Citation
Mohamat Johari, Mohamat Aidil
(2013)
General relation between sums of figurate numbers.
Doctoral thesis, Universiti Putra Malaysia.
Abstract
In this study, we seek to find relations between the number of representations of a
nonnegative integer n as a sum of figurate numbers of different types.
Firstly, we give a relation between the number of representations, ck(n), of n as
the sum k cubes and the number of representations, pk(n), of n as the sum of k
triangular pyramidal numbers, namely under certain conditions
pk(n) = c k odd (v);
where c k odd denotes the number of representations as a sum of k odd cubes and the
integer v is derived from n. Then we extend this problem by considering sums of
s-th powers with s > 3 and the associated polytopic numbers of order s.
Next, we discuss the relation between ɸ(2;k)(n), the number of representations of
n as a sum of k fourth powers, and ψ(2;k)(n), the number of representations of n as a sum of k terms of the form 8γ2 + 2γ where γ is a triangular number. When
1 ≤ k ≤ 7, the relation is
ɸ(2;k)(8n + k) = 2kψ (2;k) (n).
We extend this result by considering the relation between the number of represen-
tations of n as a sum of k 2m-th powers and the number of representations of n as
a sum of k terms determined by an associated polynomial of degree m evaluated
at a triangular number.
Thirdly, we consider the relation between sk(n), the number of representations of
n as a sum of k squares, and ek(n), the number of representations of n as a sum
of k centred pentagonal numbers. When 1 ≤k ≤ 7, this relation is
αkek(n) = sk (8n -3k)÷5 ; where αk = 2k + 2k-1 (k4)
We extend the analysis to the number of representations induced by a partition γ
of k into m parts. If corresponding number of representations of n are respectively
sγ(n) and eγ(n), then
βγeγ(n) = sγ(8n - 3k)÷5
where
βγ = 2m + 2(m-1) (( i1/4) + (i1/2)(i2/1)+(i1/1)(i3/1)
and ij denotes the number of parts of γ which are equal to j.
We end this thesis with a short discussion and proposal of various open problems
for further research.
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