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

Chandankumar, S.

Harish, M.

Mahadeva Naika, M.S.

ePrints@Bangalore University

MATEMATIQKI VESNIK66, 3 (2014), 283–293September 2014originalni nauqni radresearch paperON SOME NEW MIXED MODULAR EQUATIONSINVOLVING RAMANUJAN’S THETA-FUNCTIONSM. S. Mahadeva Naika, S. Chandankumar and M. HarishAbstract. In his second notebook, Ramanujan recorded altogether 23 P–Q modular equa-tions involving his theta functions. In this paper, we establish several new mixed modular equa-tions involving Ramanujan’s theta-functions ϕ and ψ which are akin to those recorded in hisnotebook.1. IntroductionFollowing Ramanujan, let (a; q)∞ denote the infinite product∞∏n=0(1− aqn) (a,q are complex numbers, |q| < 1) andf(a, b) :=∞∑n=−∞an(n+1)/2bn(n−1)/2, |ab| < 1,denote the Ramanujan theta-function. The product representation of f(a, b) followsfrom Jacobi’s triple product identity and is given by:f(a, b) = (−a; ab)∞(−b; ab)∞(ab; ab)∞.The following definitions of theta-functions ϕ, ψ and f with |q| < 1 are classical:ϕ(q) := f(q, q) =∞∑n=−∞qn2= (−q; q2)2∞(q2; q2)∞, (1.1)ψ(q) := f(q, q3) =∞∑n=0qn(n+1)/2 =(q2; q2)∞(q; q2)∞, (1.2)f(−q) := f(−q,−q2) =∞∑n=−∞(−1)nqn(3n−1)/2 = (q; q)∞. (1.3)The ordinary or Gaussian hypergeometric function is defined by2F1(a, b; c; z) :=∞∑n=0(a)n(b)n(c)nn!zn, 0 ≤ |z| < 1,2010 Mathematics Subject Classification: 33D10, 11A55, 11F27Keywords and phrases: Modular equations; theta-functions283284 M.S. Mahadeva Naika, S. Chandankumar, M. Harishwhere a, b, c are complex numbers such that c 6= 0,−1,−2, . . . , and(a)0 = 1, (a)n = a(a+ 1) · · · (a+ n− 1) for any positive integer n.Now we recall the notion of modular equation. Let K(k) be the complete ellipticintegral of the first kind of modulus k. Recall thatK(k) :=∫ pi/20dφ√1− k2 sin2 φ=pi2∞∑n=0(12)2n(n!)2k2n =pi2ϕ2(q), (0 < k < 1), (1.4)and set K ′ = K(k′), where k′ =√1− k2 is the so called complementary modulusof k. It is classical to set q(k) = e−piK(k′)/K(k) so that q is one-to-one, increasingfrom 0 to 1.In the same manner introduce L1 = K(`1), L′1 = K(`′1) and suppose that thefollowing equalityn1K ′K=L′1L1(1.5)holds for some positive integer n1. Then a modular equation of degree n1 is analgebraic relation between the moduli k and `1 which is induced by (1.5). FollowingRamanujan, set α = k2 and β = `21. Then we say that β is of degree n1 over α.The multiplier m is defined bym =KL1=ϕ2(q)ϕ2(qn1), (1.6)for q = e−piK(k′)/K(k).Let K, K ′, L1, L′1, L2, L′2, L3 and L′3 denote complete elliptic integrals of thefirst