On the Equivalence of Ramanujan s Partition Identities and a Connection with the Rogers Ramanujan Continued Fraction
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1 JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS 98, ARTICLE NO. 007 O the Equivalece of Ramauja s Partitio Idetities ad a Coectio with the RogersRamauja Cotiued Fractio Heg Huat Cha Departmet of Mathematics, Uiersity of Illiois at Urbaa Champaig, 409 West Gree Street, Urbaa, Illiois 680 Submitted by Bruce C. Berdt Received May, 995 DEDICATED TO THE MEMORY OF MY MOTHER A famous idetity of Ramauja coected with partitios modulo 5 is show to be equivalet to aother idetity of Ramauja. The latter idetity is used to establish a differetial equatio for the RogersRamauja cotiued fractio foud i Ramauja s lost otebook. We also prove that two other idetities of Ramauja are equivalet, oe of which is associated with Ramauja s partitio cogruece modulo 7. Last, we give a ew proof of the trasformatio formula for the Dedekid eta-fuctio, which is used i our proofs of equivalece. 996 Academic Press, Ic. Let. INTRODUCTION a; q Ł aq, q, ad f Ž q. Ž q; q.. iz 4 Note that if q e,imz0, the q f q z, where z is the Dedekid eta-fuctio X96 $8.00 Copyright 996 by Academic Press, Ic. All rights of reproductio i ay form reserved.
2 HENG HUAT CHAN I a famous mauscript o the partitio fuctio p ad the tau-fuctio Ž., recetly published with the lost otebook, Ramauja 9, p. 39 recorded the idetities q f q q Ž Ý 5 Ž q. fž q. ad q f Ž q. 5 Ž.. 5 Ý 5 q 5 f q where Ž 5. is the Legedre symbol. There are several proofs of Ž.. ad Ž.., ad refereces may be foud i our paper, where a ew proof of Ž.. is give. I provig these two idetities usig modular forms of Nebetypus, Raghava 8 remarked that his proofs throw some light o the pair of idetities beig allied. I this paper, we employ Hecke s theory of correspodece betwee Fourier series ad Dirichlet series to show that Ž.. ad Ž.. are equivalet. Thus, as Ramauja implicitly implied, Ž.. ad Ž.. are truly compaio idetities. Idetity Ž.. was employed by Ramauja 9, pp to give a short proof of his famous cogruece pž Ž mod 5.. See also our paper. I Sectio, we show that a differetial equatio satisfied by the RogersRamauja cotiued fractio follows easily from Ž... This differetial equatio was stated by Ramauja Žwithout proof ad i a slightly differet form. i 9, p Our work is motivated by the trasformatio formula for the Dedekid eta-fuctio. I Sectio 3, we utilize the Weierstrass -fuctio to provide a ew, short proof of this trasformatio formula. I Sectio 4, we prove the equivalece of Ž.. ad Ž.., ad i Sectio 5 we prove the equivalece of Ramauja s famous partitio idetity coected with partitios modulo 7 ad a compaio idetity foud i 9, p. 45. These two idetities were also discussed i Raghava s paper 8.. IDENTITY. AND ROGERSRAMANUJAN S CONTINUED FRACTION Deote the RogersRamauja cotiued fractio by q 5 q q q 3 FŽ q., q.
3 RAMANUJAN PARTITION IDENTITY EQUIVALENCE 3 It is well kow that 3; 5, pp Ž q; q 5. Ž q 4 ; q 5. 5 FŽ q. q. Ž q ; q q ; q O page 363 of his Lost Notebook satisfies the differetial equatio 9, Ramauja claimed that FŽ q. dfž q. f 5 Ž q. 5q FŽ q.. Ž.. dq f Ž q 5. We ow give a proof of.. From., we deduce that 5 log F q log q Ý log q..3 5 Differetiatig.3 with respect to q, we have Therefore, Usig., we deduce.. dfž q. q Ý. FŽ q. dq 5q 5 q ž dfž q. q 5q 5 Ý FŽ q.. dq 5 q 3. DEDEKIND TRANSFORMATION FORMULA We have already see how. ad. have coectios with partitio theory ad the RogersRamauja cotiued fractio, respectively. It is therefore iterestig to see that they are i fact equivalet. We first establish a clue which will lead to the result. LEMMA 3.. Let Im z 0. We hae iz4 iz i4 z i z Ž 3.. e fž e. e fž e.. ' iz
