Journal of Number Theory

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1 Journal of Number Theory ) Contents lists available at SciVerse ScienceDirect Journal of Number Theory wwwelseviercom/locate/jnt On Ramanujan s modular equations of degree 21 KR Vasuki GSharath Department of Studies in Mathematics University of Mysore Manasagangotri Mysore India article info abstract Article history: Received 12 October 2011 Revised 15 August 2012 Accepted 15 August 2012 Available online 22 October 2012 Communicated by David Goss In this paper we give an alternative proof of two of Ramanujan s modular equations of degree 21 which have been proved by BC Berndt using theory of modular forms Our proofs involve only the identities stated by Ramanujan 2012 Elsevier Inc All rights reserved MSC: 11F20 Keywords: Theta functions Modular equations 1 Introduction S Ramanujan has recorded several modular equations in his notebooks [9] his lost notebook [10] without proofs Ramanujan probably used a lot of methods [238] in deriving his modular equations BC Berndt [46] has proved all the modular equations recorded by Ramanujan in [9] Berndt employs three different methods to prove these modular equations first is by the Ramanujan s theory of theta functions the second by the method of parameterization the third by theory of modular forms On p 326 [4] Berndt expresses that The first two methods become rapidly more difficult as the degree of the modular equation increases while the complexity of the approach through modular forms increases only slightly as the degree increases The primary disadvantage of the third method is that modular equation must be known in advance Thus the proofs by employing modular forms are perhaps aptly called verifications On p 248 of his notebooks [9] Ramanujan has recorded six mixed modular equations of composite degrees Berndt has proved [4 pp ] all six of the above mentioned equations * Corresponding author addresses: vasuki_kr@hotmailcom KR Vasuki) sharath_gns@rediffmailcom G Sharath) X/$ see front matter 2012 Elsevier Inc All rights reserved

2 438 KR Vasuki G Sharath / Journal of Number Theory ) by using the theory of modular forms The objective of this paper is to prove two of these modular equations by employing the theory of theta functions identities stated by Ramanujan Before stating the afore mentioned modular equations we shall recall the definition of modular equations The complete elliptic integral K = K k) of first kind is defined by Kk) := π/2 θ=0 dθ 1 k 2 sin 2 θ 0 < k < 1 The number k is called the modulus of K The complementary modulus k is defined by k 2 = 1 k 2 Set K = K k ) Let L L denote the complete elliptic integrals of first kind associated with the moduli l l respectively Suppose the equality n K K = L L 11) holds for some positive integer n Then a modular equation of degree n is a relation between the moduli k l which is induced by 11) Ramanujan writes his modular equation in terms of α β α = k 2 β = l 2 The multiplier m of modular equation of degree n is defined by m = K L Let K K L 1 L 1 L 2 L 2 L 3 L 3 denote complete elliptic integrals of the first kind corresponding in pairs to the moduli α β γ δ their complementary moduli respectively Let n 1 n 2 n 3 be positive integers such that n 3 = n 1 n 2 Suppose that the equalities K n 1 K = L L n K 2 K = L 2 K n 3 L 2 K = L 3 12) L 3 hold Then a modular equation of composite degree is a relation between the moduli α β γ δ that is induced by 12) We define the multiplier m m by m := z 1 z n1 = K L 1 m := z n 2 z n3 = L 2 L 3 In this paper we derive an alternative proof of the following two modular equations: βγ 4 1 β)1 γ ) αδ 1 α)1 δ) ) 1 4 βγ 1 β)1 γ ) αδ1 α)1 δ) βγ 1 β)1 γ ) 6 m 4 = αδ1 α)1 δ) m 13) αδ βγ 4 1 α)1 δ) 1 β)1 γ ) ) 1 4 ) 1 4 αδ1 α)1 δ) 4 βγ 1 β)1 γ ) αδ1 α)1 δ) 6 m 4 = βγ 1 β)1 γ ) m 14)

