( 1) n q n(3n 1)/2 = (q; q). (1.3)

Transcription

1 MATEMATIQKI VESNIK 66, , September 2014 originalni nauqni rad research paper ON SOME NEW MIXED MODULAR EQUATIONS INVOLVING RAMANUJAN S THETA-FUNCTIONS M. S. Mahadeva Naika, S. Chandankumar and M. Harish Abstract. In his second notebook, Ramanujan recorded altogether 23 P Q modular equations involving his theta functions. In this paper, we establish several new mixed modular equations involving Ramanujan s theta-functions ϕ and ψ which are akin to those recorded in his notebook. 1. Introduction Following Ramanujan, let a; q denote the infinite product 1 aq n a, q are complex numbers, q < 1 and n=0 fa, b := n= a nn1/2 b nn 1/2, ab < 1, denote the Ramanujan theta-function. The product representation of fa, b follows from Jacobi s triple product identity and is given by: fa, b = a; ab b; ab ab; ab. The following definitions of theta-functions ϕ, ψ and f with q < 1 are classical: ϕq := fq, q = n= q n2 = q; q 2 2 q 2 ; q 2, 1.1 ψq := fq, q 3 = q nn1/2 = q2 ; q 2 q; q 2, 1.2 n=0 f q := f q, q 2 = n= 1 n q n3n 1/2 = q; q. 1.3 The ordinary or Gaussian hypergeometric function is defined by 2F 1 a, b; c; z := n=0 a n b n z n, 0 z < 1, c n n! 2010 Mathematics Subject Classification: 33D10, 11A55, 11F27 Keywords and phrases: Modular equations; theta-functions 283

2 284 M.S. Mahadeva Naika, S. Chandankumar, M. Harish where a, b, c are complex numbers such that c 0, 1, 2,..., and a 0 = 1, a n = aa 1 a n 1 for any positive integer n. Now we recall the notion of modular equation. Let Kk be the complete elliptic integral of the first kind of modulus k. Recall that Kk := π/2 0 dφ 1 k2 sin 2 φ = π 2 n= n n! 2 k2n = π 2 ϕ2 q, 0 < k < 1, 1.4 and set K = Kk, where k = 1 k 2 is the so called complementary modulus of k. It is classical to set qk = e πkk /Kk so that q is one-to-one, increasing from 0 to 1. In the same manner introduce L 1 = Kl 1, L 1 = Kl 1 and suppose that the following equality K n 1 K = L L 1 holds for some positive integer n 1. Then a modular equation of degree n 1 is an algebraic relation between the moduli k and l 1 which is induced by 1.5. Following Ramanujan, set α = k 2 and β = l 2 1. Then we say that β is of degree n 1 over α. The multiplier m is defined by for q = e πkk /Kk. m = K L 1 = ϕ2 q ϕ 2 q n1, 1.6 Let K, K, L 1, L 1, L 2, L 2, L 3 and L 3 denote complete elliptic integrals of the first kind corresponding, in pairs, to the moduli α, β, γ and δ, and their complementary moduli, respectively. Let n 1, n 2 and n 3 be positive integers such that n 3 = n 1 n 2. Suppose that the equalities n 1 K K = L 1 L 1, n 2 K K = L 2 L 2 and n 3 K K = L 3 L hold. Then a mixed modular equation is a relation between the moduli α, β, γ and δ that is induced by 1.7. We say that β, γ and δ are of degrees n1, n 2 and n 3 respectively over α. The multipliers m = K/L 1 and m = L 2 /L 3 are algebraic relation involving α, β, γ and δ. At scattered places of his second notebook [5], Ramanujan recorded a total of nine P Q mixed modular relations of degrees 1, 3, 5 and 15. For example, if P := f q3 f q 5 q 1/3 f qf q 15 and Q := f q6 f q 10 q 2/3 f q 2 f q 30, then P Q 1 Q 3 P 3 P Q = P Q The proofs of these nine P Q mixed modular relations can be found in the book by Berndt [2, pp ]. In [3,4], S. Bhargava, C. Adiga and M. S. Mahadeva