4 4 HENG HUAT CHAN Lemma 3. gives the famous trasformatio formula for the Dedekid -fuctio for which there are may proofs. For example, see. Perhaps the followig proof is ew. i Proof. We recall that 3, p. 69 if q e, the 6 6q f q Ž e e.ž e e.ž e e., Ž where e Ž, ;., e Ž, ;,. ad e3 Ž, ; Ž... Here, Ž, ; z. is the Weierstrass elliptic fuctio with periods ad. Applyig Ž 3.. with ad, we have 6e i 6 f Ž e i. Ž e e3.ž ee.ž e3e., Ž 3.3. where e Ž, ;,e., Ž ;. ad e Ž, ; Ž... 3 Next, set ad, so that q e i. Note that Ž,; z. Ž, ; z. sice the lattice geerated by ad is the same as the lattice geerated by ad. Therefore, e,;, ; e. i Similarly, e e ad e e. Applyig 3. with q e, we obtai i i 3 3 6e f Ž e. Ž e e.ž e e.ž e e. Dividig 3.3 by 3.4 ad simplifyig, we have Ž ee.ž ee3.ž e3e.. Ž 3.4. e i f Ž e i. 6 e i f Ž e i.. Takig the th root o both sides, we have e i fž e i. e i f Ž e i., where is a 4th root of uity. Settig i, we deduce that complete the proof. ' i to
5 RAMANUJAN PARTITION IDENTITY EQUIVALENCE 5 4. FUNCTIONAL EQUATIONS, HECKE S THEORY, AND THE EQUIVALENCE OF. AND. i Now, let q e. By Lemma 3., the right had side of Ž.. is Ž e f e. ' i Ž e f e. e fž e. Ž ' 5i. e fž e. 0 i 4 0 i 5 i0 i5 5 i4 i 5 i4 i f 5 i5 Ž e.. ' i 5 5 fž e. Settig q e i 5, we rewrite the equality above i the form 5 5 Ž. f 5 q 5 f 5 Ž q. q. 5' 5 f q f q This shows that i some sese the right-had side of Ž.. ca be obtaied from the right-had side of Ž.. ad vice versa. This provides the motivatio to show the equivalece of Ž.. ad Ž... To achieve this aim, it suffices to show that If we write q q 5. Ž Ž q. 5 ' 5ž 5 q Ý Ý i i e e g Ý ad hž. 5 Ý, i i 5 Ž e. 5 e the 4. may be writte as g Ž. h. Ž ' 5 ž 5 Before proceedig further, we state the followig results. LEMMA 4. ŽFuctioal Equatio for Ž s... For all complex umbers s, s s s Ž s. cos Ž s. Ž s.. Proof. See, p. 59, Theorem.7.
6 6 HENG HUAT CHAN LEMMA 4. ŽFuctioal Equatio for Ls, with a Primitive Character mod k.. For all complex umbers s, s s is is LŽ s,. Ž. Ž s. k e Ž. e G LŽ s,., k ihk where G Ý he. h Proof. See, p. 63, Theorem.. Ž c LEMMA 4.3. Suppose that for some positie costat c, a, b O.. For Im z 0, let i i u Ý ae, Ý be. 0 0 For c, set s s Ž s. Ý a, Ž s. Ý b. Suppose s ad s hae aalytic cotiuatios ito the etire complex plae. For all s, defie s Ž s. Ž. Ž s. Ž s. ad Ž s. Ž. Ž s. Ž s.. s The the followig are equialet: Ž. I For certai positie umbers A ad k ad for some complex umber s C, the fuctio s A Ža scb Ž k s is etire ad bouded i eery ertical strip, ad Ž s. CA k s Ž k s.. Ž II. k u CA Ž Ai. k Ž A.. Proof. See 7, V-6V-7. We are ow ready to prove Ž 4... The Dirichlet series associated to g is Ls, Ž s., ad the Dirichlet series associated to hž. is 5Ls, Ž s., where Ž 5.. So we may let Ž s. LŽ s,. Ž s. ad Ž s. 5LŽ s,. Ž s..
7 RAMANUJAN PARTITION IDENTITY EQUIVALENCE 7 Therefore, s s s L s, s ad s Ž s. Ž. Ž s. 5LŽ s,. Ž s.. Applyig Lemma 4. with Ž 5., ad GŽ. ' 5 6, pp. 970, we have s s s LŽ s,. Ž. Ž s. 5 cos ' 5 LŽ s,.. Ž 4.3. Now, by Lemma 4. ad Ž 4.3., Ž s. Ž s. 5 s LŽ s,. Ž s. s s s s 5 s Ž. Ž s5. cos ' 5 LŽ s,. Ž s. s s cos Ž s. Ž s. s 5 ' 5Ž s., sice Ž z. Ž z. si z, p. 50. Now, Ž s. has a pole at s with residue sice, p. 3, Theorem 4. Ž. Ž. LŽ,., 5 ' 5 4 LŽ,.. 5 ' 5 s Ž. ' Therefore, P s s s is etire. The fact that PŽ s. is bouded i every vertical strip follows from the bouds of Ž s., Ls,, pp. 707 ad Ž s. 4, p. 3 i vertical strips. More precisely, if we let s it, the give a b, there exist A, A, A,,, ad depedig o a ad b such that for t, 3 ad Ž s. A t, LŽ s,. A t, t Ž s. A3t e.