3 KR Vasuki G Sharath / Journal of Number Theory ) β γ δ are of degrees respectively over α withm = z 1 z 3 m = z 7 z 21 Now let us recall some definitions certain elementary identities which are required to prove 13) 14) After Ramanujan for q < 1 we set ϕq) := ψq) := f q) := n= n=0 n= q n2 = q; q2 ) q 2 ; q 2 ) q; q 2 ) q 2 ; q 2 ) q nn1) 2 = q2 ; q 2 ) q; q 2 ) 1) n q n3n 1) 2 = q; q) χq) := q; q 2) a; q) = ) 1 aq n n=0 One can easily show that f n := f q n ) ϕq) = f 2 5 ϕ q) = f 2 1 ψq) = f 2 2 ψ q) = f 1 f 4 f f 4 f 2 f 1 f 2 χq) = f 2 2 χ q) = f 1 f q) = f ) f 1 f 4 f 2 f 1 f 4 Lemma 11 See [9 pp ] [4 pp ]) If q = exp π K K ) z = 2 π Kthen ϕq) = z ϕ q) = z1 α) 1 4 ϕ q 2) = z1 α) 1 8 q 1 z1 8 ψq) = 2 α) 1 8 q 1 8 ψ q) = z1 2 q 1 4 ψ q 2 ) = α1 α) } 1 8 z1 2 α} 1 4

4 440 KR Vasuki G Sharath / Journal of Number Theory ) q 1 12 f q 2) = z1 2 1/3 α1 α) } 1 12 Using the above lemma Berndt converts 13) 14) into the following equivalent theta function identities: ψq 6 )ψq 14 )ϕq 3 )ϕq 7 ) q 3 ψq 2 )ψq 42 )ϕq)ϕq 21 ) ϕ q3 )ϕ q 7 )ϕq 3 )ϕq 7 ) ϕ q)ϕ q 21 )ϕq)ϕq 21 ) ψq6 )ψq 14 )ϕ q 3 )ϕ q 7 ) q 3 ψq 2 )ψq 42 )ϕ q)ϕ q 21 ) 4 f 2 6 f 2 14 q 2 f 2 2 f = q 3 ψq 2 )ψq 42 )ϕq)ϕq 21 ) ψq 6 )ψq 14 )ϕq 3 )ϕq 7 ) q3 ψq 2 )ψq 42 )ϕ q)ϕ q 21 ) ψq 6 )ψq 14 )ϕ q 3 )ϕ q 7 ) ϕ q)ϕ q21 )ϕq)ϕq 21 ) ϕ q 3 )ϕ q 7 )ϕq 3 )ϕq 7 ) 4 q2 f 2 2 f 2 42 f 2 6 f 2 14 Employing 15) to the above two identities respectively we obtain = 1 P 4 R 2 P 2 R 4 Q 6 4P 2 Q 2 R 2 P 2 Q 4 R 2 = 0 16) Q 6 R 2 P 2 Q 6 P 4 R 4 4P 2 Q 4 R 2 P 2 Q 2 R 2 = 0 17) P = q 1 2 f 1 f 21 f 3 f 7 Q = q f 2 f 42 f 6 f 14 R = q 2 f 4 f 84 f 12 f 28 Thus proving 13) 14) is equivalent to proving 16) 17) 2 Proof of 16) 17) First we shall recall the Ramanujan theta function identities which are required to prove 16) 17) Lemma 21 See [9 p 327] [5 Entry 51 p 204]) Let Then M = f 1 N = f 2 q 1 12 f3 q 1 6 f6 MN) 2 9 MN) 2 = M N ) 6 ) 6 N 21) M

