as a sum of 1. as a sum of $k$ squares is - Core

Nihonkai Math. J. vol. 3 (1992), 125-131

ON THE REPRESENTATIONS OF AN INTEGER AS A SUM OF TWO OR FOUR TRIANGULAR NUMBERS

Chandrashekar

In this note

ABSTRACT: $1\phi 1$

we show how Ramanujan’

$s$

-summation formula can be employed to obtain

formulas

an

Adiga

for

integer

the number of representations of $N\underline{\rangle}1$

as

a

sum

of

two

or four

triangular numbers.

1.

INTRODUCTION The representations of an integer

$N$

as a sum of

$k$

squares is

one of the most beautiful problem in the theory of numbers. study

of

representations

an integer as

of

sums of squares is

treated in depth in the book of E. Grosswald [4] in lattice point

problems,

The

and are useful

crystallography and certain problems

in mechanics. q-hypergeometric functions have played an important

role in the theory of representations of the numbers as sums of

squares. For instance Jacobi ‘

$s$

two and four square theorems have [2]

been proved by S.

Bhargava and Chandrashekar Adiga

the following

summation formula“ of Ramanujan [9] :

“ $1\phi 1$

–125–

on using

$\frac{(1/\alpha;q^{2})_{k}(-\Re)^{k}}{(\beta q^{2};q^{2})_{k}}z^{k}+\sum_{k=1}^{\infty}$

$=1+\sum_{k=1}^{\infty}$

$\frac{(1/\beta;q^{2})k(-\beta q)^{k}}{(\alpha q^{2};q^{2})_{k}}z^{-k}$

(1)

where $|q|$

\langle

1,

$|\beta q|$

$\langle$

$|z|$

$(a)\infty=(a;q)\infty=\prod^{\infty}$

$|1/\alpha q|$

$\langle$

,

$(1-aq^{k})$

$k=0$

and (a) $k=(a;q)_{k}\approx\frac{(a;q)\infty}{(aq^{k};q)}\infty$

This formula was first brought before the mathematical world by G.H.

pp. 222, 223]

[5.

Hardy

who

formula with many parameters“. proofs [1]

,

of

(1)

[8]).

Jacobi ‘

$s$

described

as

it

A number of direct and elementary

can be found in the literature

Letting

$\alpha\rightarrow$

$0$

,

”a remarkable

$\beta-$

$0$

(see

for example

in (1) we get the

well-known

triple product identity:

$(-qz;q^{2})_{\infty}(-q/z;q^{2})\infty(q^{2} ; q^{2})\infty$

$\sum^{\infty}$

$q^{k^{2}}z^{k}$

(2)

$=$

$k=-\infty$

where

$q,$ $a,$

$z$

are

complex

Hirschhorn [6], [7]

has

$z\neq 0$

.

given alternate proofs of Jacobi

‘

numbers

and four theorems on using (2).

–126–

with

$|q|$

\langle

1 and

M. D. $s$

two

In chapter 16 of his second notebook [9] Ramanujan introduces

the function $f(a,b)=1+\sum_{k=1}^{\infty}$

in

This

$q=e^{i\Pi T}$

(ab) $k(k-1)/2(a^{k}+b^{k})$

fact

corresponds ( putting a

where

$z$

is

complex and

,

$|ab|$

$=qe^{2iz}$

${\rm Im}(t)$

\rangle

$0$

\langle

(3)

$b=qe^{-2iz}$

,

)

1.

to

and

the classical

theta function [10] given by

$\$ 9_{3}(z,T)=1+2$

function

$f(a, b)$

.

develops the theory of theta-functions using the

then

Ramanujan

$\sum_{n=1}^{\infty}q^{n^{2}}cos2nz$

and its restrictions

$\emptyset(q)\equiv f(q, q)$

$=1+2\sum_{n=1}^{\infty}q^{n^{2}}=\frac{(-q;q^{2})_{\infty}(q^{2};q^{2})_{\infty}}{(q;q^{2})(-q^{2};q^{2})_{\infty},\infty}$

$\phi(q)\equiv f(q, q^{3})=\sum_{n=0}^{\infty}q^{n(n+1)/2}=\frac{(q^{2};q^{2})\infty}{(q;q^{2})_{\infty}}$

,

,

(4)

(5)

and $f(-q)\equiv f(-q, -q^{2})=\sum_{n=0}^{\infty}(-1)^{n}q^{n(3n-1)/2}+\sum_{n=1}^{\infty}(-1)^{n}q^{n(3n+1)/2}$

$=(q;q)_{\infty}$

.

(6)

As direct consequences of the definition (3) Ramanujan notes that

if ab

$=$

cd, then

$f(a,b)$

$f$

(

$c$

, d) $+f(-a, -b)$

$=2f$ (ac, bd)

$f$

$f(-c , -d)$

(ad, bc)

–127–

(7)

and $f$

(a , b)

$f$

(

$c$

, d)

–

$f(-c , -d)$

$f(-a , -b)$

$=2af(b/c, ac^{2}d)$

$f(b/d, acd^{2})$

.

(8)

For details of the proofs of the above identities,

one may refer

Putting $a=b=c=d=q$ in (7) and (8) we get

[1].

$\emptyset^{2}(q)+\phi^{2}(-q)=2\emptyset^{2}(q^{2})$

(9)

and $\phi^{2}(q)-\phi^{2}(-q)=8q\phi^{2}(q^{4})$

An integer

a

(

$n$

is

said

.