kind corresponding, in pairs, to the moduli√α,√β,√γ and√δ, and theircomplementary moduli, respectively. Let n1, n2 and n3 be positive integers suchthat n3 = n1n2. Suppose that the equalitiesn1K ′K=L′1L1, n2K ′K=L′2L2and n3K ′K=L′3L3(1.7)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,n2 and n3 respectively over α. The multipliers m = K/L1 and m′ = L2/L3 arealgebraic relation involving α, β, γ and δ.At scattered places of his second notebook [5], Ramanujan recorded a total ofnine P–Q mixed modular relations of degrees 1, 3, 5 and 15. For example, ifP :=f(−q3)f(−q5)q1/3f(−q)f(−q15) and Q :=f(−q6)f(−q10)q2/3f(−q2)f(−q30) ,thenPQ+1PQ=(QP)3+(PQ)3+ 4. (1.8)The proofs of these nine P–Q mixed modular relations can be found in the bookby Berndt [2, pp. 214–235]. In [3,4], S. Bhargava, C. Adiga and M. S. MahadevaMixed modular equations involving Ramanujan’s theta-functions 285Naika have established several new P–Q mixed modular relations with four mod-uli. These mixed modular equations are employed to evaluate explicit evaluationsof Ramanujan-Weber class invariants, ratios of Ramanujan’s theta functions andseveral continued fractions.Motivated by all these works, in this article, we establish several new mixedmodular equations involving Ramanujan’s theta-functions.In Section 2, we collect some identities which are useful in proofs of our mainresults. In Section 3, we establish several new P–Q mixed modular equations akinto those recorded by Ramanujan in his notebooks involving his theta-functions ϕ(q)and ψ(q).2. Preliminary resultsIn this section, we collect some identities which are useful in establishing ourmain results.Lemma 2.1. [1, Ch. 16, Entry 24 (ii), p. 39] We havef3(−q) = ϕ2(−q)ψ(q). (2.1)Lemma 2.2. [1, Ch. 16, Entry 24 (iv), p. 39] We havef3(−q2) = ϕ(−q)ψ2(q). (2.2)Lemma 2.3. [1, Ch. 17, Entry 10 (i) and Entry 11(ii), pp. 122–123] For0 < α < 1, we haveϕ(q) =√z, (2.3)√2q1/8ψ(−q) = √z{α(1− α)}1/8. (2.4)Lemma 2.4. [1, Ch. 20, Entry 3 (xii), (xiii), pp. 352–353] Let α, β and γ be ofthe first, third and ninth degrees respectively. Let m denote the multiplier relatingα, β and m′ be the multiplier relating γ, δ. Then( β2αγ)1/4+( (1− β)2(1− α)(1− γ))1/4−( β2(1− β)2αγ(1− α)(1− γ))1/4= −3mm′,(2.5)(αγβ2)1/4+( (1− α)(1− γ)(1− β)2)1/4−(αγ(1− α)(1− γ)β2(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 ofthe first, third, fifth and fifteenth degrees respectively. Let m denote the multiplier286 M.S. Mahadeva Naika, S. Chandankumar, M. Harishrelating α and β and m′ be the multiplier relating γ and δ. Then(αδβγ)1/8+( (1− α)(1− δ)(1− β)(1− γ))1/8−(αδ(1− α)(1− δ)βγ(1− β)(1− γ))1/8=√m′m,(2.7)(βγαδ)1/8+( (1− β)(1− γ)(1− α)(1− δ))1/8−(βγ(1− β)(1− γ)αδ(1− α)(1− δ))1/8= −√mm′.