A note on cubic modular equations of degree two *.

Abstract

On Page 259 of his second notebook [3], Ramanujan recorded many cubic modular equations of degree 2. In this paper we establish several cubic modular equations of degree 2 akin to those in Ramanujan's work. As an application of our results, we also establish some new P -Q etafunction identities.

Keywords and Phrases: Cubic modular equations, Eta-function identities.

1. A Family of Cubic Modular Equations

The ordinary hypergeometric series [sub.2][F.sub.1](a, b; c; x) = [[infinity].summation over (n=0)] [(a).sub.n][(b).sub.n]x.sup.b],/[(c).sub.n]

where

[(a).sub.0] = 1, [(a).sub.n] = a(a + 1)(a + 2) ... (a + n - 1), for n [greater than or equal to] 1, | x |< 1.

Let

Z(r) := Z(r; x) := [sub.2] [F.sub.1] (1/r, r-1/r; 1; x)

and

[q.sub.r]:=[q.sub.r]:=exp (-[pi] csc ([pi]/r) [sub.2][F.sub.1] (1/r, (r-1/r; 1; 1-x)/[sub.2][F.sub.1](1/r,r-1/r;1;x)).

where r = 2, 3, 4, 6 and 0 < x < 1.

Let n denote a fixed natural number, and assume that,

n [sub.2][F.sub.1](1/r,(r-1)/r;1;1-[alpha])/[sub.2][F.sub.1] (1/r,r-1/r;1;[alpha])= [sub.2][F.sub.1](1/r,(r-1)/r;1;1-[beta])/ [sub.2][F.sub.1](1/r,r-1/r;1;[beta])= (1.1)

where r = 2, 3, 4 or 6. Then a modular equation of degree n in the theory of elliptic functions of signature r is a relation between [alpha] and [beta] induced by (1.1). On Pages 257-262 of his second notebook [3, pp. 257-262], Ramanujan gives an outline of the theories of elliptic functions to alternate bases corresponding to the classical theory by way of statements of some results. Venkatachaliengar [4] examined some of these results. Proofs of all these identities can be found in [2, pp.122-123]. Recently, Adiga, Kim and Naika [1] also established some cubic modular equations in the theory of signature 3. Now we state a transformation formula which is useful in establishing several cubic equations of degree 2 in the theory of signature 3.

A Note on Cubic Modular Equations of Degree Two 3

Lemma 1.1. (see [3, p. 258]). If

[alpha] := [alpha](q) = [p(3 + p).sup.2]/[2(1 + p).sup.3] and [beta]:= [beta](q) = [p.sup.2](3 + p)/4, (1.2)

then for 0 [less than or equal to] p [less than or equal to] 1,

[sub.2][F.sub.1](1/3, 2/3; 1; [alpha]) = [(1 + p).sub.2][F.sub.1](1/3, 2/3; 1; [beta]).

For a proof of Lemma 1.1, see the work of Berndt [2, p. 112].

Theorem 1.1. If [beta] is of degree 2 over [alpha] in the theory of signature 3, then

(i)

[m.sup.3] = 3 ([beta](1-[beta])/[alpha](1-[alpha]).sup.1/3] ((1-[beta]/[alpha]).sup.1/3] - ([beta]/1[alpha]).sup.1/3) + 8/[m.sup.3] ([beta](1-[beta])/[alpha](1-[alpha]), (1.4)

(ii)[m.sup.2] [([alpha](1-[alpha])/[[beta].sup.2](1-[beta]).sup.2]).sup.1/3] = [m/sup.6]([alpha](1-[alpha])/[beta](1-[beta]))+7/3, (1.5)

(iii) [m.sup.4] ([beta](1-[beta])/[[alpha].sup.2](1-[alpha]).sup.2]).sup.1/3] = 16 9[beta](1-[beta])/[alpha](1-[alpha]))+[m.sup.6]/3, (1,6)

(iv) 8/[m.sup.3] = [alpha]/[beta]-3 ([alpha](1-[alpha].sup.2]/[[beta].sup.2] (1-[beta])).sup.1/3],

(v) [m.sup.3] = 1 - [beta]/ 1 - [alpha] - 3 ([[beta].sup.2](1 - [beta])/([alpha](1 - [alpha]).sup.2]).sup.1/3], 91.8)

(vi) [m.sup.3] = 3 [([beta(1 - [beta]).sup.2]/[[alpha].sup.2(1 - [alpha])).sup.1/3] - [beta]/[alpha] (1.9)

(vii) 8/[m.sup.3] = 3 [([[alpha].sup.2] (1 - [alpha])/[beta] [(1 - [beta]).sup.2)].sup.1/3] - 1 - [alpha]/[beta], (1.10)

(viii) m = 3 [([beta]/[[alpha].sup.2]).sup.1/3] - 4/[m.sup.2] [beta]/alpha] (1.11)