3 Mixed modular equations involving Ramanujan s theta-functions 285 Naika have established several new P Q mixed modular relations with four moduli. These mixed modular equations are employed to evaluate explicit evaluations of Ramanujan-Weber class invariants, ratios of Ramanujan s theta functions and several continued fractions. Motivated by all these works, in this article, we establish several new mixed modular equations involving Ramanujan s theta-functions. In Section 2, we collect some identities which are useful in proofs of our main results. In Section 3, we establish several new P Q mixed modular equations akin to those recorded by Ramanujan in his notebooks involving his theta-functions ϕq and ψq. 2. Preliminary results In this section, we collect some identities which are useful in establishing our main results. Lemma 2.1. [1, Ch. 16, Entry 24 ii, p. 39] We have f 3 q = ϕ 2 qψq. 2.1 Lemma 2.2. [1, Ch. 16, Entry 24 iv, p. 39] We have f 3 q 2 = ϕ qψ 2 q. 2.2 Lemma 2.3. [1, Ch. 17, Entry 10 i and Entry 11ii, pp ] For 0 < α < 1, we have ϕq = z, 2.3 2q 1/8 ψ q = z{α1 α} 1/ Lemma 2.4. [1, Ch. 20, Entry 3 xii, xiii, pp ] Let α, β and γ be of the first, third and ninth degrees respectively. Let m denote the multiplier relating α, β and m be the multiplier relating γ, δ. Then β 2 1/4 1 β 2 1/4 β 2 1 β 2 1/4 m = 3 αγ 1 α1 γ αγ1 α1 γ m, 2.5 αγ β 2 1/4 1 α1 γ 1 β 2 1/4 αγ1 α1 γ β 2 1 β 2 1/4 = m m. 2.6 Lemma 2.5. [1, Ch. 20, Entry 11 viii, ix, p. 384] Let α, β, γ and δ be of the first, third, fifth and fifteenth degrees respectively. Let m denote the multiplier

4 286 M.S. Mahadeva Naika, S. Chandankumar, M. Harish relating α and β and m be the multiplier relating γ and δ. Then αδ βγ 1/8 1 α1 δ 1 β1 γ 1/8 αδ1 α1 δ 1/8 m = βγ1 β1 γ m, 2.7 βγ αδ 1/8 1 β1 γ 1 α1 δ 1/8 βγ1 β1 γ 1/8 m = αδ1 α1 δ m. 2.8 then Lemma 2.6. [3] If L := q1/6 f qf q 9 f 2 q 3 and M := q1/3 f q 2 f q 18 f 2 q 6, then L 9 M 9 M 9 [ L 3 L 9 12 M 3 M 3 ] L 3 27L 3 M 3 1 L 3 = M 3 Lemma 2.7. [2,3] If L := q1/3 f qf q 15 f q 3 f q 5 L 3 M 3 1 L 3 M 3 48 [ L 3 M 3 M 3 L 3 ] 12 [ L 6 M 6 M 6 L 6 ] and M := q2/3 f q 2 f q 30 f q 6 f q 10, [ L 9 M 9 M 9 ] L 9 = New mixed modular equations In this section, we establish several new mixed modular equations involving Ramanujan s theta-functions ϕq and ψq. Theorem 3.1. If L := ϕqϕq9 ϕ 2 q 3 qψ qψ q 9 and M := ψ 2 q 3, then L 2 = M M Proof. The equations 2.5 and 2.6 can be rewritten as m αγ1 α1 γ 1/4 m m αγ1 α1 γ 1/4 = β 2 1 β 2 m β 2 1 β 2 Transforming the above equation 3.2 by using the equations 2.3 and 2.4, we arrive at the equation 3.1. qψ qψ q 9 Theorem 3.2. If P := ψ 2 q 3 P 2 P 2 P 2 P 2 2 = and Q := qψ q2 ψ q 18 ψ 2 q 6, then 3P Q 1 3P Q 1 P Q P Q 3P Q 3Q P P 3 Q 3 Q3 P