(2.8)Lemma 2.6. [3] If L :=q1/6f(−q)f(−q9)f2(−q3) and M :=q1/3f(−q2)f(−q18)f2(−q6) , thenL9M9+M9L9+ 12[ L3M3+M3L3]+ 27L3M3 − 1L3M3= 0. (2.9)Lemma 2.7. [2,3] If L :=q1/3f(−q)f(−q15)f(−q3)f(−q5) and M :=q2/3f(−q2)f(−q30)f(−q6)f(−q10) ,thenL3M3 +1L3M3− 48[ L3M3+M3L3]− 12[ L6M6+M6L6]−[ L9M9+M9L9]= 76. (2.10)3. New mixed modular equationsIn this section, we establish several new mixed modular equations involvingRamanujan’s theta-functions ϕ(q) and ψ(q).Theorem 3.1. If L :=ϕ(q)ϕ(q9)ϕ2(q3)and M :=√qψ(−q)ψ(−q9)ψ2(−q3) , thenL2 =M2 + 11− 3M2 . (3.1)Proof. The equations (2.5) and (2.6) can be rewritten asm′m+(αγ(1− α)(1− γ)β2(1− β)2)1/4= 1− 3mm′(αγ(1− α)(1− γ)β2(1− β)2)1/4. (3.2)Transforming the above equation (3.2) by using the equations (2.3) and (2.4), wearrive at the equation (3.1).Theorem 3.2. If P :=√qψ(−q)ψ(−q9)ψ2(−q3) and Q :=qψ(−q2)ψ(−q18)ψ2(−q6) , thenP 2Q2+Q2P 2+(P 2Q2+Q2P 2)2=(3PQ− 1PQ)(3PQ− 1PQ+3PQ+3QP+P 3Q3+Q3P 3)+ 14. (3.3)Mixed modular equations involving Ramanujan’s theta-functions 287Proof. Using the equation (2.2) in the equation (2.9), we deducea3P 12 − aP 4bQ4 + 27a2P 8b2Q8 + 12a2P 8bQ4 + 12aP 4b2Q8 + b3Q12 = 0, (3.4)where a := a(q) :=ϕ2(q)ϕ2(q9)ϕ4(q3)and b := a(q2).Using the equation (3.1), we havea :=P 2 + 11− 3P 2 and b :=Q2 + 11− 3Q2 . (3.5)Employing the equations (3.5) in the equation (3.4), we findA(P,Q)B(P,Q) = 0, (3.6)whereA(P,Q) :=(− 2P 2Q4 − 13P 6Q2 + 25P 6Q4 + 9P 4Q4 − 13P 2Q6 − 16P 8Q2+ 39P 8Q4 − 6P 10Q2 − 27P 6Q6 − 18P 8Q6 + 25P 4Q6 − 16P 2Q8 + 39P 4Q8+ 9P 10Q4 − 18P 6Q8 − 27P 8Q8 − 6P 2Q10 + 9P 4Q10 + P 2Q2 + P 6 +Q6+ P 10 + 2P 8 + 2Q8 +Q10 − 2P 4Q2)andB(P,Q) :=(− P 8 + 3P 8Q2 − P 6Q2 − P 6 + 9P 6Q4 + 9P 6Q6 −Q6 + 9P 4Q6+ 6P 4Q4 − 3P 4Q2 + 3P 2Q8 − P 2Q6 − 3P 2Q4 + P 2Q2 −Q8).Expanding in powers of q, the first and second factor of the equation (3.6), one getsrespectively,A(P,Q) = −q3(− 1− q + q2 + 4q3 − 10q4 − 4q5 + 37q6 − 28q7 − 48q8 + · · · )andB(P,Q) = q5(4− 12q − 28q2 + 144q3 − 32q4 − 764q5 + 1424q6 + 1400q7 + · · · ).As q → 0, the factor q−3B(P,Q) of the equation (3.6) vanishes whereas the otherfactor q−3A(P,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.Theorem 3.3. If P :=√qψ(q)ψ(q9)ψ2(q3)and Q :=qψ(q2)ψ(q18)ψ2(q6), thenP 2Q2+Q2P 2= 4− 3P 2 + 1P 2. (3.7)Proof. Using the equation (2.1) in the equation (2.9), we deduce27a2P 4b2Q4 + a3P 6 + 12a2P 4bQ2 + 12aP 2b2Q4 + b3Q6 = aP 2bQ2, (3.8)where a := a(q) =ϕ2(−q)ϕ2(−q9)ϕ4(−q3) and