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–