Theta-Functions and $q$-Series

Recent Titles in This Series 520 Tom Ilmanen, Elliptic regularization and partial regularity for motion by mean curvature, 1994 519 William M. McGovern, Completely prime maximal ideals and quantization, 1994 518 Rene A. Carmona and S. A. Molchanov, Parabolic Anderson problem and intermittency, 1994 517 Takashi Shioya, Behavior of distant maximal geodesies in finitely connected complete 2-dimensional Riemannian manifolds, 1994 516 Kevin W. J. Kadell, A proof of the g-Macdonald-Morris conjecture for BCn, 1994 515 Krzysztof Ciesielski, Lee Larson, and Krzysztof Ostaszewski, Z-density continuous functions, 1994 514 Anthony A. Iarrobino, Associated graded algebra of a Gorenstein Artin algebra, 1994 513 Jaume Llibre and Ana Nunes, Separatrix surfaces and invariant manifolds of a class of integrable Hamiltonian systems and their perturbations, 1994 512 Maria R. Gonzalez-Dorrego, (16,6) configurations and geometry of Kummer surfaces in P 3 , 1994 511 Monique Sable-Tougeron, Ondes de gradients multidimensionnelles, 1993 510 Gennady Bachman, On the coefficients of cyclotomic polynomials, 1993 509 Ralph Howard, The kinematic formula in Riemannian homogeneous spaces, 1993 508 Kunio Murasugi and Jozef H. Przytycki, An index of a graph with applications to knot theory, 1993 507 Cristiano Husu, Extensions of the Jacobi identity for vertex operators, and standard ^^-modules, 1993 506 Marc A. Rieffel, Deformation quantization for actions of Rd, 1993 505 Stephen S.-T. Yau and Yung Yu, Gorenstein quotient singularities in dimension three, 1993 504 Anthony V. Phillips and David A. Stone, A topological Chern-Weil theory, 1993 503 Michael Makkai, Duality and definability in first order logic, 1993 502 Eriko Hironaka, Abelian coverings of the complex projective plane branched along configurations of real lines, 1993 501 E. N. Dancer, Weakly nonlinear Dirichlet problems on long or thin domains, 1993 500 David Soudry, Rankin-Selberg convolutions for SC^+i x GL„: Local theory, 1993 499 Karl-Hermann Neeb, Invariant subsemigroups of Lie groups, 1993 498 J. Nikiel, H. M. Tuncali, and E. D. Tymchatyn, Continuous images of arcs and inverse limit methods, 1993 497 John Roe, Coarse cohomology and index theory on complete Riemannian manifolds, 1993 496 Stanley O. Kochman, Symplectic cobordism and the computation of stable stems, 1993 495 Min Ji and Guang Yin Wang, Minimal surfaces in Riemannian manifolds, 1993 494 Igor B. Frenkel, Yi-Zhi Huang, and James Lepowsky, On axiomatic approaches to vertex operator algebras and modules, 1993 493 Nigel J. Kalton, Lattice structures on Banach spaces, 1993 492 Theodore G. Faticoni, Categories of modules over endomorphism rings, 1993 491 Tom Farrell and Lowell Jones, Markov cell structures near a hyperbolic set, 1993 490 Melvin Hochster and Craig Huneke, Phantom homology, 1993 489 Jean-Pierre Gabardo, Extension of positive-definite distributions and maximum entropy, 1993 488 Chris Jantzen, Degenerate principal series for symplectic groups, 1993 {Continued in the hack of this publication)

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MEMOIRS -LlT-1"

of the

American Mathematical Society Number 315

Chapter 16 of Ramanujan's second notebook: Theta-functions and q-series C. Adiga B. C. Berndt S. Bhargava G. N. Watson

January 1985 • Volume 53 • Number 315 (second of 5 numbers) • ISSN 0065-9266

American Mathematical Society Providence, Rhode Island

1980 Mathematics Subject Classifications:

33A30, 33A25.

Library of Congress Cataloging in Publication Data Main entry under title: Chapter 16 of Ramanujan's second notebook. (Memoirs of the American Mathematical Society, ISSN 0065-9266; 315 (Jan. 1985)) Bibliography: p. 1. Functions, Theta. 2. Series, Infinite. I. Adiga, Chandrashekar, 1957II. Title: Chapter sixteen of Ramanujan's second notebook. III. Series: Memoirs of the American Mathematical Society; no. 315. QA3.A57 no. 315 [QA345] 510s[515.9'84] 84-24283 ISBN 0-8218-2316-7

Memoirs of the American Mathematical Society

This journal is devoted entirely to research in pure and applied mathematics. Subscription information. The 1994 subscription begins with Number 512 and consists of six mailings, each containing one or more numbers. Subscription prices for 1994 are $353 list, $282 institutional member. A late charge of 10% of the subscription price will be imposed on orders received from nonmembers after January 1 of the subscription year. Subscribers outside the United States and India must pay a postage surcharge of $25; subscribers in India must pay a postage surcharge of $43. Expedited delivery to destinations in North America $30; elsewhere $92. Each number may be ordered separately, please specify number when ordering an individual number. For prices and titles of recently released numbers, see the New Publications sections of the Notices of the American Mathematical Society. Back number information. For back issues see the AMS Catalog ofPublications. Subscriptions and orders should be addressed to the American Mathematical Society, P. O. Box 5904, Boston, MA 02206-5904. All orders must be accompanied by payment. Other correspondence should be addressed to Box 6248, Providence, RI 02940-6248. Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy a chapter for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgement of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication. (including abstracts) is permitted only under license from the American Mathematical Society. Requests for such permission should be addressed to the Manager of Editorial Services, American Mathematical Society, P. O. Box 6248, Providence, RI 02940-6248. Requests can also be made by e-mail to reprint-permissionCmath. ams. org. The owner consents to copying beyond that permitted by Sections 107 or 108 of the U.S. Copyright Law, provided that a fee of $1.00 plus $.25 per page for each copy be paid directly to the Copyright Clearance Center, Inc., 222 Rosewood Dr., Danvers, MA 01923. When paying this fee please use the code 0065-9266/94 to refer to this publication. This consent does not extend to other kinds of copying, such as copying for general distribution, for advertising or promotion purposes, for creating new collective works, or for resale. Memoirs of the American Mathematical Society is published bimonthly (each volume consisting usually of more than one number) by the American Mathematical Society at 201 Charles Street, Providence, RI 02904-2213. Second-class postage paid at Providence, Rhode Island. Postmaster Send address changes to Memoirs, American Mathematical Society, P. O. Box 6248, Providence, RI 02940-6248. © Copyright 1985. American Mathematical Society. All rights reserved. Printed in the United States of America. This volume was printed directly from author-prepared copy. @ The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. »= ( a ; q ) a D = T T (1 Ramanujan writes n c series

,k

^7k

xk

Tq)

where < 1 and a-,,a2,...,a ,, b-,,b2,...,b are arbitrary, except that, of course, (b. k i 0, 1

(a),.

~

k

(b/a)k(c/a)k

TbcTd); k J 0 Td/a)k(q)k

where we have applied (6.1).

a

k

'

This completes the proof of Entry 7.

An important application of Entry 7 will be made in section 38. We now prove a lemma from which Entries 8 and 9 will follow as limiting cases. LEMMA.

For

|de/abc|, |e/a|, |q| < 1, de

(8.1)

3d/b,d/c;d,de/bc;e/a).

Using Entry 2 and ( 6 . 1 ) , we f i n d t h a t , f o r

|a|, |e/a|,

|de/abc| < 1 , kk

(a)k(b)k(Ok «>k 1. Let a« a. aol A = _3 4 _j?k M x 3 + T +...+ 1

and a a

4 5 x 5 + a5 + a6 " x 7

a a

+

6 7 a + a 7 8

a x

2k-l

+

2k-2 a 2k-l a 2k-l + a 2k

CHAPTER 16 OF RAMANUJAN'S SECOND NOTEBOOK

11

Then by induction, a a -L ^l -!l ^4 2k _ ^1_ 2 X-, + 1 + x 3 + 1 +•••+ 1 X-, + 1 + A d

d

l

X-j +

a

2 3-3

1 + x 3 +~ a 4 - Bo

l ^9^Q L 6

Ax,

l + a 2a.

x3 + a3 + a4 - B

which completes the proof. LEMMA 2. Let y = n 2 - (x + I ) 2 and t = (2x + 1 ) U 2 - m 2 ) . Then (10 2)

~ 2£mx 2 2(1 y + t + 2xm +

2(2 + x)(22 - I2) 1

+ x)(]2

1

"

2(1

i2)

+

" x ) ( ]2 " 3y + t

2(2 - x)(22 - m 2 ) + 5y + t

m2)

2(k + x)(k2 - I2)

2£mx 4(x 2 -l 2 )(£ 2 -l 2 )(m 2 -l 2 ) 4(x 2 - 2 2 )(£ 2 - 22)(m2 - 2 2 ) x 2 + £ 2 + m 2 - n 2 - 1 + 3(x 2 + £ 2 + m 2 - n 2 - 5) + 5(x2 + I2 + m 2 - n 2 - 13) 4(x2 - (k - 1)2 )(£2 - (k - l)2)(m2 - (k - I) 2 ) +-.-+ (2k - l)(x2 + £ 2 + m 2 - n 2 - (2k2 - 2k + 1)) PROOF. We shall apply Lemma 1 to the left side of (10.2). With the notation being clear, we find, after elementary calculations, that 2 2 2 2 x1 + a 2 = -(x + jT + m - n - 1) and, for j >. "U x