b := a(q2).288 M.S. Mahadeva Naika, S. Chandankumar, M. HarishUsing the equation (3.1) after changing q by −q, we finda =1− P 21 + 3P 2and b =1−Q21 + 3Q2. (3.9)Employing the equations (3.9) in the equation (3.8), we find(3P 2Q2 + P 2 − 1 +Q2)(3P 4Q2 + P 4 − 4P 2Q2 −Q2 +Q4)(1− 2P 2 − 2Q2 + P 4 + 44P 2Q2 + 6P 4Q2 + 9P 4Q4 + 16PQ+ 48P 3Q3 + 6P 2Q4 +Q4)= 0. (3.10)Expanding in powers of q, factors of the equation (3.10) becomes respectively,(− 1 + q + 3q2 + 2q3 + 2q4 − 2q5 − q6 − 8q7 − 24q8 + · · · ),q4(− 4− 8q − 8q2 + 12q3 + 52q4 + 24q5 − 76q6 − 120q7 + 32q8 + · · · )and (1− 2q − 5q2 + 50q3 + 105q4 + 68q5 − 82q6 − 374q7 − 362q8 + · · · ).As q → 0, the second factor of the equation (3.10) vanishes whereas the otherfactors do not vanish. Hence, we arrive at the equation (3.7) for q ∈ (0, 1). Byanalytic continuation the equation (3.7) is true for |q| < 1.Theorem 3.4. If P :=ϕ(−q)ϕ(−q9)ϕ2(−q3) and Q :=ϕ(−q2)ϕ(−q18)ϕ2(−q6) , thenP 2Q2+Q2P 2= 4− 3Q2 + 1Q2. (3.11)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(−q)ϕ(−q9)ϕ2(−q3) and Q :=ϕ(−q4)ϕ(−q36)ϕ2(−q12) , then(P 4Q4+Q4P 4)+ 8(P 2Q2+Q2P 2)− 3(P 2Q4− 3Q4P 2)−( 1P 2Q4− 27P 2Q4)− 8( 1P 2Q2+ 9P 2Q2)− 12( 1P 2− 3P 2)+ 3(9Q4 +1Q4)= 24. (3.12)Proof. Using the equation (2.9) along with the equations (2.1) and (2.2), wededuce27P 4u4Q8v2 + 12P 4u4Q4v + 12Q8v2P 2u2 + P 6u6 +Q12v3 = P 2u2Q4v, (3.13)where u :=qψ2(q)ψ2(q9)ψ4(q3)and v := u(q4).Mixed modular equations involving Ramanujan’s theta-functions 289Invoking the equation (3.1), we findu :=1− P 23P 2 + 1and v :=1−Q23Q2 + 1. (3.14)Using the equations (3.14) in the equation (3.13), we find(3Q2P 4 + P 4 − 2Q2P 2 − P 2 + 3P 2Q4 +Q4 −Q2)(3Q2P 2 + P 2 − 1+ 4QP +Q2)(3Q2P 2 + P 2 − 1− 4QP +Q2)(P 8 + 27P 6Q8 − 72P 6Q6+ 36P 6Q4 + 8P 6Q2 − 3P 6 + 27P 4Q8 − 24P 4Q4 + 3P 4 + 9P 2Q8+ 8P 2Q6 − 12P 2Q4 − 8Q2P 2 − P 2 +Q8) = 0. (3.15)As q → 0, the last factor of the equation (3.15) vanishes whereas the other factorsdo not vanish. Hence, we arrive at the equation (3.12) for q ∈ (0, 1). By analyticcontinuation the equation (3.12) is true for |q| < 1.Theorem 3.6. If P :=ϕ(−q)ϕ(−q9)ϕ2(−q3) and Q :=ϕ(q)ϕ(q9)ϕ2(q3), then(PQ+QP)+(3PQ− 1PQ)− 4 = 0. (3.16)Proof. Using the equation (2.9) along with the equations (2.1) and (2.2), wededuce27P 8u2Q4v4 + P 12u3 + 12P 8u2Q2v2 + 12P 4uQ4v4 +Q6v6 = P 4uQ2v2, (3.17)where u :=qψ2(q)ψ2(q9)ψ4(q3)and v :=qψ2(−q)ψ2(−q9)ψ4(−q3) .Invoking the equation (3.1), we findu :=1− P 23P 2 + 1and v :=Q2 − 13Q2 + 1. (3.18)Using the equations (3.18) in the equation (3.17), we find(3P 2Q2 +Q2 − 1 + 4PQ+ P 