5 KR Vasuki G Sharath / Journal of Number Theory ) Lemma 22 See [9 p 327] [5 Entry 52 p 205]) Let Then K = f 2 L = f 1 q 1 24 f3 q 5 24 f6 KL) 2 9 ) 3 L K KL) = 2 8 K L Lemma 23 Equivalent of Entry 5xii) [9 p 231] [4 p 230] [11]) Let ) 3 22) Then X = f 1 q 1 24 f2 Y = f 3 q 1 8 f6 ) 3 8 ) 6 Y X ) = 3 X Y Lemma 24 See [1]) If α β γ δ have degrees respectively then ) 6 23) R 2 1 R = 2 Q 4 1 Q ) αδ1 α)1 δ) 48 Q = βγ 1 β)1 γ ) R = γ δ1 γ )1 δ) 48 αβ1 α)1 β) Recently ND Baruah [1] deduced the above identity from the tools known to the Ramanujan Theorem If P = q 1 2 f 1 f 21 f 3 f 7 Q = q f 2 f 42 f 6 f 14 then PQ 1 ) 3 ) 3 P PQ = 4 Q Q ) 25) P Proof Changing q to q 7 in 21) we see that M 1 N 1 ) 2 9 M 1 N 1 ) 2 = M1 N 1 ) 6 N1 M 1 ) 6 26) M 1 = f 7 q 7 12 f21 N 1 = f 14 q 7 6 f42

6 442 KR Vasuki G Sharath / Journal of Number Theory ) Multiplying 21) 26) we obtain which can be rewritten as ) 2 ) 2 } MNM 1 N 1 ) 2 81 MN MNM 1 N 1 ) M1 2 9 N 1 M 1 N 1 MN ) 6 ) 6 ) 6 ) 6 NN1 MM1 MN1 M1 N = MM 1 NN 1 NM 1 MN 1 MNM 1 N 1 ) 2 81 MNM 1 N 1 ) 2 9 PQ) 2 1 } PQ) 2 ) 6 ) 6 ) 6 ) 6 YY1 XX1 P Q = 27) XX 1 YY 1 X Y are as in Lemma 23 X 1 = f 7 Y q 7/24 f 1 = f 21 P Q are in the statement of the 14 q 7/8 f 42 theorem Now changing q to q 7 in 22) then multiplying the resultant identity with 22) we obtain ) 2 ) 2 } KLK 1 L 1 ) 2 81 KL KLK 1 L 1 ) K1 2 9 L 1 K 1 L 1 KL ) 3 ) 3 ) 6 ) 3 } LL1 KK1 LK1 KL1 = 64 8 KK 1 LL 1 KL 1 LK 1 K 1 = f 14 L q 7 1 = f 7 24 f21 q f42 Using the definitions of M N giveninlemma21x Y of Lemma 23 P Q as in the statement of the theorem the above identity can be written as MNM 1 N 1 ) 2 81 MNM 1 N 1 ) 2 9 PQ) 2 1 } PQ) 2 ) 3 ) 3 } = XX 1 YY 1 ) 3 64 XX 1 YY 1 ) X1 3 8 Y 1 28) X 1 Y 1 Similarly changing q to q 7 in 23) then multiplying the resultant identity with 23) we obtain which can be rewritten as ) 3 XX 1 YY 1 ) 3 64 XX 1 YY 1 ) 3 8 X 1 Y 1 ) 6 ) 6 ) 6 YY1 XX1 YX1 = XX 1 YY 1 1 ) 3 } ) 6 } 1 YX 1

7 KR Vasuki G Sharath / Journal of Number Theory ) ) 3 ) 3 } XX 1 YY 1 ) 3 64 XX 1 YY 1 ) X1 3 8 Y 1 X 1 Y 1 ) 6 ) 6 ) 6 ) 6 } YY1 XX1 P Q = 29) XX 1 YY 1 P Q are as in the statement of the theorem Subtracting 29) from 27) we find that MNM 1 N 1 ) 2 81 MNM 1 N 1 ) 2 9 PQ) 2 1 } PQ) 2 ) 3 ) 3 } = XX 1 YY 1 ) 3 64 XX 1 YY 1 ) X1 3 8 Y 1 X 1 Y 1 ) 6 ) 6 } P Q 2 210) Next subtracting 28) from 210) we find that 9 PQ) 2 1 PQ) 2 } ) 3 = 8 X 1 Y 1 ) 3 } ) 6 ) 6 P Q 211) Converting modular equation 24) in theta function identity by employing Lemma 11 then changing q to q we obtain X 1 Y 1 ) Cubing this on both sides we obtain ) 3 X 1 Y 1 Now from the above two identities we obtain ) ) 2 ) 2 P Q = 3 ) 3 3 X } ) 6 1Y 1 P Q = X 1 Y 1 P ) 3 X 1 Y 1 30 ) 3 ) 6 P Q = P 27 ) 6 P 9 Q ) 2 Q ) 6 P 9 Q ) 2 Q ) 2 } 36 Using this in 211) then factorizing the resulting identity we obtain ) 2 } 45 ) 4 ) 4 } Q P ) 4 ) 4 } Q P AP Q )BP Q ) = 0 212) AP Q ) = P 6 Q 6 P 2 Q 2 P 4 Q 4 4P 2 Q 4 4P 4 Q 2