5 Mixed modular equations involving Ramanujan s theta-functions 287 Proof. Using the equation 2.2 in the equation 2.9, we deduce a 3 P 12 ap 4 bq 4 27a 2 P 8 b 2 Q 8 12a 2 P 8 bq 4 12aP 4 b 2 Q 8 b 3 Q 12 = 0, 3.4 where a := aq := ϕ2 qϕ 2 q 9 ϕ 4 q 3 and b := aq 2. Using the equation 3.1, we have a := P P 2 and b := Q Q Employing the equations 3.5 in the equation 3.4, we find where and AP, QBP, Q = 0, 3.6 AP, Q := 2P 2 Q 4 13P 6 Q 2 25P 6 Q 4 9P 4 Q 4 13P 2 Q 6 16P 8 Q 2 39P 8 Q 4 6P 10 Q 2 27P 6 Q 6 18P 8 Q 6 25P 4 Q 6 16P 2 Q 8 39P 4 Q 8 9P 10 Q 4 18P 6 Q 8 27P 8 Q 8 6P 2 Q 10 9P 4 Q 10 P 2 Q 2 P 6 Q 6 P 10 2P 8 2Q 8 Q 10 2P 4 Q 2 BP, Q := P 8 3P 8 Q 2 P 6 Q 2 P 6 9P 6 Q 4 9P 6 Q 6 Q 6 9P 4 Q 6 6P 4 Q 4 3P 4 Q 2 3P 2 Q 8 P 2 Q 6 3P 2 Q 4 P 2 Q 2 Q 8. Expanding in powers of q, the first and second factor of the equation 3.6, one gets respectively, AP, Q = q 3 1 q q 2 4q 3 10q 4 4q 5 37q 6 28q 7 48q 8 and BP, Q = q q 28q 2 144q 3 32q 4 764q q q 7. As q 0, the factor q 3 BP, Q of the equation 3.6 vanishes whereas the other factor q 3 AP, Q do not vanish. Hence, we arrive at the equation 3.3 for q 0, 1. By analytic continuation the equation 3.3 is true for q < 1. qψqψq 9 Theorem 3.3. If P := ψ 2 q 3 and Q := qψq2 ψq 18 ψ 2 q 6, then P 2 P 2 = 4 3P 2 1 P Proof. Using the equation 2.1 in the equation 2.9, we deduce 27a 2 P 4 b 2 Q 4 a 3 P 6 12a 2 P 4 bq 2 12aP 2 b 2 Q 4 b 3 Q 6 = ap 2 bq 2, 3.8 where a := aq = ϕ2 qϕ 2 q 9 ϕ 4 q 3 and b := aq 2.

6 288 M.S. Mahadeva Naika, S. Chandankumar, M. Harish Using the equation 3.1 after changing q by q, we find a = 1 P 2 1 3P 2 and b = 1 Q2 1 3Q Employing the equations 3.9 in the equation 3.8, we find 3P 2 Q 2 P 2 1 Q 2 3P 4 Q 2 P 4 4P 2 Q 2 Q 2 Q 4 1 2P 2 2Q 2 P 4 44P 2 Q 2 6P 4 Q 2 9P 4 Q 4 16P Q 48P 3 Q 3 6P 2 Q 4 Q 4 = Expanding in powers of q, factors of the equation 3.10 becomes respectively, 1 q 3q 2 2q 3 2q 4 2q 5 q 6 8q 7 24q 8, and q 4 4 8q 8q 2 12q 3 52q 4 24q 5 76q 6 120q 7 32q 8 1 2q 5q 2 50q 3 105q 4 68q 5 82q 6 374q 7 362q 8. As q 0, the second factor of the equation 3.10 vanishes whereas the other factors do not vanish. Hence, we arrive at the equation 3.7 for q 0, 1. By analytic continuation the equation 3.7 is true for q < 1. Theorem 3.4. If P := ϕ qϕ q9 ϕ 2 q 3 and Q := ϕ q2 ϕ q 18 ϕ 2 q 6, then P 2 P 2 = 4 3Q2 1 Q Proof. The proof of the equation 3.11 is similar to the proof of the equation 3.7. Hence, we omit the details. Theorem 3.5. If P := ϕ qϕ q9 ϕ 2 q 3 and Q := ϕ q4 ϕ q 36 ϕ 2 q 12, then P 4 Q 4 Q4 P 2 P 4 8 P 2 P 2 3 Q 4 3Q4 1 P 2 P 2 Q 4 27P 2 Q P 2 Q 2 9P 2 Q P 2 3P 2 3 9Q 4 1 Q 4 = Proof. Using the equation 2.9 along with the equations 2.1 and 2.2, we deduce 27P 4 u 4 Q 8 v 2 12P 4 u 4 Q 4 v 12Q 8 v 2 P 2 u 2 P 6 u 6 Q 12 v 3 = P 2 u 2 Q 4 v, 3.13 where u := qψ2 qψ 2 q 9 ψ 4 q 3 and v := uq 4.