2)(3P 2Q2 +Q2 − 1− 4PQ+ P 2)(3P 2Q4+Q4 −Q2 − 2P 2Q2 − P 2 + 3P 4Q2 + P 4)(27P 8Q6 + 27P 8Q4 + 9P 8Q2+ P 8 − 72P 6Q6 + 8P 6Q2 + 36P 4Q6 − 24P 4Q4 − 12P 4Q2 + 8P 2Q6− 8P 2Q2 +Q8 − 3Q6 + 3Q4 −Q2) = 0. (3.19)As q → 0, the second factor of the equation (3.19) vanishes whereas the otherfactors do not vanish. Hence, we arrive at the equation (3.16) for q ∈ (0, 1). Byanalytic continuation the equation (3.16) is true for |q| < 1.Theorem 3.7. If P :=√qψ(q)ψ(q9)ψ2(q3)and Q :=q2ψ(q4)ψ(q36)ψ2(q12), then(P 4Q4+Q4P 4)+ 8(P 2Q2+Q2P 2)− 8(32P 2Q2 +1P 2Q2)+ 3(3P 4Q2− Q2P 4)+(27P 4Q2 − 1P 4Q2)+ 3(9P 4 +1P 4)+ 12(3Q2 − 1Q2)− 24 = 0. (3.20)290 M.S. Mahadeva Naika, S. Chandankumar, M. HarishProof. The proof of the equation (3.20) is similar to the proof of the equation(3.12). Hence, we omit the details.Theorem 3.8. If P :=√qψ(q)ψ(q9)ψ2(q3)and Q :=√qψ(−q)ψ(−q9)ψ2(−q3) , then(P 2Q2+Q2P 2)−(3PQ+1PQ)(PQ− QP)+ 2 = 0. (3.21)Proof. The proof of the equation (3.21) is similar to the proof of the equation(3.16). Hence, we omit the details.Theorem 3.9. If L :=ϕ(q3)ϕ(q5)ϕ(q)ϕ(q15)and M :=ψ(−q3)ψ(−q5)qψ(−q)ψ(−q15) , thenM =1 + L1− L. (3.22)Proof. The equations (2.7) and (2.8) can be rewritten as1 +√mm′(αδ(1− α)(1− δ)βγ(1− β)(1− γ))1/8=√mm′− mm′(αδ(1− α)(1− δ)βγ(1− β)(1− γ))1/8. (3.23)Transforming the above equation (3.23) by using the equations (2.3) and (2.4), wearrive at equation (3.22).Theorem 3.10. If P :=ψ(−q3)ψ(−q5)qψ(−q)ψ(−q15) and Q :=ψ(−q6)ψ(−q10)q2ψ(−q2)ψ(−q30) , then(P 2Q2+Q2P 2)+(PQ+QP)−(PQ+1PQ)+(√PQ− 1√PQ)(√P 3Q3+√Q3P 3+√PQ+√QP)= 0. (3.24)Proof. Using the equation (2.2) in the equation (2.10), we finda2P 4b2Q4 + a4P 8b4Q8 − a6P 12 − 48a4P 8b2Q4 − 12a5P 10bQ2− 48a2P 4b4Q8 − 76a3P 6b3Q6 − b6Q12 − 12b5Q10aP 2 = 0, (3.25)where a := a(q) :=ϕ(q3)ϕ(q5)ϕ(q)ϕ(q15)and b := a(q2).Using the equation (3.22), we havea := a(q) :=P − 1P + 1and b := b(q) :=Q− 1Q+ 1. (3.26)Mixed modular equations involving Ramanujan’s theta-functions 291Employing the equations (3.26) in the equation (3.25), we find(PQ+ P − 1 +Q)(P 2 − P − 2PQ+ PQ2 +Q2)(P 2Q+ P 2 −Q− 2PQ+Q2)(12P 4Q4 − P 2Q2 − 5P 4Q3 − 5P 4Q2 − 5P 3Q4 − P 3Q3− P 3Q2 − 5P 2Q4 − P 2Q3 − P 6 −Q6 − 3P 8 − 3Q8 + 3P 6Q2 + 7P 5Q2− 3P 6Q+ 3P 2Q6 + 7P 2Q5 − 3PQ6 − 7P 4Q6 − 2P 4Q5 + 5P 3Q6+ 7P 3Q5 − 12P 5Q5 − 2P 5Q4 + 7P 5Q3 − 7P 6Q4 + 5P 6Q3 + P 6Q6− 5P 6Q5 − 5P 5Q6 + P 7Q7 − P 7Q6 + 5P 7Q5 − P 6Q7 + 5P 5Q7 + 7P 7Q4+ 7P 4Q7 + 3Q7 +Q9 + 3P 7 + P 9 + 3P 9Q2 + P 9Q3 − 3P 3Q8 − 3P 3Q7+ P 3Q9 + 3P 9Q− 9P 8Q2 + 3P 7Q2 − 9P 8Q+ 9P 7Q− 3P 8Q3− 3P 7Q3 + 3Q9P 2 − 9Q8P 2 + 3Q7P 2 + 9Q7P − 9Q8P + 3PQ9)(P 4 +Q4 − P 2Q− PQ2 − PQ+ P 4Q− P 3Q3 + P 3Q2 + P 3Q+ P 2Q3 + PQ4 + PQ3 − P 3 −Q3) = 0. (3.27)As q → 0, the last factor of the equation (3.27) vanishes whereas the other factorsdo not vanish. Hence, we arrive at the equation (3.24) for q ∈ (0, 1). By analyticcontinuation the equation (3.24) is true for |q| < 1.Theorem 3.11. If P :=ψ(q3)ψ(q5)qψ(q)ψ(q15)and Q :=ψ(q6)ψ(q10)q2ψ(q2)ψ(q30), then(PQ+QP)=(P − 1P)+ 2. (3.28)Proof. Using the equation (2.1) in the equation (2.10), we deducea2P 2b2Q2 + a4P 4b4Q4 − a6P 6 − 48a4P 4b2Q2 − 12a5P 5bQ− 48a2P 2b4Q4 − 76a3P 3b3Q3 − b6Q6 − 12b5Q5aP = 0, (3.29)where a := a(q) :=ϕ(−q3)ϕ(−q5)ϕ(−q)ϕ(−q15) and b := a(q2).Changing q to −q in the equation (3.22), we finda :=1 + PP − 1 and b :=1 +QQ− 1 . (3.30)Using the equations (3.30) in the equation (3.29), we find(− P + PQ− 1−Q)(− P 2 + P 2Q+ 2PQ−Q−Q2)(P 2 + P − 2PQ+Q2 − PQ2)(− P 4 − P 3 −Q3 − PQ2 − PQ3 + PQ4 − P 2Q+ P 2Q3− P 3Q+ P 3Q2 + P 3Q3 + P 4Q−Q4 + PQ)(− 3P 8 − P 6 − 3P 7 − P 9− P 2Q2 + P 2Q3 − 5P 2Q4 + P 3Q2 − P 3Q3 + 5P 3Q4 − 5P 4Q2 + 5P 4Q3+ 12P 4Q4 + 3QP 6 + 9QP 7 + 9QP 8 + 7Q3P 5 − 5Q3P 6 − 3Q3P 7 + 3Q3P 8292 M.S. Mahadeva Naika, S. Chandankumar, M. Harish− 7P 5Q2 + 3P 6Q2 − 3P 7Q2 − 9P 8Q2 − 3P 9Q2 + P 9Q3 + 2P 5Q4 − 7P 6Q4− 7P 7Q4 + 3P 9Q+ 2P 4Q5 − 7P 4Q6 − 7P 4Q7 + 5P 6Q5 + P 6Q6 + P 6Q7− 12P 5Q5 + 5P 5Q6 + 5P 5Q7 + 5P 7Q5 + P 7Q6 + P 7Q7 + 3PQ6 + 9PQ7+ 9PQ8 + 7P 3Q5 − 5P 3Q6 − 3P 3Q7 + 3P 3Q8 + 3PQ9 − 7P 2Q5 + 3P 2Q6− 3P 2Q7 − 9P 2Q8 − 3P 2Q9 + P 3Q9 − 3Q8 −Q6 − 3Q7 −Q9) = 0.(3.31)As q → 0, the second factor of the equation (3.31) vanishes whereas the otherfactors do not vanish. Hence, we arrive at the equation (3.28) for q ∈ (0, 1). Byanalytic continuation the equation (3.28) is true for |q| < 1.Theorem 3.12. If P :=ϕ(−q3)ϕ(−q5)ϕ(−q)ϕ(−q15) and Q :=ϕ(−q6)ϕ(−q10)ϕ(−q2)ϕ(−q30) , then(PQ+QP)=(Q− 1Q)+ 2. (3.32)Proof. The proof of the equation (3.32) is similar to the proof of the equation(3.28). Hence, we omit the details.Theorem 3.13. If P :=ϕ(−q3)ϕ(−q5)ϕ(−q)ϕ(−q15) and Q :=ϕ(−q12)ϕ(−q20)ϕ(−q4)ϕ(−q60) , then(P 2Q2+Q2P 2)+ 3(Q2 +1Q2)− 4(PQ+1PQ)+ 3( PQ2− Q2P)−(PQ2 +1PQ2)− 2(P − 1P)+ 4(Q− 1Q)= 0. (3.33)Proof. Using the equations (2.1) and (2.2) in the equation (2.10), we deduce48Q2b2P 4a4 −Q4b4P 4a4 −Q2b2P 2a2 + 48Q4b4P 2a2 + P 6a6+ 76Q3b3P 3a3 + 12QbP 5a5 +Q6b6 + 12Q5b5Pa = 0, (3.34)where a := a(q) :=ψ(q3)ψ(q5)ψ(q)ψ(q15)and b := a(q4).Changing q to −q in the equation (3.22), we finda :=1 + PP − 1 and b :=1 +QQ− 1 . (3.35)Employing the equations (3.35) in the equation (3.34), we deduce(Q2P 2 − 2QP 2 + P 2 + 2P + 1− 4QP + 2Q− 2Q2P +Q2)(QP 