8 444 KR Vasuki G Sharath / Journal of Number Theory ) BP Q ) = P 6 Q 6 P 2 Q 2 P 4 Q 4 4P 2 Q 4 4P 4 Q 2 Now from the definitions of P Q wehave P = q 2q 2 q 3 4q 4 3q 5 7q 7 6q 8 7q 7 6q 8 7q 9 12q 10 6q 11 10q 12 26q 13 12q 14 18q ) Q = q q 3 q 5 q 7 q 9 2q 13 2q ) Using 213) 214) in the definition of AP Q ) BP Q ) we see that AP Q ) = 2q 3 2q 4 6q 5 18q 6 12q 7 16q 8 92q 9 90q q q q ) BP Q ) = 56q q 19 76q q q 22 48q q q ) We can see that as q 0 q 3 AP Q ) 0 as q 3 BP Q ) = 0 Thus BP Q ) = 0 in some neighborhood of q = 0 hence by analytic continuation BP Q ) = 0in q < 1 Thus P 6 Q 6 P 2 Q 2 P 4 Q 4 4P 2 Q 4 4P 4 Q 2 = 0 which is equivalent to 25) The identity 25) is due to S Bhargava et al [7] they employed 13) 14) to prove 25) Proof of 16) Changing q to q 2 in 25) we obtain Q 6 R 6 Q 2 R 2 Q 4 R 4 4Q 2 R 4 4Q 4 R 2 = 0 217) Multiplying 25) by R 2 217) by P 2 then subtracting one from the other gives the required result Proof of 17) Multiplying 25) by R 4 217) by P 4 then subtracting one from the other gives the required result Acknowledgment Authors are thankful to DST New Delhi for awarding research project [No SR/S4/MS:517/08] under which this work has been done

9 KR Vasuki G Sharath / Journal of Number Theory ) References [1] ND Baruah On some of Ramanujan s Schläfli-type mixed modular equations J Number Theory ) [2] BC Berndt Ramanujan s modular equations in: Ramanujan Revisited Academic Press Boston 1988 pp [3] BC Berndt Introduction to Ramanujan s modular equations in: Proceedings of the Ramanujan Centennial International Conference Annamalaingar December 1987 The Ramanujan Mathematical Society New York 1991 [4] BC Berndt Ramanujan s Notebooks Part III Springer-Verlag New York 1991 [5] BC Berndt Ramanujan s Notebooks Part IV Springer-Verlag New York 1994 [6] BC Berndt Ramanujan s Notebooks Part V Springer-Verlag New York 1998 [7] S Bhargava C Adiga MSM Naika A new class of modular equations in Ramanujan s alternative theory of elliptic functions of signature 4 some new P Q eta-function identities Indian J Math 45 1) 2003) [8] KG Ramanathan Ramanujan s modular equations Acta Arith ) [9] S Ramanujan Notebooks 2 volumes) Tata Institute of Fundamental Research Bombay 1957 [10] S Ramanujan The Lost Notebook Other Unpublished Papers Narosa New Delhi 1988 [11] KR Vasuki On some Ramanujan s Schläfli-type modular equations Aust J Math Anal Appl 3 2) 2006) 1 8