S£L

NATIONAL BUREAU OF STANDARDS REPORT

‘ 199U

THE COEFFICIENTS OF CERTAIN INFINITE PRODUCTS

by

M, Newman

U. S. DEPARTMENT OF COMMERCE

NATIONAL BUREAU OF STANDARDS
U. S. DEPARTMENT OF COMMERCE

Charles Sawyer, Secretary

NATIONAL BUREAU OF STANDARDS

A. V. Astin, Director THE NATIONAL BUREAU OF STANDARDS

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NATIONAL BUREAU OF STANDARDS REPORT

NBS PROJECT NBS REPORT

1102-21-1101* October ll;, 19£2 199i*

THE COEFFICIENTS OF CERTAIN INFINITE PRODUCTS

by

M, Newman

, Approved for public release by the

The publication, re

. Director of the National Institute of

unless permission is

25, d, c. Suchper Standards and Technology (NIST)

cal ly prepared If t On October 9, 2015

in part, Is prohibited

landards, Washington

sport has been speclfieport for Its own use.

The coefficients of certain infinite products

Introduction , The Identities

(a) IT

00

(l-xn

) = I°°(-un x

(3n2+n >/2 » s I H: - e©

(b) TT°° (I-xn )3 = y °°(-l) n

(2n+l) x

(nS+n)/2

vr - t L-

*i -.o

are classical, (a) being the so-called 'pentagonal number theorem*

of Euler, and (b) being an identity of Jacobi's veil-known in the

theory of elliptic modular functions. It is remarkable that for

OQ

no other values of r are the coefficients of TT (i -xn

) r known

explicitly. It will be the purpose of this paper to give various

recursion formulas for these coefficients, which are consequences of

identities between functions on certain modular subgroups. In

particular, the coefficients will be investigated thoroughly for

r ~ 2,4,6. A typical theorem is that every integer occurs as coefficient for r =s 2 .

1. Preliminary identities . In what follows all sums will be extended

from 0 to oo and all products from 1 to oo , unless otherwise indicated.

¥e denote the coefficient of x

n in TT (l-xn

) r by Pr (n); i.e..
2

(1.1) TT (i-xn

) r = X p r( n ) xn .

Noting that TT U-xn

) r =TT (i-xn

) a .7T( l-xn) r ”a

, we obtain the identity

( 1 . 2 ) Pr(n) pr-a(n -3)

J=o

Differentiating (l.l) logarithmically, we also obtain

(1.5) n PP (n) = -r V'a-fa) Pr (n-j),^(o) » J . *”3*/ d I

J

(1.3) is especially suited to numerical calculation.

Using (a), (b) , and (1.2), we can obtain the following explicit

formulas for ?2(n), F^(n), P,g(n):

(1A) P

z ("O -JT(-l)

x+y

n = (3x2 ± x)/2 + (3y

2

± y)/2

(..S') P,(*0 " 2>DX+y (2x+l)

n = ( x

2+x )/2 + (3y2 ± y )/2

(1.6) ('H.) = V(-l)x+2T (2x+l)(2y+l)

n - (x2+x)/2 + (y

2+4)/2

It is understood in formulas (1. 4) -(1-6) that (x,y) runs over
all positive integral solutions of the defining equations,

formulas, though complicated, are. arithmetically enlightening and

will he studied in ‘detail. We set <x> = £x + ij, so that <x >

is the integer nearest to x. A little consideration shows that

we may rewrite formulas (l .4) -(l .6) as follows:

, <(u+v) /6 )

(1.7) t2 (n) = ^ (-1)

o c> U +V1 - 2ynn4»P

( 1 . 8 )

(1.9)

P'^(n)

PgW

i

Q(u+3v)/&3

(-1) ‘ v

u

2+3v2 ~ 2 An + 4, u, v odd

I (-1)

(U+V)

./2

uv

2 P U-hV^ “ u

It is understood in formulas (1.7) -(1.9) that (u , v) runs over al

positive integral solutions of the defining equations

.

ep IT p

p

PlCL

Let a be the canonical decomposition of If &

is odd for- some f « -1(4), then the equation x +y2 * a has no

solutions in integers. If e. is odd for some f g -1(3). then

the equation x

2+3y2 - a has no solutions in integers n We can say

therefore

:
( 1 .10) Let r = 2,4,6. Let -~1 n+1 = TT P be the canonical

decomposition of n+1. If is odd for some f

9

g -l(mr ), where

mr = 4,3,4 for r = 2,4,6, then P r (n) - 0,

Hence we need only consider n such that tit n+1 = Z Qp'", where

is a product of primes = 1 (mr ) and Qr is a product of primes

g -I (mr )

We note that if f = -1(4), then x

2

+y

2 ~ 0(p) implies x=y=0 (p)

