transformation formula of ητ and Theorem.3. Acknowledgements We sincerely thank Professor Shaun Cooper and the anonymous referee for uncovering several misprints in the earlier version of this paper. References [Bailey 92a] W. N. Bailey, A note on two of Ramanujan s formulae, Quart. J. Math., Oxford Ser , MR 3,72d Zbl [Bailey 92b] W. N. Bailey, A further note on two of Ramanujan s formulae, Quart. J. Math., Oxford Ser , MR,38f Zbl [Berndt and Ono 999] B. C. Berndt and K. Ono, Ramanujan s unpublished manuscript on the partition and tau functions with proofs and commentary, Sém. Lothar. Combin , 63 Art. B2c. Reprinted as pp in The Andrews Festschrift, edited by D. Foata and G.N. Han, Springer, Berlin, 200. MR 2000i:0027 Zbl [Berndt et al. 2000] B. C. Berndt, H. H. Chan, J. Sohn, and S. H. Son, Eisenstein series in Ramanujan s lost notebook, Ramanujan J. : 2000, 8. MR 200j:08 Zbl [Chan 99] H. H. Chan, New proofs of Ramanujan s partition identities for moduli and 7, J. Number Theory 3: 99, 8. MR 96g:2 Zbl [Chan 996] H. H. Chan, On the equivalence of Ramanujan s partition identities and a connection with the Rogers-Ramanujan continued fraction, J. Math. Anal. Appl. 98: 996, 20. MR 96m:09 Zbl [Chan and Liu 2003] H. H. Chan and Z.-G. Liu, Analogues of Jacobi s inversion formula for the incomplete elliptic integral of the first kind, Adv. Math. 7: 2003, MR 200b:33027 Zbl

14 6 HENG HUAT CHAN AND ZHI-GUO LIU [Ramanujan 988] S. Ramanujan, The lost notebook and other unpublished papers, Narosa, New Delhi, and Springer, Berlin, 988. MR 89j:0078 Zbl [Serre 973] J.-P. Serre, A course in arithmetic, Graduate Texts in Math. 7, Springer-Verlag, New York, 973. MR MR #896 Zbl [Shen 99] L.-C. Shen, On the additive formulae of the theta functions and a collection of Lambert series pertaining to the modular equations of degree, Trans. Amer. Math. Soc. 3: 99, MR 9a:33038 Zbl [Whittaker and Watson 927] E. T. Whittaker and G. N. Watson, A course of modern analysis, th ed., Cambridge University Press, Cambridge, 927. MR 97k:0072 Received September 26, 200. Revised January 6, 200. HENG HUAT CHAN DEPARTMENT OF MATHEMATICS NATIONAL UNIVERSITY OF SINGAPORE KENT RIDGE, SINGAPORE 9260 REPUBLIC OF SINGAPORE matchh@nus.edu.sg ZHI-GUO LIU EAST CHINA NORMAL UNIVERSITY DEPARTMENT OF MATHEMATICS SHANGHAI P.R. CHINA zgliu@math.ecnu.edu.cn liuzg8@hotmail.com