7 Mixed modular equations involving Ramanujan s theta-functions 289 Invoking the equation 3.1, we find u := 1 P 2 1 Q2 3P 2 and v := 1 3Q Using the equations 3.14 in the equation 3.13, we find 3Q 2 P 4 P 4 2Q 2 P 2 P 2 3P 2 Q 4 Q 4 Q 2 3Q 2 P 2 P 2 1 4QP Q 2 3Q 2 P 2 P 2 1 4QP Q 2 P 8 27P 6 Q 8 72P 6 Q 6 36P 6 Q 4 8P 6 Q 2 3P 6 27P 4 Q 8 24P 4 Q 4 3P 4 9P 2 Q 8 8P 2 Q 6 12P 2 Q 4 8Q 2 P 2 P 2 Q 8 = As q 0, the last factor of the equation 3.15 vanishes whereas the other factors do not vanish. Hence, we arrive at the equation 3.12 for q 0, 1. By analytic continuation the equation 3.12 is true for q < 1. Theorem 3.6. If P := ϕ qϕ q9 ϕ 2 q 3 and Q := ϕqϕq9 ϕ 2 q 3, then P Q Q P 3P Q 1 P Q 4 = Proof. Using the equation 2.9 along with the equations 2.1 and 2.2, we deduce 27P 8 u 2 Q 4 v 4 P 12 u 3 12P 8 u 2 Q 2 v 2 12P 4 uq 4 v 4 Q 6 v 6 = P 4 uq 2 v 2, 3.17 where u := qψ2 qψ 2 q 9 ψ 4 q 3 and v := qψ2 qψ 2 q 9 ψ 4 q 3. Invoking the equation 3.1, we find u := 1 P 2 3P 2 1 and v := Q2 1 3Q Using the equations 3.18 in the equation 3.17, we find 3P 2 Q 2 Q 2 1 4P Q P 2 3P 2 Q 2 Q 2 1 4P Q P 2 3P 2 Q 4 Q 4 Q 2 2P 2 Q 2 P 2 3P 4 Q 2 P 4 27P 8 Q 6 27P 8 Q 4 9P 8 Q 2 P 8 72P 6 Q 6 8P 6 Q 2 36P 4 Q 6 24P 4 Q 4 12P 4 Q 2 8P 2 Q 6 8P 2 Q 2 Q 8 3Q 6 3Q 4 Q 2 = As q 0, the second factor of the equation 3.19 vanishes whereas the other factors do not vanish. Hence, we arrive at the equation 3.16 for q 0, 1. By analytic continuation the equation 3.16 is true for q < 1. qψqψq 9 Theorem 3.7. If P := and Q := q2 ψq 4 ψq 36, then P 4 Q 4 Q4 P 4 ψ 2 q 3 P 2 8 P P 4 Q 2 1 P 4 Q P 2 Q 2 1 P 2 Q 2 9P 4 1 P 4 ψ 2 q 12 3P 4 3 Q 2 Q2 P Q 2 1 Q 2 24 =

8 290 M.S. Mahadeva Naika, S. Chandankumar, M. Harish Proof. The proof of the equation 3.20 is similar to the proof of the equation Hence, we omit the details. qψqψq 9 qψ qψ q 9 Theorem 3.8. If P := ψ 2 q 3 and Q := ψ 2 q 3, then P 2 P 2 3P Q 1 P P Q Q Q 2 = P Proof. The proof of the equation 3.21 is similar to the proof of the equation Hence, we omit the details. Theorem 