On some P - Q modular equations of degree 45

On page 330 of his second notebook

The ordinary hypergeometric series is defined by where (a) 0 := 1 and (a) n := a(a + 1)(a + 2) Suppose that holds for some n ∈ Z + , then the relation induced by between α and β by (1.1) is called a modular equation of degree n in signature r.The case r = 2 is called classical case.Ramanujan has recorded many modular equation in his Notebooks [18,19] both in classical and alternate theories (r = 3, 4 and 6), proof of all modular equations recorded by Ramanujan can be seen in [4,5,6].
In the unorganized pages of his second notebook [18], Ramanujan recorded 23 identities, so called P -Q eta function identities or P -Q modular equations.These are identities involving quotients of eta function, which are designated by P or Q by Ramanujan.Proofs of these P -Q identities employing various modular equations of Ramanujan or via the theory of modular forms have been given by Berndt and L.-C.Zhang [9].Similar 14 identities involving ratio of Dedikind's eta function found on page 55 of Ramanujan's Lost notebook [19] have been proved by Berndt [7] employing the above methods.These P -Q modular equations play a very important role in evaluations of class invariants, continued fractions and ratios of theta functions, for more details, see [1,2,3,11,13,15,24,25,26,27,30].
On page 330 of his notebooks [18], Ramanujan has recorded a P -Q modular equation of degree 45, proof of which has been given by Berndt [5,Entry 39,p. 192] by employing theory of modular forms.In Section 2 of this paper, we give a simple proof of Entry 39 [5] and also establish certain new identities involving ratios of theta functions in degree 45.
Ramanujan [18] has recorded many Russell [20,21], Schläfli [22]and Weber [29] type modular equations, proofs of these can be seen in [6].For a wonderful introduction to Ramanujan's work on modular equations, refer [17].In his notebooks [18], Ramanujan has recorded several modular equations in the theory of signature 3, details of these can be seen in [6,8].For recent works on the same, one may refer [12,10,14,28,16,23].Motivated by these, in Section 3 of this paper, employing the P -Q identities obtained in Section 2, we deduce certain new mixed modular equations in the theory of signature 3 which are analogous to Ramanujan-Schläfli and Ramanujan-Weber type modular equations in classical theory.

P -Q Modular Equations
In this Section, we establish certain P -Q modular equations.We first recall the following Lemmas: Theorem 2.1.[18, p. 330][5, p. 192] If Proof.Let B and B 5 be as defined in Lemma 2.2, we set B 3 := f 3 q 1/4 f 9 and B 15 = f 15 q 5/4 f 45 , then Now from Lemma 2.2 and (2.4), we have Solving this for P 3 B 6

G. Sharath
From Lemma 2.1, we have .
Changing q → q 5 in the above identity, we see that .
Next, multiplying the above two identities and then employing (2.4), we deduce that . Now using (2.9) and (2.10) in the above identity and upon simplification, we obtain 2 " ) .
Squaring the above on both sides, substituting the values of K 1 and K 2 and then factorizing, we arrive at where and From the definitions of P and Q, we have and Using these in G(P, Q) and H(P, Q), we find that and Clearly q 47/2 G(P, Q) → 0 as q → 0, whereas q 47/2 H(P, Q) 6 → 0 as q → 0.
Hence by analytic continuation G(P, Q) = 0 in |q| < 1.Thus, which is equivalent to the required result.
Remark.Comparing the definition [18, p. 330] of u and v and that of P and Q of Theorem 2.1, we see that P = v u and Q = u, using these in Theorem 2.1, we deduce Entry 39 [5, p. 192 .
Proof.Following (2.3), we set C 9 = f 9 q 3/2 f 45 , then we note that Now, from Lemma 2.3 and (2.11), we have Solving the above for where From (2.12), we have Changing q to q 3 in (2.12) and (2.13), we respectively obtain where Multiplying (2.12) and (2.15), we deduce that 4(RS . Similarly, multiplying (2.13) and (2.14), we see that Now, adding the above two identities, we find that 2 Squaring the above on both sides, substituting the values of K 1 and K 2 in the resulting identity and then factorizing, we arrive at By definitions of R and S, it follows that RS 6 = −1 and RS + 1 RS 6 = 1, hence the third factor must be zero, which is equivalent to the required result.
Changing q to q 5 in Lemma 2.1, we have

17)
where A 5 = f 5 q 5/12 f 15 and Now, rearranging the terms in Lemma 2.1 correspondingly as in (2.17) and then multiplying the resultant identity with (2.17), we obtain , where P and Q are as defined in Theorem 2.1.The above identity can be rewritten as Next, using (2.16) in Theorem 2.1, we see that Cubing the above identity throughout and after simplification, we have where S = M + 9 M + 3. Solving the above identity for Squaring the above identity on both sides yields 20) and (2.21), we see that Squaring the above identity and simplifying, we obtain Again squaring the above identity and after simplification we arrive at where Now by the definitions of M and N , we have M = q 2 (1 − q − q 2 + q 6 + 2q 7 + q 9 − 3q 10 − q 12 − q 15 + • • •) and N = q 4/3 (1−q −q 2 +2q 5 −q 6 +q 9 +2q 10 −3q 11 −2q 12 +2q 14 +3q 15 −• • •).
Using these in the definition of U (M, N ) and V (M, N ), we see that Clearly q 16 U (M, N ) → 0 as q → 0, where as q 16 V (M, N ) 6 → 0 as q → 0. Hence by analytic continuation, we have U (M, N ) = 0, which is equivalent to the required result.
Now, multiplying the identity in Lemma 2.1 and (2.24) and then employing (2.16) and (2.22) in the resulting identity, we obtain G. Sharath # and (iv) where