,

and if p = -1(3 ), then x

2+3y2 = 0(p) implies x=y=0 (p)

Making use of the above, we obtain the following reduction

formulas:

o ^ Q2 ( v ) /*o ^

(1.11) p2 ( (^^2 -1 )/12 ) = Z (-1)

“

. __ rQii(u+3v)/6j

(1.12) P4( Q4' -U/6) = S, Z (-1)

L

A-t Zv'L - H ^ Lf , <*)V T<bct

(1.1?) P6 ( C ‘A'-U/'O = %

2

p6( % -l)/1*)-

In particular,

(1.14) P2 ((Q2 -1)/12) = (-1)

n

P 1 24 /* + i

(1.15) P4UQ4 -l)/6) = (-1)

(1.16) P6 ((Q6

2

-1)A) = Q6

2

[2Q4/5 ]

Q,

A similar set of explicit formulae can be written down when
5

£ is a prime. We note only that if % is a prime, then a ^

»JL{ J < Qo(°< )/6 ^

2 ((£ Qg

c -1)/12) = 2(-l)

^

where 0 are the uniquely determined positive integers such that

?a

=o< 4

t

*

2 . The prIn c 1 pal identities . We are going to extend the scope of

the previous identities by recursion formulas derived from identities between modular functions. The following theorem has been

proved by the author in his paper £ lj{£)

Theorem 1. Suppose that r is even, 0<r£24, Let pbe a prime sue]

that r(p-l) ~ 0(24)

.

Set $

= r(p -l)/24. Then

(2.1) £ Pr

(np +$)xn = Pr (S)TRl~xn

) r

- p

X

x

SrT(l~xn Py\

By comparing coefficients of corresponding powers of x, we find

(a. 2} Pr (np+$) - P IS) p (n) -p

*-/

Pr ((n-S)/p),(

2

)

fLth the convention that Pr (x) is zero if x is not an integer.

(^) lumbers in square brackets refer to bibliography at end of paper

(“) The case r»24 reduces to Mordell 1 s identity for Ramanujan ? s function 'C(n) •
We propose to derive some more identities of this type for

r b 2,4,6 and for various associated p ' s

,

For this purpose

we will require a number -theoretic lemma, the proof of which

is straightforward and will ‘be omitted.

Lemma 1. Suppose that r is 2,4, or 6. Let

p

"oe a prime such

that r(p+l) g, 0(24). Set &

~ r(p'--l)/24, - f C

A/ f]

np, a a 0,1,,,, . Then is the least value of n for

which is the sum of two pentagons (r»2); the sum of a

triangle and a pentagon (r~4)

the sum of two triangles (r«6)

Lemma 1 leads easily to a proof that the functions S r defined

in £l 1 are constant , under- the conditions imposed above (^)

¥e S ’ i is lead to the following theorem:

Theorem 2. With the notation of lemma 1, we have

( .5) 2 Pplnp + A)^ - (.pW/S-lry (1 . /C)!'

By comparing coefficients of corresponding powers of x, we find

. • ) Mnp + A) = (- p' 2 -1

pp

(n/p) .

(3) 58

’ ‘ici nts of the noJo terms are essentially

» */?J -1. These coefficients are of necessity

aero, in view of the formulas (1-4) -(l. •*) .
!

> We vi 11 consider . Deductions from :h- principal formula s f recursion formulas (2,2) and (2.4) more closely. It will be

understood that a formula derived from a theorem is subject

to the hypotheses of that theorem.

We see from (2.4) that

(5.1) P

p (nf +A) - 0. (n,p)

0,1,...), and

A+-

Sething n =* *y-t+i ~ P/^4 + A ^ = ?r {JU t

) , formula (2.4) reads At+ i = (-p) xy "“ 1 At . This

implies that Pp(/^>P + r(p -lj/24) = (-p)^* 7 ~ A/ ^ Pr (/^o )

In particular, choosing UQ ~ 0,1,2 we obtain

(?. 2 )

. 3

)

3 A)

,2*£ , . , P

r (r(P -l)/24) = l-p)

/2-l)t

Pr (p + r(p -l)/24) - -r(-P) U/ ’

2.t Xt , ?r (|'P + r(f -l)/24) = r(p-3)/2 .l-p)

?/2-l)

The

Dorai

recursion formula (2

discuss! on . Set ting

.2) requires a slightly more ela

T-, r \ - ^r/2-1

a = Pr kv )

. d a

p , n =JJL ,

i Bia \ u —

A . - ^ Si L~* a ..e. » t+2 *‘t+l + b A

tion depends <m the

equation - ax 4-

0,1,...), and At = Pr (yt^), (2.2) reads

a o. The solution of th is difference ©qua

nature of the roots of the characteristic

d S3 o. This in turn depends on the dis-

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