3.9. If L := ϕq3 ϕq 5 ϕqϕq 15 and M := ψ q3 ψ q 5 qψ qψ q 15, then M = 1 L 1 L Proof. The equations 2.7 and 2.8 can be rewritten as m αδ1 α1 δ 1/8 m 1 = m βγ1 β1 γ m m αδ1 α1 δ 1/ m βγ1 β1 γ Transforming the above equation 3.23 by using the equations 2.3 and 2.4, we arrive at equation Theorem If P := ψ q3 ψ q 5 qψ qψ q 15 and Q := ψ q6 ψ q 10 q 2 ψ q 2 ψ q 30, then P 2 P 2 P Q Q P P Q 1 P Q 1 P P Q 3 P Q Q 3 Q 3 P 3 P Q Proof. Using the equation 2.2 in the equation 2.10, we find a 2 P 4 b 2 Q 4 a 4 P 8 b 4 Q 8 a 6 P 12 48a 4 P 8 b 2 Q 4 12a 5 P 10 bq 2 Q = P 48a 2 P 4 b 4 Q 8 76a 3 P 6 b 3 Q 6 b 6 Q 12 12b 5 Q 10 ap 2 = 0, 3.25 where a := aq := ϕq3 ϕq 5 ϕqϕq 15 and b := aq2. Using the equation 3.22, we have a := aq := P 1 P 1 and b := bq := Q 1 Q

9 Mixed modular equations involving Ramanujan s theta-functions 291 Employing the equations 3.26 in the equation 3.25, we find P Q P 1 Q P 2 P 2P Q P Q 2 Q 2 P 2 Q P 2 Q 2P Q Q 2 12P 4 Q 4 P 2 Q 2 5P 4 Q 3 5P 4 Q 2 5P 3 Q 4 P 3 Q 3 P 3 Q 2 5P 2 Q 4 P 2 Q 3 P 6 Q 6 3P 8 3Q 8 3P 6 Q 2 7P 5 Q 2 3P 6 Q 3P 2 Q 6 7P 2 Q 5 3P Q 6 7P 4 Q 6 2P 4 Q 5 5P 3 Q 6 7P 3 Q 5 12P 5 Q 5 2P 5 Q 4 7P 5 Q 3 7P 6 Q 4 5P 6 Q 3 P 6 Q 6 5P 6 Q 5 5P 5 Q 6 P 7 Q 7 P 7 Q 6 5P 7 Q 5 P 6 Q 7 5P 5 Q 7 7P 7 Q 4 7P 4 Q 7 3Q 7 Q 9 3P 7 P 9 3P 9 Q 2 P 9 Q 3 3P 3 Q 8 3P 3 Q 7 P 3 Q 9 3P 9 Q 9P 8 Q 2 3P 7 Q 2 9P 8 Q 9P 7 Q 3P 8 Q 3 3P 7 Q 3 3Q 9 P 2 9Q 8 P 2 3Q 7 P 2 9Q 7 P 9Q 8 P 3P Q 9 P 4 Q 4 P 2 Q P Q 2 P Q P 4 Q P 3 Q 3 P 3 Q 2 P 3 Q P 2 Q 3 P Q 4 P Q 3 P 3 Q 3 = As q 0, the last factor of the equation 3.27 vanishes whereas the other factors do not vanish. Hence, we arrive at the equation 3.24 for q 0, 1. By analytic continuation the equation 3.24 is true for q < 1. Theorem If P := ψq3 ψq 5 qψqψq 15 and Q := ψq6 ψq 10 q 2 ψq 2 ψq 30, then P Q Q = P P P Proof. Using the equation 2.1 in the equation 2.10, we deduce a 2 P 2 b 2 Q 2 a 4 P 4 b 4 Q 4 a 6 P 6 48a 4 P 4 b 2 Q 2 12a 5 P 5 bq 48a 2 P 2 b 4 Q 4 76a 3 P 3 b 3 Q 3 b 6 Q 6 12b 5 Q 5 ap = 0, 3.29 where a := aq := ϕ q3 ϕ q 5 ϕ qϕ q 15 