A note on P-Q modular equations *.

Abstract

Ramanujan has recorded many P-Q modular equations in his notebooks. In this paper, we establish certain P-Q modular equations, using some of Ramanujan's modular equations.

Keywords and Phrases: Theta-functions, Modular equations, Eta-function identities.

1. Introduction

In his 'lost' notebook [11] and in the unorganized portions of his second notebook [10], Srinivasa Ramanujan (1887-1920) has recorded several P-Q modular equations or eta-function identites. Berndt and Zhang [7] proved some of Ramanujan eta-function identities in [7]. Proof of all the identities recorded by Ramanujan are given in Chapter 25 of [4] and [6]. Baruah [2], Madhusudan, Naika and Vasuki [8], as well as Naika [9], have given alternative proofs of some of these identities.

Each modular equation is equivalent to a certain theta-function identity, but a theta-function identity may not have an equivalent modular equation. Modular equations can be used to evaluate class invariants and certain q-continued fractions, including the Rogers-Ramanujan continued fractions, thetafunctions and certain other quotients and products of theta-functions [5].

In this paper, we establish certain P-Q eta-function identites by using Ramanujan eta-funciton identities. Some of our results seem to be new.

2. Preliminary Definitions and Notations

In Chapter 16 of his second notebook ([1], [10]), Ramanujan develops a theory of theta-functions. His theta-function is defined by

f(a, b) := ([infinity].summation over (n = -[infinity])][a.sup.n(n-1)/2] [b.sup.n(n-1)/2], [absolute value of ab] < 1.

If we set a = [qe.sup.2iz], b = [qe.sup.-2iz] and q = [e.sup.[pi]i[tau]], where z is complex and Im [tau] [greater than or equal to] 0, then f(a,b) = [[??].sub.3](z, [tau]), where [[??].sub.3](z, [tau]) denotes one of the classical theta-functions in its standrd notation[12]. Following Ramanujan, we define

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

and

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

where

[(a; q).sub.[infinity]] := [[infinity].[PI] over (n = 0)](1 - [aq.sup.n]), [absolute value of q] < 1.

Note that f(-q) := [q.sup.-1/24][eta](z), where q = [e.sup.2[pi]i[tau]] and [eta] denotes the Dedekind eta-function.

Let [(a).sub.k] := a(a + 1)(a + 2) ... (a + k - 1) and define the ordinary hypergeometric function.

[sub.2][F.sub.1](a, b; c; z) := [[infinity].summation over (k = 0)] [(a).sub.k][(b).sub.k]/[(c).sub.k]k! ([z.sup.k]), [absolute value of z] < 1.

Suppose that

[sub.2][F.sub.1](1/2, 1/2; 1;1 - [beta])/[sub.2][F.sub.1](1/2, 1/2; 1; [beta]) = n [sub.2][F.sub.1](1/2, 1/2; 1;1 - [alpha])/([sub.2][F.sub.1(1/2, 1/2; 1; [alpha]), (2.1)

for some positive integer n. A relation between the moduli [square root of [alpha]] and [square root of [beta]] induced by (2.1) is called as modular equation of degree n and [beta] is said to have degree n over [alpha]. Let [z.sub.1] := [sub.2][F.sub.1](1/2, 1/2; 1; [alpha]) and [z.sub.n] := [sub.2][F.sub.1](1/2, 1/2; 1; [beta]), where n is the degree of the modular equation. Lastly, the multiplier is defined by

m = [z.sub.1]/[z.sub.n].

