Estimates of five restricted partition functions that are quasi polynomials
Tóm tắt
A function
defined on
is said to be a quasi polynomial if,
is a polynomial in
for each
, where
is a positive integer. In this article, we show that the below given restricted partition functions are quasi polynomials: (i)
-number of partitions of
with exactly
parts and least part being less than
, (ii)
-number of distinct partitions (partitions with distinct parts) of
with exactly
parts and least part being less than
, (iii)
-number of partitions of
with exactly
parts and
least parts, (iv)
-number of partitions of
with exactly
parts and one largest part and (v)
-number of partitions of
with exactly
parts and difference between least part and largest part exceeds
. Consequently, following estimates were derived: (i)
(ii)
(iii)
(iv)
(v)
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