2j+l + a2j+l + a2j+2 = (2j + l ) y + t + 2(j - x)(j 2 - m 2 ) + 2(j + 1 + x)((j + I ) 2 - £ 2 ) = -(2j + l)(x2 + I2 + m 2 - n 2 - 2 j 2 - 2j - 1). Thus, by Lemma 1, we find that the left side of (10.2) is equal to -2£mx 4(l 2 -x 2 )(l 2 -£ 2 )(l 2 -m 2 ) 4(2 2 - x 2) (2 2 - I2) (2 2 - m 2 ) 2 2 2 2 2 2 2 2 -(x + £ + m - n - 1) - -3(x + £ + m - n - 5) - -5(x2 + £ 2 + m 2 - n 2 - 13) 4((k - l ) 2 - x2)((k - I ) 2 - £2)((k - l ) 2 - m 2 ) -• (2k - l)(x2 + I2 + m 2 - n 2 - 2k2 + 2k - 1) which is equivalent to the continued fraction on the right side of (10.2). ENTRY 10. Let x, £, m, and Defi ne

n be real with y = n 2 - (x + 1) > 0.

12

C. ADIGA, B. C. BERNDT, S. BHARGAVA, AND G. N. WATSON

p = r(^(x + Jl-m + n + l))r(%U + il-m-n^l))r(^(x~Jl + m + n + l ) ) r ( ^ U - ^ + m-n-t-l)) r ( ^ ( x - £ - m + n + l))r(%(x-Jl-m-n + l))r(%(x + jl + m + n + l))r(^(x + il + m - n + l))* Then 1 - P=

4(x 2 - 1 2 ) U 2 - l 2 ) ( m 2 - I 2 )

2^nx

1 + P

Z

x

Z

+ l

2

+ m - n

2

- 1 + 3(x 2 + ^ 2 + m2 - n 2 - 5)

., 4(x 2 - 2 2 ) ( £ 2 - 2 2 )(m 2 - 2 2 ) 5(x 2 + I1 + m2 - n 2 - 13) PROOF.

+ •••

We shall apply Entry 40 from Chapter 12 in Ramanujan's second note-

book [60, vol. 2, pp. 151, 152], [24], which has been proved by Watson [76]. Let R = nr(^(a ± B ± Y ± 6 ± £ +

D),

where the product contains 8 gamma functions and where the argument of each gamma function contains an even number of minus signs. Let Q = nr(^(a ± 3 ± y ± 6 ± £

+ 1)),

where the product contains 8 gamma functions and where the argument of each gamma function contains an odd number of minus signs. one of the parameters nn -\\ UU,J;

1

- Q/R 1 + Q/R " x

Suppose that at least

B, y, 5, e is equal to a nonzero integer.

Then [76]

8agy6£ 4 ^ A ^ 4 ^ .4 ^ 4 ^ , x , 2 ^ 2 , 2 .2 , 2 , , 2 1{2 (a + 3 + Y + 6 + £ + l ) - ( a + (D 3 + y + 6 + £ - l )

2 -9 2T}

UQ/

64(a 2 - 1 2 )(3 2 - 1 2 ) ( Y 2 - 1 2 )(6 2 - l 2 ) ( e 2 - 12)_ +

3{2(a

4

+

3 4 + Y 4 + 64 + e 4 + l )

- (a 2 + 3 2 + yZ

+

62 + £2 - 5 ) 2 - 6 2}

64(a2 - 2 2 )(B 2 - 2 2 ) ( Y 2 - 2 2 )(5 2 - 2 2 )(e 2 - 2 2 ) + 5(2(a 4 + 3 4 + Y 4 + 6 4 + £ 4 + 1) - (a2 + 3 2 + Y 2 + S 2 + £ 2 - 13) 2 - 14 2 } +••• In (10.3), let a = x, $ = n - e, y = £, and 6 = m, where tive integer.

In the quotient

are independent of e ent of £

e

is a posi-

Q/R of 16 gamma functions, we observe that

and 8 depend upon

8

£. The quotient that is independ-

is precisely equal to P, while the quotient that depends upon

£

is eciual to r(%(x-£+m+n-2£+l) )r(Û-£+m-n+2£+l)) T(%(x+£-m-n+2£+l)) r(%(x+£-m+n-2£+l)) ' By Stirling's formula, the quotient above tends to Hence,

1 as

e

tends to °°.

CHAPTER 16 OF RAMANUJAN'S SECOND NOTEBOOK do.4)

l i r a

|^R

=

^ P

13

.

We next examine the r i g h t side of (10.3) as

e tends t o

~.

An elemen-

tary calculation shows that 2 ( x 4 + ( n - e ) 4 + £ 4 + m4 + e 4 + l ) - (x 2 + (n - e ) 2 + £ 2 + m2 + e 2 - ( 2 j 2 + 2j + I ) ) 2 = 4(n 2 - x 2 - I2 as

e

tends t o

°°,

- m2 + 2 j 2 + 2j + l ) e 2 + 0 ( e ) ,

where

0 0,

then the

(10.2) are positive i f

jth j

numerator

Observe that i f and denominator

i s s u f f i c i e n t l y large.

Perron's text [53, p. 47, Satz 2 . 1 1 ] , i f we l e t continued f r a c t i o n on the l e f t side converges.

x, £, m, and

n

are real

on the l e f t side of

Hence, by a theorem in k

tend t o

»

in (10.2), the

Therefore, (10.5) converges,

and the proof i s complete. The following beautiful theorem i s a q-analogue of Entry 33 i n Chapter 12 [60, p. 149], [ 2 4 ] .

14

C. ADIGA, B. C. BERNDT, S. BHARGAVA, AND G. N. WATSON ENTRY 11. If

"a)JbL (-a) lb)

(a

|q|, |a| < 1, then

U-b)oo

+ (a)J-b)

FIRST PROOF.

- b

= a

q(a - bq2)(aq2 - b)

(a - bq)(aq - b)

1 - q +

1 -q

J

1 - q5

+

We shall employ Heine's [41] continued f r a c t i o n f o r a quo-

t i e n t of two contiguous basic hypergeometric s e r i e s, namely, f o r 2

]

I t follows from (11.1) that (11.2)

2 3 2 2 . h oî(bq/a,bq /a;q ;q ;a ) a - D* i q 2_ 0,

(0 (11

-5)

Vl,n-Bm+l,n

m

[I

for

m= 0

b „m+l w m+1

' aq ^q

bXD

)(q

2m+1

)(l-q

may be v e r i f i e d d i r e c t l y .

2m+2

-aVl , 3

2

)(l.q ^ )

if

n > 0.

Thus, assume that

m _> 1 .

First,

n = 0, ^

^L

m

2k,

ai, )ir d-q") m+1,0 " m+1,0 = (1 - q k=l

A

For

n = 0,

=

q ( 1n

The case

if

B

M

-ITd k=l

2k

-q' K )N = 0.

n > 0,

A - B m+1 ,n m+1 ,n (1 -

q

2m

)

(bq m/ /a) VUH a / 9n

m-1 2 n ^

(q

J^r) iq

J2n+1

2m

)2n

© K

_2n+2kN

M

(bqm+

k=l

(q"

q

A

'

{1

-^

q

,,rt

m

a

/a)

)

q

k=l

.

q 2iw2n+lj

(1 - | q 2 n « ) ( 1 - q21""1)}

2m+l W 1 '' "

x M (1 - q

T T (i - q2n+2k)

)'2n o.

( 1

/T b m + l w u m+2, x ( l - - q )(bq /a)2n.2 / 2m+4^ Ti (q ^2n-2 ~

2n —

q

2m+2 w i 2m+3* '' ~q

( f• r•, (i - q2n+2k)} lf

2 n w m+1 b\ m )(q - -)q . a

Equality (11.5) now readily f o l l o w s . Now multipl y both sides of (11.5) by

2n a*1"

and sum on

deduce that ni/

.

HH-IW

m+1

, ,

N -, - D , = q (a : bq )(aq - b) "m+1 m+1 ,-, (1 - q -7^+t\ ^ + l ) ( 1 _ q 2m+Z )(1 _

aw-3) um+2'

n, 0 ,

to


17

or (1

N .i _ ZnH-ljf"m+1 D

m, - bq"'"')(aq . m+1 x,^ m+1- b J \= q'"(a

m+1

2m+2 , (1 -q )D , H 2m+3v m+1 (1 - q ) D m+2

>

9

71

or

m+1

where

n (1

"q

2m+3x m+2 'D m+2

m ^L 0. This is the desired recursion formula for N./D. .. Now return to (11.4) and, beginning with

m = 0, apply (11.6) succes-

sively to find that

(-a)oo (b)oo + (a)J-b) oo oo a - b

q(a - bq 2 )(aq 2 - b)

(a - bq)(aq - b)

1 - q +

1-q

3

. +

q'"(a - bq1 "^') (aq , i r r ' - b ) . . /-, 2m+3x Nm+2 +•••+ (1 - q D mf2

, 5 1 - q

From the definition of Hm and D , we observe that m m as

m tends t o

«>.