(ix) [m.sup.2] = 3 (1 - [alpha]/[(1 - [beta]).sup.2]) - 2/m (1 - [beta]/1 - [alpha]), (1.12)

(x) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.13)

and

(xi) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.14)

Proof of (1.4). From (1.2), by elementary calculations, we have

1-[alpha] = [(1-p).sup.2](1+p)/2[(1+p).sup.3] and 1-[beta]=(1-p)[(2+p).sup.2]/4 (1.15)

Using (1.2) and (1.15) in (1.4), we find that

3([beta](1-beta])/[alpha](1-[alpha])).sup.1/3] ([(1-[beta]/[alpha]).sup.1/3] - [([beta]/1 - [alpha]).sup.1/3]) + 8/[m.sup.3] ([beta] (1 - [beta]/[alpha](1 - [alpha])) = [(1 + p).sup.3] = [m.sup.3].

This completes the proof of (1.4).

The proofs of the identities (1.5) to (1.15) are similar to the proof of (1.4). We omit the details.

2. P-Q Eta-Function Identities

Following Ramanujan's work, we define

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and

f (-q) = f (-q, -[g.sup.2]) = [[infinity].summation over (n= -[infinity] [(-1).sup.n] [q.sup.(n (3n-1)/2]

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

In this section we obtain some new P - Q eta-function identities on employing modular equations in Section 2 and the following lemma: Lemma 2.1. For 0 < x < 1,

b(q) = [(1 - x).sup.1/3]z = [f.sup.3]( - q)/f(- [sup.3]) and c(q) = [x.sup.1/3]z = [eq.sup.1/3][f.sup.3](- [q.sup.3])./f(f - q).

For a proof of Lemma 2.1, see [2, p.109].

Theorem 2.1. (see [3, p. 327]).Let

P = f ([-q.sup.2])/[q.sup.1/24] f ([-q.sup.3]) and Q - f (-q)/[q.sup.5/24]. (2.2)

Then

[(PQ).sup.2] - 9/[(PQ).sup.2] = [(Q/P).sup.3] - [(P/Q).sup.3] (2.3)

Proof. Using (2.1) in (1.4) and then using (2.2), we obtain

1 = [P.sup.5]/Q + 9P/[Q.sup.5] + [8P.sup.6]/[Q.sup.6]. (2.4)

On simplification, we obtain (2.3).

Theorem 2.2. Let

P = [[psi].sup.4](q)/[q[[psi].sup.4]([q.sup.3]) and Q = [[psi].sup.4] ([q.sup.2])/[q.sup.2][[psi].sup.4]([q.sup.6]). (2.5)

Then

[P.sup.2] (P - 9 / P - 1) = Q [(Q - 9/ Q - 1).sup.2]. (2.6)

Proof. Using (2.1) in (1.13), we find that

[[phi].sup.4](-q)/[[phi].sup.4]([-q.sup.3])=[[psi].sup.4](q) - 9q[[psi].sup.4][q.sup.3] (2.7)

Using Entry 24(ii) and (iv) of Chapter 16 of Ramanujan's second notebook [3, p. 198] in (2.7), we obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Using (2.5) in (2.8) and (2.9), we obtain the required result.

Theorem 2.3. Let

P = [phi](-q)/[phi]([-q.sup.3] and Q = [phi]([-q.sup.2]/[phi](-[q.sup.6]). (2.10)

Then

P [(P - (P - 9/ P - 1).sup.2] = [Q.sup.2](Q - 9/ Q - 1). (2.11)

The proof of Theorem 2.3 is similar to the proof of Theorem 2.2, so we omit the details.

Remark. The P -Q eta-function identities (2.6) and (2.12) appear to be new in the literature.

Acknowledgments

The author is grateful to Prof. H. M. Srivastava for his valuable suggestions. 8 M. S. Mahadeva Naika

Received June 23, 2004, Accepted January 14, 2005.

References

[1] C. Adiga, T. Kim and M. S. Mahadeva Naika, Modular equations in the theory of signature 3 and P - Q identities, Adv. Stud. Contemp. Math. 7 (2003), 33-40.

[2] B. C. Berndt, Ramanujan's Notebooks, Part V, Springer-Verlag, New York, 1994.

[3] S. Ramanujan, Notebooks (2 volumes), Tata Institute of Fundamental Research, Bombay, 1957.

[4] K. Venkatachaliengar, Development of Elliptic Functions According to Ramanujan, Technical Report 2, Madurai Kamaraj University, Madurai, 1988.

* 2000 Mathematics Subject Classification. Primary 33D15, 33D20; Secondary 11S23.

M. S. Mahadeva Naika ([dagger])

Department of Mathematics, Bangalore University, Central College Campus

Bangalore 560 001, Karnataka State, India

([dagger]) E-mail: msmnaika@redi_mail.com