and b := aq2. Changing q to q in the equation 3.22, we find a := 1 P P 1 1 Q and b := Q Using the equations 3.30 in the equation 3.29, we find P P Q 1 Q P 2 P 2 Q 2P Q Q Q 2 P 2 P 2P Q Q 2 P Q 2 P 4 P 3 Q 3 P Q 2 P Q 3 P Q 4 P 2 Q P 2 Q 3 P 3 Q P 3 Q 2 P 3 Q 3 P 4 Q Q 4 P Q 3P 8 P 6 3P 7 P 9 P 2 Q 2 P 2 Q 3 5P 2 Q 4 P 3 Q 2 P 3 Q 3 5P 3 Q 4 5P 4 Q 2 5P 4 Q 3 12P 4 Q 4 3QP 6 9QP 7 9QP 8 7Q 3 P 5 5Q 3 P 6 3Q 3 P 7 3Q 3 P 8

10 292 M.S. Mahadeva Naika, S. Chandankumar, M. Harish 7P 5 Q 2 3P 6 Q 2 3P 7 Q 2 9P 8 Q 2 3P 9 Q 2 P 9 Q 3 2P 5 Q 4 7P 6 Q 4 7P 7 Q 4 3P 9 Q 2P 4 Q 5 7P 4 Q 6 7P 4 Q 7 5P 6 Q 5 P 6 Q 6 P 6 Q 7 12P 5 Q 5 5P 5 Q 6 5P 5 Q 7 5P 7 Q 5 P 7 Q 6 P 7 Q 7 3P Q 6 9P Q 7 9P Q 8 7P 3 Q 5 5P 3 Q 6 3P 3 Q 7 3P 3 Q 8 3P Q 9 7P 2 Q 5 3P 2 Q 6 3P 2 Q 7 9P 2 Q 8 3P 2 Q 9 P 3 Q 9 3Q 8 Q 6 3Q 7 Q 9 = As q 0, the second factor of the equation 3.31 vanishes whereas the other factors do not vanish. Hence, we arrive at the equation 3.28 for q 0, 1. By analytic continuation the equation 3.28 is true for q < 1. Theorem If P := ϕ q3 ϕ q 5 ϕ qϕ q 15 and Q := ϕ q6 ϕ q 10 ϕ q 2 ϕ q 30, then P Q Q = Q P Q Proof. The proof of the equation 3.32 is similar to the proof of the equation Hence, we omit the details. Theorem If P := ϕ q3 ϕ q 5 ϕ qϕ q 15 and Q := ϕ q12 ϕ q 20 ϕ q 4 ϕ q 60, then P 2 3 Q 2 1 Q 2 4 P Q 1 P 3 P Q Q 2 Q2 P P 2 P Q 2 1 P Q 2 2 P 1 P 4 Q 1 = Q Proof. Using the equations 2.1 and 2.2 in the equation 2.10, we deduce 48Q 2 b 2 P 4 a 4 Q 4 b 4 P 4 a 4 Q 2 b 2 P 2 a 2 48Q 4 b 4 P 2 a 2 P 6 a 6 76Q 3 b 3 P 3 a 3 12QbP 5 a 5 Q 6 b 6 12Q 5 b 5 P a = 0, 3.34 where a := aq := ψq3 ψq 5 ψqψq 15 and b := aq4. Changing q to q in the equation 3.22, we find a := 1 P P 1 1 Q and b := Q Employing the equations 3.35 in the equation 3.34, we deduce Q 2 P 2 2QP 2 P 2 2P 1 4QP 2Q 2Q 2 P Q 2 QP 2 P 2 P Q Q 2 P Q 2 3Q 8 6P 4 15P 7 6P 8 45Q 8 P 4 3QP 3 18QP 4 45QP 5 60QP 6 45QP 7 18QP 8 Q 2 P 8Q 2 P 2 10Q 2 P 3 28Q 2 P 4 68Q 2 P 5 64Q 2 P 6 34Q 2 P 7 12Q 2 P 8