3. Main Theorems 

Theorem 3.1. Let

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Then

[(PQ).sup.2] + 9/[(PQ)].sup.2] = [(Q/P).sup.3] +5 [(Q/P).sup.2] +5 [(Q/P) -5 (P/Q) +5 [(P/Q).sup.2] - [(P/Q).sup.3]. (3.1)

Proof. From Entry 10(i) of Chapter 17 [3, p. 122], we have

PQ = [([z.sub.1] [z.sub.5])/[z.sub.3] [z.sub.15]).sup.1/2] (3.2)

and

P/Q = [([z.sub.1] [z.sub.15]/[z.sub.3][z.sub.5]).sup.1/2]. (3.3)

Let

([z.sub.3][z.sub.5]/[z.sub.1][z.sub.15]).sup.1/2] = t = [(m'/m).sup.1/2], (3.4)

where m' = [z.sub.5]/[z.sub.15]. From (3.2), (3.3) and (3.4), it follows that

[(PQ).sup.2] = [(mt).sup.2]

and

P/Q = 1/t. (3.5)

From equation 11.17 of Chapter 20 [3, p. 385], we have

([mt.sup.2]) + 9/[(mt).sup.2] = [[t.sup.6] + [5t.sup.5] + [5t.sup.4] - [5t.sup.2] + 5t - 1] / [[t.sup.3]]. (3.6)

Using (3.5) in(3.6), we readily deduce (3.1), to complete the proof of the theorem.

It is worthwhile to note that Theorem 3.1 can be used to evaluate the Ramanujan's cubic continued fraction.

Theorem 3.2. Let

P = f(-q)/[q.sup.1/24]f(-[q.sup.2])

and

Q = f(-[q.sup.7])/[q.sup.7/24]f(-[q.sup.14]).

Then

[(PQ).sup.3] + 8 [(PQ).sup.3] + 7 = [(Q/P).sup.4] + [(P/Q).sup.4]. (3.7)

Proof. Let

R = f(q)/[q.sup.1/24]f[-q.sup.2])

and

S = f([q.sup.7])/[q.sup.7/24]f(-[q.sup.14]),

and [beta] be of degree 7 over [alpha]. By Entry 12(i) and (iii) in Chapter 17 [3, p. 124], we deduce that

R = [2.sup.1/6][([alpha](1 - [alpha])).sup.-1/24]

and

S = [2.sup.1/6][([beta](1 - [beta])).sup.-1/24],

which implies

1/2[square root of 2] [(16[alpha][beta](1 - [alpha])(1 - [beta])).sup.1/8] = [(RS).sup.-3] (3.8)

and

[[([beta] (1 - [beta])/[alpha](1 - [alpha])].sup.1/6] = [(R/S).sup.4]. (3.9)

From Entry 19 of Chapter 19 [3, p. 315], we have if

M = [(16[alpha][beta](1 - [alpha])(1 - [beta])).sup.1/8]

and

N = [([beta] (1 - [beta])/[alpha](1 - [alpha]))].sup.1/6],

then

N + 1/N + 7 = 2[square root of 2](M + 1/M). (3.10)

From (3.8), (3.9) and (3.10), we arrive at

[(R/S).sup.4] + [(S/R).sup.4] + 7 = 8/[(RS).sup.3] + [(RS).sup.3].

Replacing q with -q, we see that [(RS).sup.3] transforms to -[(PQ).sup.3] and [(R/S).sup.4] transforms to [(-P/Q).sup.4]. Thus we obtain

[(PQ).sup.3] + 8/[(PQ).sup.3] + 7 = [Q/P.sup.4] + [(P/Q).sup.4].

This comletes the proof.

Theorem 3.3. If

P = f[-q.sup.2])/[q.sup.5/24]f(-[q.sup.7])

and

Q = f(-q)/[q.sup.13/24]f(-[q.sup.14]),

then

[(PQ).sup.4] + [7.sup.4]/[(P/Q).sup.4]

= [(Q/P).sup.9] + [(2P/Q).sup.9] +5 [Q/P.sup.6] +5 (P/Q).sup.6] -8 (Q/P).sup.3] + [(2P/Q).sup.3] +18. (3.11)

Proof. We set

R := f(-q)/[q.sup.1/24]f([-q.sup.2]),

S := f(-[q.sup.7])/[q.sup.7/24]f([-q.sup.14]),

M := [f.sup.2](-q)/[q.sup.1/2][f.sup.2](-[q.sup.7])

and

N := [f.sup.2](-[q.sup.2])/q[f.sup.2](-[q.sup.14]). (3.12)