Hence, l e t t i n g

m tend to

°°

N / D m approaches 1 m m rr

above, we complete the

proof. ENTRY 12.

For

/ 2 3 4, (,Z 3 4x (a q ;g ) J b q ;q ) (a2q;q4)Jb2q;q4)oo

a

|aq/b|, |bq/a|, |q| < 1,

_

1

(a - bq)(b - aq)

o Q (a - bq 3 )(b - aq3 )

1 - ab + (1 - ab)(q 2 + 1) + (1 - ab)(q 4 + 1)

+•••"

We are very grateful t o R. A. Askey and D. M. Bressoud f o r helpful suggestions in proving t h i s beautiful continued f r a c t i o n .

Entry 12 i s a q-ana-

logue of Entry 25 i n Chapter 12 [60 , v o l . 2, p. 147], [ 2 4 ] . lemmas w i l l be needed f o r our proof.

Four a n c i l l a ry

The f i r s t i s a q-analogue of Euler's

continued f r a c t i o n f o r a quotient of two contiguous ordinary hypergeometric series.

Euler's continued f r a c t i o n was rediscovered by Ramanujan and can be

found i n Chapter 12 of the second notebook [60, v o l . 2, p. 147], [24, Entry 22]. LEMMA 1 .

If

| c / a | , | b | , | q | , |x| < 1 ,

then

18

C. ADIGA, B . C . BERNDT, S. BHARGAVA, AND G. N. WATSON

(1 - c)

91(aq,b;bq

/a;-q/a)

(b;q).(q 2 /a;q).

-

(q k + 1 ;q),(-q k + 2 ;q),

( b q ^ q j j q ^ k=0 (q

(b;q)Jq 2 /a;q) m

(bqVa;q) (q;q) CO

'

' 'CO

-

k+2

/a;q) oo (-q

k=0 ( q "

^

/a;q) oo

(q 2 k 4 ;q Z ) 00

k+1

k+1

k

Va^L

(b;q).(q2/a;q),(q4;q2),

7

k+1

7 0 7

-

I

(q2/a2;q2)k

• a. 7

(bq7a;q)oo(q;q)oo(qVai:;q^)oo k=0 ( q W ) k

k+]

(1 - q

k+1

)(1 - q

k

/a)b

(b;g).(q2/a;q),(q2;q2), - (1/a2;q2)k ^ k k —2~;—~— 2—2— ^ — 2 — 2 — u - q ) U - q / a ) b

(bqVa-.q^tq-.q^d/a^q')^ k=0 (q';q') k

(b;q)co(q 2 /a;q)oo _

—.—

« £

0/a2;q2)k —? J] . (1 + ) q

b(bq Z /a;q) c o (q; q 2 ) o o (l/a Z ;q Z ) c o k=0 ( q 2 ; q \

I

a

2k,

+ ii_l b . J

Applying (2.1) three times on the r i g h t side above, we complete the proof


20

2 PROOF OF ENTRY 12. In Lemma 1, replace a, b, c, q, and x by a q, 2 2 aq/b, abq , q , and -qb/a, respectively. Observe that, for each nonnegative integer n, 1 - C q n + M (1 - m ^ l ) q a

and - M q

(1

_ cg! ){1 . b n, n a

are transformed, respectively, into (1 - ab)(q2n+2 + 1) and (a - bq2n+1)(b - a q 2 n + 1 ) . Hence, Lemma 1 yields

(12.1)

2 2 2 o / - . ( a q,aq/b;abq ;q ;-qb/a) 3 4 9 A = (1 - abq^) L ' 2 2^-jCa q,aq /b;abq ;q ;-qb/a ) = (1 - ab)(q 2 + 1) i (a - bq 3 )(b - aq3) (a - bq 5 )(b - aq5 ) (1 - ab)(q 4 + 1) + (1 - ab)(q 6 .+ 1) +•

After some elementary manipulation, we f i n d that 1 (a - bqMb L _ a q l + =__L_

1

_

ab

(a - bq 3 )(b - aq3)

(a - bq)(b - aq)

1 - ab + (1 - ab)(q

2

+ ! ) + ( ! -

ab)(q 4 + 1)

+•••'

Comparing t h i s with Entry 12, we discover that i t remains to show that

(12.2)

2

q ,

,23 4 W k 2 3 4N _ 1 M ;q J J b q ;q )o

1

Ta - bq)(b - aq) ^ -, 7 -^ ^ ^LL+ 1 - ab

Secondly, apply Lemma 3 with

=

~7~2 47772 47 (a q;q ) J b q;q )oo

a, b,

respectively, to deduce that

(12.3)

2

2 2 2 ^ ( a q,aq/b;abq ;q ;-qb/a)

( a 2 q V ) (b2q3;q4) (abq2;q2)J-qb/a;q2)

(q 2 ;q 4 )

and q

replaced by

2 a q , aq/b, and


and

21

T h i r d l y , we invoke Lemma 4 with a , b , and q replaced by aq/b, a q , 2 q , respectively , to f i n d that

(12,4)

2

^lâq

3

2

/ b , aq;abq

4 2 ;q

;

~qb/a)

(a2q;q2L(bq3/a;q2L a 2 q(abq 4 ;q 2 )

2

/q;q4)o

(q 2 ;q 4 ) c o (b 2 /a 2 q 2 ;q 4 ) o o 4 a 2 q ; q 4 )

(bW),, (b 2 q 3 ;q 4 ) b -7-1-1^ , 2 3 4 + aq H , 2 5 4x (a q ;q X ) (a q ;q )

b

- 0 +£) a Hq

r(b

;

(a 2 q;q 2 ) o o (bq 3 /a;q 2 ) o o (aq + b)

,(1 - ab) ( b 2 q 3 ; q 4 ) t t

T T 7 . 4 2~, 2 4^ , . 2 , 2 2 i \T~2 47 a q (abq ;q ) (q ;q ) (b /a q ;q ) ^ (a q;q )

(b 2 q;q 4 ) 0 , 2 3 4N (a q ;q )

Combining (12.3) and (12.4) and using the notation (12.1) , we deduce that

A =

a2q(a2q3;q4)Jb2q3;q4)Jb/aq;q2)oo (bq 3 /a;q 2 ) { ( 1 - ab)(a 2 q 3 ;q 4 ) (b 2 q 3 ;q 4 )

- (a 2 q;q 4 ) (b 2 q;q 4 ) }

Hence,

(a - bq)(b - aq)

+ ]

_

gb

(a - bg)(b - ag)(bq 3 /a;q 2 )J(l - a b ) ( a V ^ J b V ^ 4 ) . . - (a 2 q;q 4 Ub 2 g;q 4 )J a2q(a2q3;q4) (b2q3;q4) (b/aq;q2)

+ 1 - ab (a 2 g;q 4 ) c o (b 2 q;q 4 ) oo , 2 3 4> , . 2 3 4v (a q ;q ) (b q ;q )

~ ~ ~ ~

22


Thus, the proof of (12.2) i s completed, ENTRY 13. m

n

If

|q| < 1,

and so Entry 12 i s established as well.

then

V ( ^knk(k+1)/2 - ] ffl ^(q - q) "T+ i + — r ~ + k=o

PROOF.

aq i

a(q 4 - q ) + i +••••

Let

(13 2) f(b,a> = (aq)

-

^Jo^W-

Then i t i s easy t o v e r i f y that (13.3)

f ( b , a ) = f(b,aq) - aqf(bq,aq)

and (13.4)

f ( b , a ) = f(bq,a) + bqf(bq 2 ,aq).

From (13.3) and (13.4), f(bq,a) = f(bq 2 ,a) + bq 2 f(bq 3 ,aq) f(bq 2 ,aq) + (bq 2 - aq)f(bq 3 ,aq), Using (13.4) , the equality above, and i t e r a t i o n , we f i n d that (13.5)

l(b'a)

= 1 +

f(bq,a)

^

=

]

+

H ^ a ) _ f(bq\aq) 1

-j

1

+

M

!

bq

2

]

- aq

+

bq

f ( b q ,aq) f(bq J ,aq)

3

bq

+

i

4

2 - aq

+IibgW) f(bq^,aq^) 2 3 4 2 b£ bq - aq bq bq - aq 1 + 1 + 1 + 1 +...' +

!

bq 2 - aq

N

The i n f i n i t e continued f r a c t i o n above converges since f ( b q 2 n , a q n ) / f ( b q 2 n + 1 , a q n ) -- 1 + 0(q 2 n + 1 ) Now set f ( a , a ) = 1.

b = a

as

n

i n (13.5) and employ Entry 9.

tends t o

».

Observe t h a t , by Entry 9,

Taking the reciprocal of both sides of (13.5) we deduce (13.1).


23

A generalization of Entry 13 appears i n Ramanujan's " l o s t " notebook; see Andrews' paper [ 7 , eq. ( 1 . 4 ) ]

and Hirschhorn's paper [44] f o r

proofs.

Adiga

and Bhargava [ 1 ] have established, with a unified approach, some continued f r a c t i o n expansions in Ramanujan's " l o s t " notebook, including those mentioned above, as well as a related continued f r a c t i o n of Hirschhorn [ 4 3 ] .