2 − P 2− P −Q+Q2P −Q2)(3Q8 − 6P 4 − 15P 7 − 6P 8 + 45Q8P 4 + 3QP 3+ 18QP 4 + 45QP 5 + 60QP 6 + 45QP 7 + 18QP 8 −Q2P + 8Q2P 2+ 10Q2P 3 − 28Q2P 4 − 68Q2P 5 − 64Q2P 6 − 34Q2P 7 − 12Q2P 8Mixed modular equations involving Ramanujan’s theta-functions 293+Q3P − 8Q3P 2 − 12Q3P 3 + 16Q3P 4 + 38Q3P 5 + 24Q3P 6 + 4Q3P 7− 5Q4P + 3Q4P 2 − 9Q4P 3 − 13Q4P 4 + 33Q4P 5 + 17Q4P 6 − 19Q4P 7− 7Q4P 8 − 7Q5P + 19Q5P 2 + 17Q5P 3 − 33Q5P 4 − 13Q5P 5 + 9Q5P 6+ 3Q5P 7 + 5Q5P 8 + 4Q6P 2 − 24Q6P 3 + 38Q6P 4 − 16Q6P 5 − 12Q6P 6+ 8Q6P 7 +Q6P 8 + 3QP 9 − 3Q2P 9 +Q3P 9 + 3Q8P 6 − 18Q8P 5+ 15Q9P 4 − 6Q9P 5 +Q9P 6 +Q6 − P 3 − 20P 6 − 15P 5 − P 9 + 45Q8P 2+ 34Q7P 2 + 15Q9P 2 − 64Q7P 3 + 68Q7P 4 − 28Q7P 5 − 10Q7P 6 + 8Q7P 7+Q7P 8 − 60Q8P 3 − 20Q9P 3 + 3Q7 +Q9 − 6Q9P − 18Q8P − 12Q7P )(P 4 +Q4 + 3P 2 + P − 4QP − 4QP 2 + 2Q2P − 2Q2P 3 + 4Q3P 2− 4Q3P 3 − 3Q4P + 3Q4P 2 −Q4P 3 + 3P 3) = 0. (3.36)As q → 0, the last factor of the equation (3.36) vanishes whereas the other factorsdo not vanish. Hence, we arrive at the equation (3.33) for q ∈ (0, 1). By analyticcontinuation the equation (3.33) is true for |q| < 1.Acknowledgement. The authors are grateful to the referee for his/her valu-able 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, 1991.[2] B. C. Berndt, Ramanujan’s Notebooks, Part IV, Springer-Verlag, New York, 1994.[3] S. Bhargava, C. Adiga, M.S. Mahadeva Naika, A new class of modular equations in Ramanu-jan’s alternative theory of elliptic function of signature 4 and some new P–Q eta-functionidentities, Indian J. Math. 45 (1) (2003), 23–39.[4] S. Bhargava,C. Adiga, M.S. Mahadeva Naika, A new class of modular equations akin to Ra-manujan’s P–Q eta-function identities and some evaluations therefrom, Adv. Stud. Contemp.Math. 5 (1) (2002), 37–48.[5] S. Ramanujan, Notebooks (2 volumes), Tata Institute of Fundamental Research, Bombay,1957.(received 14.12.2012; in revised form 24.06.2013; available online 10.09.2013)M. S. Mahadeva Naika, M. Harish, Department of Mathematics, Bangalore University, CentralCollege Campus, Bengaluru-560 001, Karnataka, IndiaE-mail : msmnaika@rediffmail.com, numharish@gmail.comS. Chandankumar, Department of Mathematics, Acharya Institute of Graduate Studies, Solde-vanahalli, Hesarghatta Main Road, Bengaluru-560 107, Karnataka, IndiaE-mail : chandan.s17@gmail.com

On some new mixed modular equations involving Ramanujan's theta-functions

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On some new mixed modular equations involving Ramanujan's theta-functions

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