Thus,

N/M = [S.sup.2]/[R.sup.2], MN = PQ, RS = Q/P. (3.13)

Employing (3.12) in Entry 55 of Chapter 25 [4, p. 209], we obtain

MN + 49/MN = [(N/M).sup.3] + [(M/N).sup.3] -8 (N/M) -8 (N/M). (3.14)

Squaring both sides of (3.14) and then using (3.13), we obtain

[(MN).sup.2] + [7.sup.4]/[(MN).sup.2]

= [(S/R).sup.12] + [(R/S).sup.12] -16 [(S/R).sup.8] -16 [(R/S).sub.8] +48 [(S/R).sup.4] +48 [(R/S).sup.4] +32. (3.15)

Employing (3.12) in (3.7), we have

[(S/R).sup.4] + [(R/S.sup.4] = [(RS).sup.3] + 8/[(RS).sup.3] +7. (3.16)

Squaring both sides of (3.16) and then simplifying, we arrive at

[(S/R).sup.8] + [(R/S).sup.8] = [(RS).sup.6] + 64/[(RS).sup.6] + 14 [(RS).sup.3] + 112/[(RS).sup.3] + 63. (3.17)

Multiplying (3.16) and (3.17) and then simplifying, we obtain

[(S/R).sup.12] + [(RS).sup.12]

= [(RS).sup.9] + 512 / [(RS).sup.9] + 21[(RS).sup.6] + 1344/[(RS).sup.6] + 168[(RS).sup.3] + 1344 / [(RS).sup.3] + 658. (3.18)

Now using (3.16), (3.17)and (3.18) in (3.15) and on applying (3.13), we deduce(3.11).

Thus we complete the proof of the theorem.

Theorem 3.4.

i) If

P = f(-q)/[q.sup.1/24]f(-[q.sup.2])

and

Q = f([-q.sup.3])/[q.sup.1/8]f(-[q.sup.6]).

then

[(PQ).sup.3] + 8/[(PQ).sup.3] = [(Q/P).sup.6)] - [(P/Q).sup.6]. (3.19)

ii) If

P = f(-q)/[q.sup.1/24]f(-[q.sup.2])

and

Q = f(-[q.sup.5])/[q.sup.5/24]f(-[q.sup.10]),

then

[(PQ).sup.2] + 4/[(PQ).sup.2] = [(Q/P).sup.3] - [(P/Q).sup.3]. (3.20)

iii) If

P = f(-[q.sup.2])f([-q.sup.3])/[q.sup.1/12]f(-q)f(-[q.sup.6])

and

Q = f(-[q.sup.10])f(-[q.sup.15]))/[q.sup.5/12]f(-[q.sup.5])f(-[q.sup.30]),

then

[(PQ).sup.2] + 1/[(PQ).sup.2] = [(Q/P).sup.3] + [(P/Q).sup.3] + 5. (3.21)

iv) If

P = f(-q)f(-[q.sup.2])/[q.sup.1/4]f(-[q.sup.3])f(-[q.sup.6])

and

Q = f(-[q.sup.5])f(-[q.sup.10]))/([q.sup.5/4]f(-[q.sup.15])f(-[q.sup.30]),

then

[(PQ).sup.2] + 81/[(PQ).sup.2]

= [(Q/P).sup.3] + [(P/Q).sup.3] -5 [(Q/P).sup.2] -5 [(P/Q).sup.2] - 5 (Q/P) -5 (P/Q) +20. (3.22)

v) If

P = f(-q)f(-[q.sup.6])/[q.sup.3/4]f(-[q.sup.10])f(-[q.sup.15])

and

Q = f(-[q.sup.2])f(-[q.sup.3]))/[q.sup.5/4]f(-[q.sup.5])f(-[q.sup.30]),

then

(PQ) + 25/(PQ) = [(Q/P).sup.4] + [(P/Q).sup.4] -5 [(Q/P).sup.2] -5 [(P/Q).sup.2] -2. (3.23)

vi) If

P = f(-q)f(-[q.sup.6])/[q.sup.7/6]f(-[q.sup.5])f(-[q.sup.30])