The proof

of Entry 13 that we have given i s d i f f e r e n t , and perhaps simpler, than others given i n the l i t e r a t u r e .

L a s t l y, we remark that the special case

a = 1

of

Entry 13 i s due t o Eisenstein [ 3 2 ] . After s t a t i n g Entry 13, Ramanujan gives formulas f o r the denominator of the

nth

1,

and (13 7)

'

D

n I

=

2n+1

ak M

q

(n+Dk k - I T d

, . -qn+1"J),

n > 0.

To prove (13.6) and (13.7), we shall employ a f a m i l i a r recursion formula fo r p a r t i a l denominators [72, p. 15] along with induction on F i r s t , from ( 1 3 . 1 ) , i t i s obvious that

D-, = 1

and

are in agreement with (13.7) and ( 1 3 . 6 ) , respectively.

n. Dp = 1 + aq,

Proceeding by induction

and using the aforementioned recursion formula, we f i n d that D

2n = D2n-1

+

^""Sn-Z

n-1

k nk

k=0

[ql

k j=l

n

k nk

k

k^l ^

= 1+

n k=0 Replacing

„ .

0

k

„ .

k

„k nk k-1

j

9o

,

n

k

n

+

k

n

^k^T

k-1 j=l

(qn-k-qn)}Tf(l-qn-J) j=l

.

j=0

k by

j - 1,

we complete the proof of (13.6).

The proof of (13.7), which begins with the recursion formula D

2n+1 = D2n

+

^

" "n,D2n-V

.

j=l

k-1 ( n - 1 ) ( k - l )

k=l

I lV !l{(l-qn-k) [q,

Kq,

iq;

k=0

j'Jl

k=l

-, n-1 ,k ( n - l ) k

which

24


is yery similar to the proof of (13.6), and so we omit the details. ENTRY 14. (14.I!

Q

If

n < 1

0 < a < q1_n,

and

then

0.

Using (16.1), iteration, and the special cases pointed out above, we find that P. = ! = i +

V

F

i

a

,q

= i + aa

] +

W 1

o

+

1

+

1 +

*9h n

+

1

- * , + F'n- ,1/ F nn

2 " '

ao2

a

1 +•••+

n-1 1

+

H

which is the required r e s u l t . Entry 17 offers another famous discovery of Ramanujan known as "Ramanujan's summation of the

-^-j-"

I t was f i r s t brought before the mathematical world by

hardy [39, pp. 222, 223] who described i t as "a remarkable formula with many parameters."

Hardy did not supply a proof but indicated that a proof could be

constructed from the q-binomial theorem.

The f i r s t published proofs appear t o


27

be by W. Hahn [37] and M. Jackson [47] i n 1949 and 1950, respectively.

Other

proofs have been given by Andrews [ 3 ] , [ 4 ] , Andrews and Askey [ 1 1 ] , Askey [ 1 3 ] , and Ismail [ 4 5 ] . The short proof of Entry 17 that we o f f e r below has been motivated by Askey's paper [13] and appears to be new. We emphasize that Entry 17 i s an extremely useful r e s u l t , and several app l i c a t i o n s of i t w i l l be made i n the sequel. 14 and 17, see Askey's paper [ 1 3 ] ,

For a connection between Entries

Further applications of Entry 17 have been

made by Andrews [ 7 ] , Askey [ 1 4 ] , and Moak [ 5 2 ] . has been found by Andrews [ 8 , Theorem 6 ] .

A generalization of Entry 17

Entry 17 serves as a bridge connect-

ing the two primary topics of Chapter 16, q-series and theta-functions. ENTRY 17. Suppose that

(17.1)

1+

-

I

k=l

|3q| < |z| < l / | a q | .

(l/a;q2),(-aq)k

^-i

(3q ;q ; k

f(-qz;q 2 ) (-q/z;q 2 )

Then

(l/3;q2)k(-Bq)k

-

zK + I

k=l

"
3q2;q2)J

" l(-aqz;q 2 )J-3q/z;q 2 )Jl(aq 2 ;q 2 )j3q 2 ;q 2 )J " PROOF. Let f(z) denote the former expression in curly brackets on the right side of (17.1). Since f(z) is analytic in the annulus, |3q| < |z| < 1/|aq|, we may set f(z) =

I c,zk, k=-oo K

From the d e f i n i t i o n of

|3q| < |z| < l / | a q | . f,

i t i s easy to see that

(3 + qz)f(q 2 z) = (1 + a q z ) f ( z ) , Equating c o e f f i c i e n t s of

| 3/q| < | z | < l / | a q | .

k z , -°° < k < °°,

on both sides, we f i n d that

(3q'KCk + q^ K C ^ = Ck + otqc k _ r

(17.2) Hence, C. = k

n 2k-2, . aqU - q /a)c. 1 ~ ^ - , 1 - 6aZK

1 < k < »,

and c

(3q(l - q 2 k " 2 / 6 ) c _ k + 1

-k =

: 7k I - aq

> ! i

k

< ->

28

C. ADIGA, B . C . BERNDT, S. BHARGAVA, AND G. N. WATSON

where, to get the latter equality, we replaced

k by

1-k

in (17.2)

Iterating the last two equalities, we deduce that, respectively, (17.3)

ck =

(-aq) (l/a;q ),cn j—^ O- , (3q ;q ).

1 < k < oo,

and (-3q)k(l/3;q2),c c

-k

, 2 2^ laq ;q 'k

'

'-

K

'

Examining (17.1), we see that, to complete the proof, it suffices to show that (aq2;q2)

(17.4) cr

(Bq 2 ;q 2 )

~° " ( q 2 ; q 2 ) > 3 q 2 ; q 2 L

Now l e t

|Bq| ,

z = -1/aq,

lim (1 + aqz)f(z) = lim (1 + aqz)*(z) = lim z-^-l/aq z-^-l/aq n^°

by Abel's theorem.

Using t h e d e f i n i t i o n of

(l/cr.q2)^

(q 2 ;q 2 UaBq 2 ;q 2 ) 00

(BqV^

H)ncn (aq)

—-,

f(z) and ( 1 7 . 3 ) i n ( 1 7 . 5 ) , we

o b t a i n ( 1 7 . 4) and hence ( 1 7 . 1 ) i n t h e r e g i o n (l/a;q2)oo(aq2;q2)oo

and

we f i n d that

|B/q| < |z| < l / | a q | .

The proof of Entry 17 i s now completed by a n a l y t i c c o n t i n u a t i o n . Entry 17 can be reformulated in a more compact s e t t i n g . the d e f i n i t i o n of {c)

k

for every and

=

(c;

^k

integer

-az/q,

(^qj^ =

, k.

k

by defining

,

>

In Entry 17, now replace

respectively.

o

Lastly, replace

written in the form (a). . (az) (q/az) (q) (b/a) k kV V OO ' OO ^ OO OO

I TET* ^ k " ( z U b /az)Jb ) o o (q/a) c

k= _»

where

|b/a| < |z| < 1.

We f i r s t extend

q

a, B, and z by

2 1/a, b/q ,

by q. Then (17.1) can be

29

CHAPTER 16 OF RAMANUJAN'S SECOND NOTEBOOK COROLLARY.

If

|nq| < |z| < 1/|nq|, then

2

«> (l/n;q )k(-nq)k(zk + z'k)

1 + I

(^z;q 2 M-q/z;q 2 Mq 2 ;q 2 Mn 2 q 2 ;q 2 ) o

(nq 2 ;q 2) k

k=l PROOF.

(-nqz;q 2M-nq/z;q2Mnq2;q2 )f

Set a = 3 = n in Entry 17.

The remainder of Chapter 16 is devoted to the function (18.1)

f(a.b) = 1 +

I

(ab) k(k - 1)/2 (a k

+b

k

) =

[

k=l where

|ab| < 1.

is complex and

I f we set Im(x) > 0,

ak(k+l)/2bk(k-l>/2f

k=-°° a = qe then

1Z

, b = qe~

1Z

,

f(a,b) = # ( Z , T ) ,

and

q = eniT,

where

#3(Z,T)

where

z

denotes

one of the classical theta-functions in i t s standard notation [77, p. 464]. Thus, a l l of Ramanujan's theorems on #3(Z,T).

f(a,b)

may be reformulated in terms of

I t seems preferable, however, to retain Ramanujan's notation.

Not

only w i l l the reader f i n d i t easier to follow our presentation in conjunction with Ramanujan's, but Ramanujan's theorems are more simply and elegantly stated in his notation. ENTRY 18. (i)

We have

f(a,b) = f ( b , a ) , f ( l , a ) = 2f(a,a3),

(ii) (iii)

f ( - l , a ) = 0,

and, i f (iv)

n

i s an integer, =an(n+1)/2bn(n-1)/2f(a(ab)n,b(abrn).

f(a,b)

Ramanujan remarks that ( i v ) i s approximately true when teger.

n

is not an i n -

We have not been able to give a mathematically precise formulation of

t h i s statement. PROOF.

Repeated use of ( i v ) w i l l be made i n the sequel.

F i r s t , ( i ) i s obvious.