11 Mixed modular equations involving Ramanujan s theta-functions 293 Q 3 P 8Q 3 P 2 12Q 3 P 3 16Q 3 P 4 38Q 3 P 5 24Q 3 P 6 4Q 3 P 7 5Q 4 P 3Q 4 P 2 9Q 4 P 3 13Q 4 P 4 33Q 4 P 5 17Q 4 P 6 19Q 4 P 7 7Q 4 P 8 7Q 5 P 19Q 5 P 2 17Q 5 P 3 33Q 5 P 4 13Q 5 P 5 9Q 5 P 6 3Q 5 P 7 5Q 5 P 8 4Q 6 P 2 24Q 6 P 3 38Q 6 P 4 16Q 6 P 5 12Q 6 P 6 8Q 6 P 7 Q 6 P 8 3QP 9 3Q 2 P 9 Q 3 P 9 3Q 8 P 6 18Q 8 P 5 15Q 9 P 4 6Q 9 P 5 Q 9 P 6 Q 6 P 3 20P 6 15P 5 P 9 45Q 8 P 2 34Q 7 P 2 15Q 9 P 2 64Q 7 P 3 68Q 7 P 4 28Q 7 P 5 10Q 7 P 6 8Q 7 P 7 Q 7 P 8 60Q 8 P 3 20Q 9 P 3 3Q 7 Q 9 6Q 9 P 18Q 8 P 12Q 7 P P 4 Q 4 3P 2 P 4QP 4QP 2 2Q 2 P 2Q 2 P 3 4Q 3 P 2 4Q 3 P 3 3Q 4 P 3Q 4 P 2 Q 4 P 3 3P 3 = As q 0, the last factor of the equation 3.36 vanishes whereas the other factors do not vanish. Hence, we arrive at the equation 3.33 for q 0, 1. By analytic continuation the equation 3.33 is true for q < 1. Acknowledgement. The authors are grateful to the referee for his/her valuable remarks and suggestions which considerably improved the quality of the paper. REFERENCES [1] B. C. Berndt, Ramanujan s Notebooks, Part III, Springer-Verlag, New York, [2] B. C. Berndt, Ramanujan s Notebooks, Part IV, Springer-Verlag, New York, [3] S. Bhargava, C. Adiga, M.S. Mahadeva Naika, A new class of modular equations in Ramanujan s alternative theory of elliptic function of signature 4 and some new P Q eta-function identities, Indian J. Math , [4] S. Bhargava,C. Adiga, M.S. Mahadeva Naika, A new class of modular equations akin to Ramanujan s P Q eta-function identities and some evaluations therefrom, Adv. Stud. Contemp. Math , [5] S. Ramanujan, Notebooks 2 volumes, Tata Institute of Fundamental Research, Bombay, received ; in revised form ; available online M. S. Mahadeva Naika, M. Harish, Department of Mathematics, Bangalore University, Central College Campus, Bengaluru , Karnataka, India msmnaika@rediffmail.com, numharish@gmail.com S. Chandankumar, Department of Mathematics, Acharya Institute of Graduate Studies, Soldevanahalli, Hesarghatta Main Road, Bengaluru , Karnataka, India chandan.s17@gmail.com