and

Q = f(-[q.sup.2])f(-[q.sup.3]))/[q.sup.5/6]f(-[q.sup.10])f(-[q.sup.15]),

then

(PQ) + 25/(PQ) = [(Q/P).sup.6] + [(P/Q).sup.6] +5 [(Q/P).sup.3] +5 [(P/Q).sup.3] -2. (3.24)

vii) If

P = f(-q)/[q.sup.1]/12f(-[q.sup.3])

and

Q = f(-[q.sup.7])/[q.sup.7/12]f(-[q.sup.21]),

then

[(PQ).sup.3] + 27/[(PQ).sup.3] = [(Q/P).sup.4] - [(P/Q).sup.4] -7 (Q/P).sup.2] + 7[(P/Q).sup.2]. (3.25)

Proof. As the pattern of proof of this theorem is identical with our proof of Theorem 3.3, we just give only the references of the required modular equations.

To prove (3.19), we employ the modular equations Entry 51 [4, p. 204] and Entry 52 [4, p. 205].

To prove (3.20), we employ the modular equations Entry 53 [4, p. 206] and Entry 54 [4, p. 207].

To prove (3.21x), we employ the modular equations Entry 59 [4, p. 214] and Entry 61 [4, p. 218].

To prove (3.22), we employ the modular equations Entry 60 [4, p. 215] and Entry 61 [4, p. 218].

To prove (3.23), we employ the modular equations Entry 61 [4, p. 218] and Entry 65 [4, p. 230].

To prove (3.24), we employ the modular equations Entry 59 [4, p. 214] and Entry 65 [4, p. 230].

To prove (3.25), we employ the modular equations Entry 68 [4, p. 336] and Entry 70 [4, p. 336].

We note that Theorem 3.2 and Theorem 3.4(i), (ii) can be employed to find values of the Ramanjan-Weber class invariants.

Acknowledgements

The authors would like to thank Dr. C. Adiga (University of Mysore, Mysore) and Prof. Kiran Reddy (Acharya Institute, Bangalore) for their constant encouragement.

Received July 23, 2003, Accepted March 8, 2004.

References

[1] C. Adiga, B. C. Berndt, S. Bhargava and G. N. Watson, Chapter 16 of Ramanujan's Second Notebook: Theta-functions and q-series, Mem. Amer. Math. Soc. No. 315, 53 (1985).

[2] N. D. Baruah, On some of Ramanujan's identities for eta-functions, Indian J. Math. 42 (2000), 253-266.

[3] B. C. Berndt, Ramanujan's Notebooks, Part III, Springer-Verlag, New York, 1991.

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

[5] B. C. Berndt, Ramanujan's Notebooks, Part V, Springer-Verlag, New York, 1998.

[6] B. C. Berndt, Modular equations in Ramanujan's lost notebook, Number Theory, Hindustan Book Company, Delhi, 1999, 55-74.

[7] B. C. Berndt and L. -C. Zhang, Ramanujan's identities for eta-functions, Math. Ann. 292 (1992), 561-573.

[8] H. S. Madhusudhan, M. S. Mahadeva Naika, and K. R. Vasuki, On some Ramanujan's P-Q identities, Hardy-Ramanujan J. 24 (2001), 2-9.

[9] M. S. Mahadeva Naika, P-Q eta-function identites and computation of Ramanujan-Weber class invariants (Preprint).

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

[11] S. Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa, New Delhi, 1988.

[12] E. T. Whittaker and G. N. Watson, A Course of Modern Analysis, Fourth Ed., Cambridge University Press, Cambridge, 1966.

K. R. Vasuki ([dagger]) and T. G. Sreeramamurthy

Department of Mathematics, Acharya Institute of Technology,

Sodevanahalli, Chikkabanavara Post Office,

Hesaragatta Road, Bangalore 560090, India

* 2000 Mathematics Subject Classification. Primary 33D90, 11A55; Secondary 11F20.

([dagger]) E-Mail: vasuki [e-mail]