Secondly,

f(l.a) = 2 (

+

ak{k+1)/2+

f

k=l -

211 + I aa t k= k=l

k(k+l)/2

f

k=2

a"*"- 1 " 2

30

C. ADIGA, B. C. BERNDT, S. BHARGAVA, AND G. N. WATSON f

h1 +

/•

OO

I

^ k=l k even

1 + I a k=l = 2! 1 +

+

CO

I > ak(k+l)/2

k=l k odd

k(2k+l)

+

r ak(2k-l. k=l

K K u/ J K K+l y aa k(k-l)/2 ^ " ^ ((aa 3jk(k+l)/2 ) ^ ^^ k=l

+

y ak(k+l)/2(a3jk(k-l)/2" k=l

2 f(a,a 3x Thirdly, f(-l,a) =

I (-i)k(k + l)/2 a k(k-l)/2 + J k=2 k=l

upon the replacement of

Fourthly, replacing

k by

M ) k(k-l)/2 a k(k + l)/2 =

0f.

k + 1 in the first sum on the right side.

k by

k + n on the far right side of (18.1), we

find that

f(a,b) =

I

k=-

a

(k+n)(k+n+l)/2.(k+n)(k+n-l)/2

^(n+1)/2 b n(n-l)/2

= a n(n + l)/2 b n(n-l)/2

J a k(k+2n+l)/2 b k(k+2n-l)/2 k=-°°

£ {a(ab)n}k(k+1)/2{b(ab)-n}k(k-1>/2 , k=-°°

which completes the proof of (iv).

ENTRY 19. We have


31

f(a,b) = ( - a ^ b J J - ^ a b J J a ^ a b ) ^ . PROOF.

In Entry 17, let qz = a, q/z = b, and a = 3 = 0.

In the notebooks [60, vol. 2, p. 197], Ramanujan informs us how he proved Entry 19 by remarking that "This result can be got like XVI. 17 Cor. or as follows. We see from iv. that if a(ab) n or b(ab) be equal to -1 then f(a,b) = 0 and also if (ab) n = 1, f(a,b){l - (|)n/2> = 0 and hence f(a,b) = 0. Therefore (-a;ab) , (-b;ab) , and (ab;ab) are the factors of CO

CO

CO

f(a,b)." (We have slightly altered Ramanujan's notation.) The product in Entry 19 converges only when |ab| < 1, but there is even a more serious objection to Ramanujan's argument. It is not clear that the only factors of f(a,b) are (-a;ab) , (-b;ab) , and (ab;ab) . CO

CO

OO

Entry 19 i s Jacobi's famous t r i p l e product i d e n t i t y , f i r s t established in his Fundamenta Nova [ 4 8 ] .

See the texts of Andrews [ 6, pp. 2 1 , 22] and Hardy

and Wright [40, pp. 282, 283] for other proofs. 2 ENTRY 20. I f a3 = TT, Re(a ) > 0, and n

i s any complex number, then

^ f(e- a 2 + n a ,e- a 2 - n a ) = e n 2 / 4 / g f ( e - e 2 + i n 3 , e - e 2 - i n S ) . Entry 20 is a formulation of the classical transformation formula for the theta-function # 3 ( Z , T ) [77, p. 475]. This entry is also recorded in Chapter 14 [60, vol. 2, p. 169, Entry 7]. A proof via the Poisson summation formula is sketched in [21]. ENTRY 21. If |q|, |a|, |b| < 1, then (21.1)

Log(-a;q) ro -

V_ f ^ — f ? k=l k(I - q )

and (21.2)

Log f(a,b) = Log(ab;ab)

PROOF. Log(-a;q)

+ I -^ [a. + k=l k(l - a V )

For |q|, |a| < 1, = I Logd + aqn ) = I I {~]) n=0 n=0 k=l r (-1)k"1ak ^ K k=l n=0

f

k,n

,{^

]

K

^ (-l)k-]ak k=l k(l - q K )

b }

.

32

C. ADIGA, B. C. BERNDT, S. BHARGAVA, AND G. N. WATSON Equality (21.2) follows immediately from Entry 19 and (21.1). ENTRY 22.

(i)

If

| q | < 1,

*(q) = f(q,q) = 1 + 2

then

~

(WlJqV^ k2 X q = * j—^— , k=l (q;q^)J-q W L

(q2;q2) . . -. 3. r k(k+l)/2 ^ i|;(q) = f ( q , q ) = Z q = ? , k=0 (q;qX

,.M (n)

f(-q) E f(-q,-q2) = I

(iii)

(-1)kq

k (

^)/2

+

k=0

I

(-1)kqk{3k+1)/2

k=l

= (q;q>«,> and (iv)

X (q)

= (-q;q2)oo-

Observe that f ( - q ) = e"71

>

where we have used (22.2) again. Lastly, by Entry 22(iv),

x(q>x(-q) = (-q;q2)Jq;q2)co = ( q V L = x(-q2). ENTRY 25. We have (i) *(q) + *(-q) = 22(q) + / ( - q ) = 2^ 2 (q 2 ),

and (vii)

* 4 (q) " * 4 (-q) = 16q * 4 ( q 2 ) . PROOF OF (i). By Entry 22(i),

*(q) + *(-q) = 2 + 2

£ (qk + (-q)k ) k=l 2

oo

= 2+4 I

k=l

q 4 k = 2*>(q4).

PROOF OF (ii). By Entries 22(i) and (ii), *(q) - *(-q) = 4

I q(2k"1) k=l r

A

, 8,k(k-l)/2

. ,, 8*

k=1 PROOF OF ( i i i ) .

The f i r s t equality i s a ready consequence of Entry 22(i]

By Entries 22(i ) and ( i i ) and (22.2),

(qVU-qVlJ-qVL = (qV&qV). OO

4

'

'

CO

4X , 2 2X

On the other hand, by Entry 22(ii) and (22.2),


37

i 2 2,2 l(e~ ) = S e a / V e ~ 3 ). zva \p

aB = TT2, then e-a/12 4^-

f(.e-2a}

= e "S/12

e'a/244^

f(e"a) = e " 6 / 2 4 %

^

f(

.e-2B)f

f(e^),

a(3 = TT, then


40 and

e«/24x(e-a)=e6/24x(e-6).

(v)

PROOF OF ( i ) .

Set

n = 0

in Entry 20.

PROOF OF ( i i ) . Set n = a in Entry 20 and observe that 2 2 f ( l , e ~ 2 a ) = 2iMe~2a ) by Entries 1 8 ( i i ) and 2 2 ( i i ) . PROOF OF ( i i i ) .

By Entries 2 7 ( i ) , 2 7 ( i i ) , and 2 5 ( i i i ) ,

2a ^ e ^ M e " 2 " ) = 3ea

where

7

W

3

M-e"3 )

a3 = IT. Multiplying both sides by

ip(e~

p

2 )

and using Entry 2 4 ( i i ) ,

we f i n d that 2 2 2 2 2 o ~t ~^ \i/~~2a \ , / -23 \ oâ / 4 r 3 / -23 \ 2a 4{;ij (V-VL}(PV)2 = 7 ] - (-ap k=l

;p M-bp

;p M p ;p )m

= T f fUp^.bp""*). k=l COROLLARY.

We have

f(-q2,-q3)f(-q,-q4)

= f(-q)f(-q5)

and f(-q,-q6)f(-q2?-q5)f(-q3,-q4)

transformation

t h e t a - f u n c t i o n s , and many gen-

e r a l i z a t i o n s has been presented by Berndt [ 1 8 ] , [ 1 9 ] , ENTRY 28.

discussion

= f(-q)f2(-q7).

that

42

on."

C. ADIGA, B. C. BERNDT, S. BHARGAVA, AND G. N. WATSON After the last equality, Ramanujan [60, vol. 2, p. 199] remarks "and so By these words, he implies that

(28.1)

~\\- f(- q k ,-q 2 n + 1 - k ) = f t - q J f ^ t - q 2 0 * 1 ) , k=l

where n is any positive integer. The corollary records the cases n = 2, 3 of (28.1). We shall now establish (28.1). Employing Entries 19, l(iii), and 22(iii), we find that - r r ft n k n 2 n + 1 " k \ - r r Jrn k .n 2 n + 1 ^ / > i - k . M ^ , 2n+i 2n+i> \ II T(-q ,-q ) = II l(q ;q JJq ;q )Jq ;q lA } k=l k=l l »f(-q)fn-Vq2n+1>: ENTRY 29. (i)

If

ab = cd,

then

f ( a , b ) f ( c , d ) + f ( - a , - b ) f ( - c , - d ) = 2f(ac,bd)f(ad,be)

and (ii)

f(a,b)f(c,d) - f(-a,-b)f(-c,-d) = 2 a f ( | , | a b c d ) f ( | , | abed).