(10)

a $(a+1)/2$ ,

to be a triangular if $n=$

The purpose of this note is to show how Ramanujan’ $s11$

Z.

summation

formula

of representations

can be used to obtain formulas for the number of

an integer

$N$

) 1

as a sum of two or

four

triangular numbers.

2.

MAIN IHEOBLS

THEORE Yl 1. form

If

$N\underline{\rangle}1$

, the number of representations of

$n(n+1)/2+m(m+1)/2$ ;

$n,m\underline{\rangle}0$

$N$

in the

is

$T_{2}(N)=d_{1}(4N+1)-d_{3}(4N+1)$

where

$d_{i}(4N+1)$

For a proof of

is the number of divisors

$d$

this Theorem one may refer

However, Theorem 1 also follows from (1)

–128–

of

$4N+1,$

$d\equiv i(mod 4)$

N. J. Fine

, (4),

(5)

.

[3, p. 73].

and (10).

THEOREM 2.

form

If

,

$N\underline{\rangle}1$

the number of representations of

$n(n+1)/2+m(m+1)/2+i(i+1)/2+k(k+1)/2$ ; $T_{4}(N)$

$n,m,$

$N$

$i,k\underline{\rangle}0$

in the is

$.=$

$\sum_{d|2N+1}d$

PROOF.

Putting $a=\beta=-1$ in (1)

some simplification

and after

of the right side we get

$(-qz;q^{2})$

$(-q/z;q^{2})$

$(q^{2} ; q^{2})^{2}$

$\frac{\infty\infty\infty}{(qz;q^{2})(q/z;q^{2})_{\infty}(-q^{2};q^{2})_{\infty}^{2},\infty}$

$=\frac{1+qz}{1-qz}-2\sum_{n=1}^{\infty}$

$q^{2n}[(zq)^{n}-(zq)^{-n}]1+q^{2n}$

Dividing both sides by $1+qz/1-qz$ and letting changing

$q^{2}$

to

$-q$

$z\rightarrow$

$-1/q$

and

then

and using (4), we get $nq^{n}$

$\phi^{4}(q)=1+8$

(11)

$\sum^{\infty}$

$n=1$

$1+(-q)^{n}$

Multiplying (9) and (10) and using (4) and (5) we have $\phi^{4}(q)-\emptyset^{4}(-q)=$

16q

$\phi^{4}(q^{2})$

.

using (11) in (12) we get

$q\phi^{4}(q^{2})=$

$(2n+1)q^{2n+1}1-q^{4n+2}$ $\sum_{n=0}^{\infty}$

–129–

(12)

Changing

$q$

to

1/2

in this and using (5) we have

$q$

$1-q^{2n+l}(2n+1)q^{n}$

$[ \sum_{n=0}^{\infty} q^{n(n+1)/2} ]^{4}=\sum_{n=0}^{\infty}$

$\sum^{\infty}$

$\sum^{\infty}$

$n=0$

$m=0$

$=$

$\sum$

$D_{N}=$

(13)

$D_{N}q^{N}$

$=\sum_{N=0}^{\infty}$

where

$(2n+1)q^{2nm+m+n}$

$(2n+1)$

$=$

$d|2N+l\sum d$

$2N+1=(2n+1)(2m+1)$ Now,

comparing

the

coefficients of

$q^{N}$

on both sides of

(13)

we

get the required result.

ACKNOWLEDGEMENT.

The

author

is thankful to the referee for his

comments and suggestions.

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. 315, 53 (1985), 1-85.

2.

S. Bhargava and Chandrashekar Adiga, Simple Proofs of Jacobi’ Two

and

(U.K. )

3.

Four Square Theorems,

$s$

Int. J. Math. Educ. Sci. Technol.

1-3, (1988) , 779-782.

Nathan J. Fine, Basic Hypergeometric Series and Applications,

Mathematical

Surveys

and

Monographs, No. 27, Amer. Math. Soc.

(1988).

–130–

4.

E. Grosswald,

Representations of Integer as Sums of Squares,

Springer-Verlag, New York,

(1985).

5.

G. H. Hardy, Ramanujan, Chelsea, New York,

6.

M. D. Hirschhorn,

7.

Simple

A

9.

Jacobi

‘

Theorem, Amer. Math. Monthly, 92 (1985),

579-580.

of

Jacobi f

M. D. Hirschhorn,

Proof

Simple

A

Theorem, Proc. Amer. Math. Soc.

8.

of

Proof

(1978).

M. E.H. Ismail,

A

Amer. Math. Soc.

, 63

S.

Ramanujan,

, 101

(1987),

Notebooks

Two

$s$

Four Square

436-438.

Simple Proof of Ramanujan ‘ (1977),

Square

$s$

$s1\phi_{1}-sum$

, Proc.

185-186. (2 Volumes),

Tata

Institute

of

Fundamental Research, Bombay (1957).

10. E. T. Whittaker and G. N. Watson,

A Course of Modern Analysis,

Fourth Edition, University Press, Cambridge, (1966).

DEPARTMENT OF MATHEMATICS UNIVERSITY OF MYSORE MANASA GANGOTRI MYSORE 570 006 INDIA.

Received Aug. 10,1992

–131–