8 8 HENG HUAT CHAN Thus, PŽ s. is uiformly bouded for t ad a b. PŽ s. is clearly bouded i the remaiig regio where t sice it is a etire fuctio. Now, sice coditio Ž. I of Lemma 4.3 is satisfied with a0 0, b, A 5, k, ad C 5' 5, we coclude that 0 which is 4.. ž 5 ž 5' 5 i 5 g 5 h h, 5 ' 5 ž 5 5. EQUIVALENCE OF TWO OTHER RAMANUJAN IDENTITIES I the aforemetioed mauscript, Ramauja 9, p. 45 wrote dow the two idetities q f q q qf Ž q. f q 8q Ž Ý 3 7 Ž q. fž q. ad q f Ž q qf Ž q. f q 8. Ž Ý 7 7 q f q We will apply the ideas illustrated i the previous two sectios to show the equivalece of 5. ad 5.. By Lemma 3., we observe that qf Ž q. f Ž q. 8q f 7 q 7 f q f 3 Ž q. f 3 q 7 f 7 Ž q. iq 8i, ' 3 3' f q Ž. where q e i 7. Hece, to show the equivalece of these two ideti- ties, it suffices to show that Ž q. i q q q q Ý Ý '
9 RAMANUJAN PARTITION IDENTITY EQUIVALENCE 9 If we write ad i Ž e. i Ý i 3 g e 7 Ž e. i e h 87 Ý, i 7 e the, by 5.3, we eed to show that i gž. h. Ž ' 7 ž 7 The associated Dirichlet series for g Ž. is Ž s. Ls, Ž s., while the associated Dirichlet series for h Ž. is Ž s. 7Ls, Ž. Ž s., where is the Legedre symbol Ž 7.. Therefore, ad s Ž s. Ž. Ž s. LŽ s,. Ž s. s Ž s. Ž. Ž s.ž 7. LŽ s,. Ž s.. By Lemmas 4. ad 4. ad the fact that GŽ. i 7 6, pp. 970, we easily verify that Ž 3 s. 7 7 Ž s.. s ' Now, the fuctio Ž s. has a pole at s 3, with residue Sice, p. 3, Theorem 4. 3 Ž. 3 L 3,. 3 3 LŽ 3,., 3 7 ' 7 s we see that P s s 7 Ž87 Ž 3s.. is etire. The fuctio P Ž s. is also bouded i every vertical strip Žthe argumet is similar to that '
10 0 HENG HUAT CHAN give i Sectio 5.. Thus, Ž I. of Lemma 4.3 is satisfied with a0 0, b0 8, k3, A 7, ad C 7. Therefore, we coclude that i ž 7 g 7 h i h, ' 7 ž 7 which is Ž REMARK. Garva has recetly give a short ad elegat proof of Ž 5... See. ACKNOWLEDGMENT The author thaks B. Berdt for his ecouragemet ad suggestios. REFERENCES. T. M. Apostol, Itroductio to Aalytic Number Theory, Spriger-Verlag, New York, H. H. Cha, New proofs of Ramauja s partitio idetities for moduli 5 ad 7, J. Number Theory, to appear. 3. K. Chadrasekhara, Elliptic Fuctios, Spriger-Verlag, New York, E. T. Copso, Theory of Fuctios of a Complex Variable, Oxford Uiv. Press, Lodo, G. H. Hardy ad E. M. Wright, A Itroductio to the Theory of Numbers, Oxford Uiv. Press, New York, E. Ladau, Elemetary Number Theory, Chelsea, New York, A. Ogg, Modular Forms ad Dirichlet Series, Bejami, New York, S. Raghava, O certai idetities due to Ramauja, Quart. J. Math. Oxford Ž. 37 Ž 986., S. Ramauja, The Lost Notebook of Other Upublished Papers, Narosa, New Delhi, S. Ramauja, Some properties of p, the umber of partitios of, Proc. Cambridge Philos. Soc. 9 Ž 98., C. L. Siegel, A simple proof of Ž. ' i,mathematika Ž L. C. Washigto, Itroductio to Cyclotomic Fields, Spriger-Verlag, New York, G. N. Watso, Theorems stated by Ramauja Ž IX.: Two cotiued fractios, J. Lodo Math. Soc. 4 Ž 99., 337.
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