Many of the identities of Entry 25 above and Entry 30 below are instances of Entry 29. Formula (ii) was discussed by Hardy [39, p. 223] who also briefly sketched a proof. Since the proofs of (i) and (ii) are similar, we give only the proof of (i). Less elementary proofs of Entries 29(i), (ii) may be found in the treatise of Tannery and Molk [71]. PROOF OF (i). Letting *(* K \ ^ A\ f(a,b)f(c,d)

V l

p = ab = cd, we see that Jm p

+n

)/2-(m+n)/2 m n a c .

m,n=-°° Thus, settin g

m - n = 2j

and m + n = 2k,

we f i n d that


43

f(a,b)f(c,d) + f(-a,-b)f(-c,-d) = 2

= 2 = 2

y (m2+n2)/2-(m+n)/2amcn m5n=-*° m+n even I p^+k2-kaj+kck^ j5k=-00 I

pk(k-1)(ac)V(d+1)(bc)-j

= 2f(ac,bd)f(ad,bc). Several of the identities of Entry 25 are special cases of the formulas in Entry 30. ENTRY 30. We have (i) f(a,ab2)f(b,a2b) = f(a,b)iKab), (ii) f(a,b) + f(-a,-b) = 2f(a 3 b,ab 3 ), (iii) f(a,b) - f(-a,-b) = 2 a f ( ^ a 4 b 4 ) , (iv) f(a,b)f(-a,-b) = f(-a 2 ,-b 2 M-ab), (v)

f2(a,b) + f2(-a,-b) = 2 f ( a 2 , b 2 M a b ) ,

and (vi) f2(a,b) - f2(-a,-b) = 4af(|,f a 2 b 2 )i«a 2 b 2 ). In the proofs below, we set p = ab. PROOF OF (i). Using in turn Entries 28, 24(iv), and 24(iii), we deduce that f(a,ab2)f(b,a2b) =

f

^

b

\f_^

f(a,bM-p)/(p) f(-p)f(-pZ)

=f( a ,b)*(p).

PROOF OF (ii). Using the definition (18.1) of f(a,b),

we find that

44


f(a,b) + f(-a,-b) = 2

=2

oo

I p^-1)/^ k=-°° k even

1 p k ( 2 k - l ) a 2 k = 2 " (p 4,k(k-l)/2 (a 3 b) k k=-°o k=-°° = 2f(a 3 b,ab 3 ).

PROOF OF (iii). Proceeding as in the proof above, we have f(a,b) - f(-a.-b) = 2

=2

I pk(2k+l)a2k+l k=-co

I k=-°° k odd =2a

pMk-l)/2ak

I (p 4)k(k + l)/2 (a/b) k k=-co

~ 2af (l'b

p}

-

PROOF OF (iv). By Entries 19 and 22(i), f(a,b)f(-a,-b) = (a 2 ;p 2 Ub 2 ;p 2 )Jp;p)f 2 2 2 = f(-a ,-b ) ? ? = f ( - a S - b 2 M - p ) . Alternatively, if we set c = -a and d = -b in Entry 29(i), we easily obtain the desired result. PROOF OF (v). Putting c = a and d = b in Entry 29(i), we easily achieve the sought result. PROOF OF (vi). Set c = a and d = b in Entry 29(ii) and use Entries I8(ii) and 22(ii). COROLLARY.

If ab = cd, then

f(a,b)f(c,d)f(an,b/n)f(cn,d/n) - f(-a,-b)f(-c,-d)f(-an,-b/n)f(-cn,-d/n) = 2af(c/a,ad)f (d/an,acn)f(n,ab/n)ijj(ab). PROOF. For brevity, set


and

a = f(a,b)f(c,d),

a' = f(-a,-b)f(-c,-d),

3 = f(an,b/n)f(cn,d/n),

3' = f(-an,-b/n)f(-cn,-d/n),

L = a3 - a'3'. By Entries 29(i) and (ii), we readily find that a + a' = 2f(ac,bd)f(ad,bc), a - a' = 2af(p£ abcd)f( ~ abed), 3 + 3' = 2f(acn2,-^[)f(ad,bc), n

and 2 3 - 3* = 2 a n f ( - ~ ^ - abcd)f(]j,£ abed), en Substituting these in the obvious identity 2(a3 - a'3') = (a + a')(3 - 3' ) + (a - a')(3 + 3*) and using Entries 29(i) and (ii), we find that L = af(ad,bc)f(|j,jj abcd)J2nf(ac,bd)f(-^,-^- abed) ^ en + 2f(|,| abcd)f(acn2,^f)} = af(ad,bc)f(j[,| abcd){[f(n,^)f(acn,^) - f(-n,- f

)f(-acn,- ^ ) ] + [f(n,^)f(acn,^)

• f(-n.- f

)f(-acn.- £ ) } }

= 2af(ad,bc)f(£,j| abcd)f(n,^)f(acn,^). If we apply Entry 30(i) and use the hypothesis equality above may be written as L = 2af(c/a,ad)n»(ab)f(n,^)f(^, acn),

ab = cd, we find

46


which is what we wanted to prove. ENTRY 31 . Let U . a n(n+l)/2 b n(n-l)/2 each integer n. Then

and

= a n(n-l

y

)/2bn(rH-l )/2

fQr

U n+r n-1 x V n-r f(U,,V,) = I U f U ' U 1 1 r r r r=0

(31.1)

Ramanujan writes Entry 31 in the form (31.2)

f d J p V ^ = f(U n ,V n ) + l ^ f

+ V,f

Vl

VI

V]

Vl +

• V]

Vl

U

U

'

l

Vftf Tl

n+1 U

+

l

Vv U

n-2 2

V

where the sum on the right side evidently contains Entry 18(iv), for r > 1, (31.3)

V r f ( - ^ ,9- 4 ^ | = U_ J n-r r Vr

2 U n-r Vn+r ^

n+2

• v2

n terms. However, by

Vr n-r-

U 2n-r V r = IL n-rf 1Un-r 'U n-r

This shows that the sums on the right sides of (31.1) and (31.2) are equal. PROOF. n-1 r=0

We have rU

^ V

00

1

"-1

fUn+vOk(k+1)/2

For ]- i Y " r p l

r

u

u

r

r >

k=-oo r=0

r

^

u

Vrlk(k'1)/2

r J

2

Y

V

u T-k u k(k+l)/2 v k(k-l)/2

Y

n-1 y

a(nk+r)(nk+r+l)/2b(nk+r)(nk+r-l)/2

k=-°° r=0 =

l

EXAMPLES. (1!

a

k(W)/2bk(k-l)/2.f(uv)i

We have

*(q) = ^(q9) + 2qf(q3,q15) = ^ ( q 2 5 ) + 2qf(q15,q35) + 2q4f(q5,q45)


47

and *(q) = f(q 3 ,q 6 ) + q^(q9)

(ii)

* f(q 6 ,q 10 ) + qf(q 2 5 q 14 )

= f(q 10 ,q 15 ) + qf(q 5 5 q 2 °) +

qW')

= f(q 15 ,q 21 ) + q^(q9) + q 3 f(q 3 ,q 3 3 ). PROOF. The two equalities of part (i) follow from Entry 31 and (31.3) by setting a = b = q and n = 3, 5, respectively. The four equalities in part (ii) follow from Entry 31 by setting 3 3 (a,b,n) = (l,q,3),(q,q ,2), (l,q,5), and (q,q ,3), respectively. The examples above are, in fact, special cases of the following general formulas: */„\ -L V n(n^\ • ,(q 2 ) = 2 ^

!

l

and 2 7 *(-q) " ^(q 2 ) = -2qf \\$ ] PROOF. Putting we find that

.

a = q, b = q 3 , and c = d = q 2 in Entries 29(i) and (ii),

*(q)*(q2) + * ( - q M - q 2 ) = 2f 2 ( q 3 , q 5 ) and * ( q M q 2 ) - *(-q)^(-q2) = 2qf 2 (q,q 7 ). These equalities reduce to the desired identities upon using the fact o iK-qM-q ) = vK-q) *(q), which is deducible from Entries 25(iii) and (iv). EXAMPLE (v). f(q,q5) = *(-q 3 ) X (q). PROOF. By Entries 19,22(ii), and 22(iv), f(q,q5) = (-qiq^J-q^q^Jq^q 6 ^ o •(-q;q 2 ) ENTRY 32. We have

(q 6 ;q6 ) , 3 fi" = x ( q W - q 3 ) (-q ;q )„

50

C. ADIGA, B. C. BERNDT, S. BHARGAVA, AND G. N. WATSON ;4, » ' ( q ? » ' ( q ) - i - . » (-q) V t q T ~

where we have used Entry 22(i) and where G(n)

(-zq;q2) f-q/z;q2) {zq;qZ)Jq/z;q\

Comparing (34.3) and (34.4), we see that it suffices to show that Log G(n) = 4 J ^ W ^ k • 1 )n) k=l (2k - 1)(1 - q 4k ~^) Like the calculation of Log F(n) in the previous proof, the proof of the equality above is quite straightforward. Ramanujan now states two "corollaries". We have not been able to discern why the appellation "corollary" has been given to these two results. Moreover, the "corollaries" are incorrect. We shall give two corrected versions of each corollary. Firstly, we prove versions where the "right sides" are corrected; secondly, we establish reformulations when the "left sides" are corrected. Most likely, the first versions are what Ramanujan had in mind. COROLLARY (i) (FIRST VERSION). real, then

I

(34.5) ^ L o g

k = -oo

If |q| < 1 and z = e 2 i n , where

n

is

q k(3k " 2) sin{2(3k - l)n}

sin(2n)

k(3k-2) J (3k - l)q k=-°°

r qk sin2 (kn) + ? q 4k sin 2 (2kn) L 9 1 / 2 . av k=l k(l - q^K) k=l k(l - q K ) PROOF. By Watson's quintuple product identity [74] (see also [5] or [70]), (q 2 ;q 2 ) (zq;q 2 L( q /z;q 2 )(z 2 ;q 4 ) o o ( q 4 /z 2 ;q 4 L v

k=-I

3kd+k, 3k -3k -3k-l 3k+l, q (z q -z q

56


. y

3k2-2kz3k-l _

:_oo

3k2-2kz-3k+l]

J

}

|( = -oo

I q3k "2ksin{2(3k - l ) n } , k=-°°

= 2iz

where we obtained the penultimate equality by replacing previous latter summands.

k by k - 1 in

Thus, upon dividing both sides above by 1 - z

find that oo

CiA C\ (34.6)

2

I q 3 k " 2 k s i n { 2 ( 3k - 1 )n} k= fill) -°° 2 f m sin(2n) I (3k - 1 )q 3k -2k

where F(n) = (qz;q 2 )

(q/z;q2)Jq4z2;q4)Jq4/z2;q4)

Now a straightforward calculatio n yield s

9

Log F(n)

V q cos(2kn) ^

,

9

"

2kx K

k=l k ( l - q^ )

v ^

q 4k cos(4kn) ,

4kv

k=l k ( l - q^ K)

so that

(34

7,

1 4

Log

F(n) _ 1 FfOT " 2

g k ( l - cos(2kn))

y k=l

k(l - q

1

r

q 4 k ( l - cos(4kn))

2

k=l

k (l - q 4 k )

Combining (34.6) and (34.7), we readily a r r i v e at (34.5). COROLLARY ( i ) (SECOND VERSION).

If

|q| < 1

r e a l , then

[34.8)

{Log

[l k=l

V L

( - q ) k ( k - ^ / 2 C O S ( 2 k -^(-q)cos n

q k sin 2 (kn) , ? V K

k=l k(l + q ) PROOF. Let

Dn'

L

k=l

q 4k sTn 2 (2kn) Air

k(l - q 4 K )

and

z = e2ln,

where

2


57

(zq;q 2 )Jq/z;q 2 )Jz 2 q 4 ;q 4 Uq 4 /z 2 ;q 4 )Jq 2 ;q 2 ) 2

" _ (zq 2 ;q 2 )Jq 2 /z ; q 2 ) oo (q;q 2 ]5?;q 4 )f Then a straightforward, but rather lengthy, calculation shows that (34.9)

LogP(z)=4

k • 2,, > 4k . 2,„, , [ ^ Sln < n> + 4 I /2sin(6k+l)n

I (34.14)

{Log .inn V

I (6k k=-°°

q k sin 2 (kn) , v K

k=l k(l - q ) PROOF.

l)q^3k

+

+k

>/2

g k sin 2 (2kn)

k=l k(l - q^ K )

Applying Watson's quintuple product i d e n t i t y , we f i n d that

(q;q) (qz;q) ( l / z ; q ) (qz 2 ;q 2 ) £

q(3r+k)/2(z3k

J

= 2i2-V2

.

(q/z2;q2) 2-3k-l}

q(3k^k)/2sin(6k

+ 1)n_

k=-°° Dividing both sides by (1 - 1/z), we deduce that

(34.15)

k

l

g f e | = ='°° sin n

q(3k2+k)/2sin(6k+l)n T—

m

I

(6k + l ) q

t3k

+K)U

k=-a>

where G(n) = (qz;q) (q/z;q) (qz 2 ;q 2 )

(q/z2;q2)

,

n

is


59

Proceeding in the same fashion as in the first proof of Corollary (i), we find that sin (k (kn) , j g sin2(k n ) (34.16) JLogf[^-i_ ^3 sin ^k + I ^ ^ k

k=l

2

k

k(l - q )

2

k=l k(l - q " )

Taking (34.15) and (34.16) together, we deduce (34.14). Although we know of no specific statement of the quintuple product i d e n t i t y by Ramanujan, Corollaries ( i ) and ( i i ) appear to be good evidence that he knew this identity. COROLLARY ( i i )

(SECOND VERSION).

If

|q| < 1

and

z = e2in,

r e a l , then

2 + (34.17) i L o gVf V(-q) ^ k=-» I 1^H?*^ + 2q^ cos(2n) K

[

"\]

+ q4K >

__ ^ gksin2(kn) , £ qksin2(2kn) _ k=l k(l - q k ) k=l k(l + q k ) PROOF. Let (^;q)Jq/z;qUz 2 q;q 2 Ug/z 2 ;q 2 ) c o (q 2 ;q 2 )f

.

(z 2 q 2 ; q 2 ) co (q 2 /z 2 ; q 2 ) M (q; q ) 2 ( q; q 2 )f Then an elementary calculatio n yields

(34.18)

LogQ(z)=4

? 9 ^ M k=l k ( l - qK )

4

+

? q k sin 2 (2kn) k=l k ( l + q )

Put G(n) = ( q ; q ) 2 ( q ; q 2 ) 2 Q(z) Using the f a c t o r i z a t i o n ( z W ^

= (zq;q)oo(-zq2;q2)J-zq;q2)a>

and a similar factorization for (q2/z2;q2)/2 k=l

Then for each positive integer (35.2)

^Q

PROOF.

n

i

2 n

k=1

n,

i2k ? l !-: ] |12 2 k p „2kQw2n-2k*

We shall write Entry 33(ii) in the form

Log

(35.3)

I (-l) k " 1 q k ( k " 1 ) / 2 (sin(2k - l)e)/0 k=l I (-l) k - 1 (2k-l)q k ( k " 1 )/ 2 k=l qK(l - cos(2k6

2 L°9i

k=l

k(l - q*)

R,

and equate coefficients of like powers of 0. Expanding

sin(2k - 1)0 in its

Maclaurin series and then inverting the order of summation, we readily find that (35.4)

L = ^ LL uoy g I i=lT. n 2 l n £ 0 T 2 H T ! ^2n

&

On the other hand, using a familiar expansion for Log

sin 0 ,

which, in fact,

Ramanujan derived in Chapter 5 [60, vol. 2, p. 52], [25, p. 64, Entry 16], we find that, for |0| < TT ,

(35.5)

i

v

R = i2 Î

( 1)JB

-

2.i(2e)2J : T (-nj+122j v ^ ' V + .L I =] - T Z 3 T T - Î —

( 2 j)( 2j )i

w2 j

) i ! y j ( B2J + y k2iivi]e2j = I H "T2JTT x J/: v r 4jJ j=l

= I

j=l

"

"

k=l

1 - q

( - 1 ) j + 1 2 2 j pP e, 2 j (2j)! 2j

Using (35.4) and (35.5) in (35.3), we deduce that, for |e| < IT,


62

rV

,2n = expj I

D i f f e r e n t i a t i n g both sides with respect to 7 H ) ^ Q2n-1 _ y ( - ] ) j + 1 2 ^ + 1 L 2n - 1 ! 9 ~ j.=/1, ( 2 j - l1))i! n=l = Equating c o e f f i c i e n t s of

F

[2371

oo

n

2j9 J

9, p P r

we f i n d that Q

9 22 jj D

2 M ? (-1)* Q 2k Q 9 ,^L0 T2k7T JZkTl 2k

o2j+lp

n

K I (,!)n+l J 2 j g 2 n - 2j „2n-1 (2j-l)!(2n-2j)! n=l j=l ,2n-l , n >_ 1 , on both sides, we readily deduce

(35.2). ENTRY 3 5 ( i i ) .

For each positive integer

° - 2")Bn 2n

where B define

+

y

l-1)k+\n-V

k=l

1 + q*

denotes the nth Bernoulli number.

n, l e t

For each nonnegative integer n,

+ Y (-1) (2k - 1) q 2k-l k=l i - q i °° / !xk-l 2k-l 1 + y (-U q 4 £ ?k-l 4 k=l 1 - q^K '

I F

2"

where then

E?

(35.6)

denotes the 2n th Euler number.

Hn

If n is any positive integer,

2n - l|022kP Q k=l 2k - 1| 2k 2n-2k'

PROOF. Write Entry 34(i) in the form

(35.7)

(.l) k - 1 q 2k ^cos(2k - 1)9^ 2k-l i - q ^(q) k-1 k. 2 I M T V O -cos(2ke))E R k=l k(l + q*)

fsec 9 + 4 I k=l L = -Log = -Log(sec e) +

As in the previous proof, we shall expand both sides in powers of 9 and equate coefficients. Using (35.1), we find that, for |e| < TT/2,

CHAPTER 16 OF RAMANUJAN'S SECOND NOTEBOOK (35.8) 00

y

sec e+ 4 ? H ^ Y ^ W k k=l 1 - q2k~] (-1)

/ -n

0

2n

U n j

n=0 00

E0

2n

-

n

n = 0

(35.9)

/(q)

°°

- 1)8

/ T\k-1 2k-l

("])

y

°°

9

/ i\k-l/o,,

/ T\n/ol

lN

2n

V (-1) (2k - 1)

1 - q^ K '

k=l ri

Putting

°°

A

63

Un;

n=0

0

fl2n

*

T x 2n_2k-1 ^ «

in Entry 3 4 ( i ) , we f i n d that = 1+4

(

X k=l

"1} " q 2k-l 1 - q

'

Combining (35.7)-(35.9), we conclude that (35.10)

92n .

Zn (z)]

L = -Log I

On the other hand, using a well-known expansion for Log(sec 6), which, in fact, Ramanujan found in Chapter 5 [60, vol. 2, p. 52], [25, p. 64, Entry 17], we find that - (-l)j+122j(1 - 2 2j )B„. R=^ ^ ^ H B^

(35.11)

(-1)J'+122J'+1P yl - " 2j „2j 6 (2j)! • j=l Hence, putting (35.10) and (35.11) in (35.7), we deduce that v

(-I)V

K

n=Q

I.

w

2n

,

B 2n

(2TTT 9 t? \\ n

_

-

-

f

e x p ex

y

^

P I Z._ —

( - 1 ) J 2 2 j + 1 P „ . ,.-. ; 2j „ 2 j

(T7TTT 2j)!

D i f f e r e n t i a t i n g both sides with respect to -

ni1

( 1)nQ

-

2n

(2n - 1)!

fl2n-l_

6

~

" I} _ "

y n ^

(-1)J22J+1P2j

(2j - 1)T

, un M )

e

8, 9

•

we f i n d that (-l)kQ2kfl2k

» k

I

Q

(2k)! °

" 22j+lp2jQ2n-2j j?! I 2 j - 1 ) ! ( 2 n - 2 j ) !

e

2n-l

64


Equating coefficients of

0

, n >_ 1, on both sides, we easily deduce (35.6).

Ramanujan [60, vol. 2, p. 202] has an erroneous factor of

(-l)n~k

in the

sum on the right side of (35.6). Define OO

L = l - 24 I

|(

-&

k=l 1 - q K

K q M = 1 + 240 I . k=l 1 - q K

and 00

5k

= 1 . 504 I JS-9.

N

k=l 1 - q k '

Ramanujan remarks that Sn n

E

I (-l) k+1 (2k- 1 ,2n + l q k(k-l)/2 f k=l

can always be expressed in terms of

L, M,

u

and

Ramanujan [57], [59, pp. 136-162] proved that a polynomial in Q,

and

R,

L, M,

and

respectively.)

N.

N.

Now by Entry 35(i),

Pp, ^'•••'Pon' n — ^*

can be written as a polynomial in

P2J_ 1, can be expressed as

(In [57], L, M,

Hence,

S

S

In an epic paper,

and

N

are denoted by

P,

can be represented as a polynomial in

L, M, and N. EXAMPLES. N

Let Q , n >_ 2, be defined as in Entry 35(i).

be defined as above.

(1)

3Q2 = L,

(ii)

50 bg

4

Let L, M, and

Then

= 5L 2 - 2M 3

and

(111)

7Q6 - 3Bl3 - *?L+16N

PROOF.

The three desired equalities follow by putting

respectively, in Entry 35(i).

n = 1, 2, and 3,

The calculations are straightforward.


65

Note that Entries 36(i), (ii) below reduce to Entries 29(i), (ii), respectively, if p = 1. ENTRY 36. (i)

If

p = ^

then

5

S ~ l{f(a,b)f(c,d) + f(-a,-b)f(-c,-d)} oo

=

I (ad) k ( k + 1 ) / 2 (bc) k ( k - 1 ) / 2 f(acp k , " , k=-°° p

and (ii)

D E l{f(a,b)f(c,d) - f(-a,-b)f(-c,-d)} I a2k+1(ad)k(k-1)/2(bc)k(k+1)/2f(^,âbcd). c k=-°° ap

= PROOF.

We shall prove just (i), since the proof of (ii) is similar.

Putting s =

n - m = 2k

y

m,n=-°° m+n even

=

I

(cd)^ m

(cd)J

2

and +n

n + m = 2j, - v >^ 0.

gj- = A 2 /B 2 .

Then

66

C. ADIGA, B. C. BERNDT, S. BHARGAVA, AND G. N. WATSON S

= J (Ajk q 2uk 2 f ( ABq2 p + 4v k>

^

AB

Now let k = yn + m, -« < n < °°, 0 £ m _< y -1. Thus,

m=0 n=-°° W

Ab

I

Next, apply Entry 18(iv) with a = ABq2w+4vm, and

b = q 2 v " 4 v m /AB,

n replaced by vn. Hence, $ =

u-1 -1

2 2 2 r„ûn+m f A l ^ + m q 2o../..~.,~\2 y ( y n + m ) 2 ( A B r v n q . 2 y v 2 nn2 ^ mn

-°°

m=0 n=-°°

^

^ ^

fAfQq2um2 B

i

1. J

m=0

• -ÂB—J ? i

n=-co

xf(SBq^

kn{V'v) n (y+v)

B

VH

4«

q

2y(y 2 -v 2 )n 2 + 4mn(y 2 -v 2 )

'

3 ^ ) .

In summary, we have shown that (36.1) S = l{f(Aq v+V ,q M+V /A)f(Bq y - V ,q 1J - V /B) + f(-Aq^ V J -q^ V /A)f(-Bq y - V ,-q^ v /B)} v ]

~

I

m=0

fA]mn2pm2 f A ^ n (2y+4m)(y 2 -v 2 ) B ^ „(2y-4m)(y 2 -v 2 ) i\P-v 3y+v f ( ABq 2y+4vm 9

a

2y-4vm

AB

We now examine Entry 36(ii) under the same substitutions as above. Thus, letting k = yn + m, -«> < n < °°, 0 £ m £ y - 1, we find that n

v

M y+vv2k+lfB], kk o2 iy2k 2 f B

D = k=-°° I (AqM )

M q

f

Aq

-4vk-2v A

'Bq

4vk+2v+4yl


= A V I ( AB )^ n+m q^ +v )( 2 ^ n+2m+1 ) +2 ^^ n+m ) 2 m=0 n=-°°

f fA n4v(yn+m)+2v+4y J8 -4v(yn+m)-2v T Bq 'A q Apply Entry 18(iv) with a -gq and with

,

b =^q

n replaced by vn. Therefore,

D =A V I (AB)yn+mq(y+v)(2yn+2m+l)+2y(yn+m)2fA]""vn m=0 n=-°° v

xq

-2yv2n2-(2y+2v+4vm)vn,;fA 4y+2v+4vm B -2v-4vm f^q ,Aq

= A y y ( AB) m q( 2m+ ", )(^ +v)+2ym m=0 q 2y(y

y

A(y-v)n B(y+v)n

n=2

2

2

2

2

-v )n +2(2m+l)(y -v )n f A 4y+2v+4vm B -2v-4vm Bq 'A q

In summary, we have shown that (36.2) D =l{f (Aq^.q^/Ajf (Bq^.q^/B) - f (-Aq^+Vqy+V/A)f ( W V q P " V 2

2

= A V / AB) m Q (2m+1)(p+v)+2ym 2 ( y - v M+V (2y+4m+2)(y 2 -v 2 ) q ( 2 y ' 4 m ' 2 ) ( y " V } 1 m=0 ^ A^V+V Y f [A

X f

(B

q

4y+2v+4vm B -2v-4vm"

' Aq

We now record a couple of special cases of (36.1). (36.1), we find that

(36.3) =

^(q^>(q^) +

Letting

A=B= 1

«(-q^M^)}

U

^ q 2ym 2 f(q (2y+4m)(y 2 -v 2 ) jq (2y-4m)(y 2 -v 2 ) )f(q 2y+4vm jq 2y-4vm )> m=0

Next, putting A = q y v and B = q y " v in (36.1) and using Entries 18(ii), (iii), we find that

68


2H^+ZVM^-ZV)

(36.4) =

y

^ q 2ym 2 +2vm f(q (2y+4m)(y 2 -v 2 ) 5q (2y-4m)(n 2 -v 2 ) )f( 4y+4vm -4vm ^ m=0 Adding Entries 30(ii) and (iii) yields f(a,b) = f(a 3 b,ab 3 ) + a f(|,£a 4 b 4 ). a = q2vm+y/2

Putting (36.5)

and b = q "

2

^ ^

we

see that

f ( q 2vm + y/2 j q -2vm + y/2 )

= f(q4vm+2^

-4vm+2yx

+

2vm+y/2f , -4vm 4vm+4yx

Multiplying (36.4) by q y / , adding the resulting equality to (36.3), and us (36.5), we deduce that

(36.6) ^ ( q ^ M q ^ ) + ^-q^M-q^)} + 2q^ W y+ ^)*(q 2y " 2V ) =

y

" ] q 2ym 2 f(q (2y+4m)(y 2 -v 2 ) 5q (2y-4m)(y 2 -v 2 ) }

m=0 x { f ( q 2y + 4vm 5 q 2y-4v m )

= V

qWf

+ q 2vm + y/2 f (q 4y+4vni 9q -4vm )}

(q(^m)(/-v2)jq(2y4m)(y2-v2))f(q2vm+y/25q-2vm+y/2)i

m=0 Looking back at the proofs of (36.1) and (36.2), we observe that we can replace

m by m + jy for any integer

right sides of (36.1) and (36.2).

j and not alter the summands on the

Note that (36.3) and (36.6) also remain

unchanged if m is replaced my -m. Finally, observe that, with the use of Entry 18(iv), we may replace

m by -m on the right side of (36.4) as well.

These observations are useful in simplifying these formulas somewhat 2 To ill illustratethe remarks above, consider (36.4). Replacing q deduce that

^+VM