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Statements
(
1
-
x
)
∑
m
,
n
=
0
∞
p
m
(
≤
k
,
𝒟
,
n
)
x
m
q
n
=
∑
m
=
0
k
[
k
m
]
q
q
m
(
m
+
1
)
/
2
x
m
=
∏
j
=
1
k
(
1
+
x
q
j
)
,
1
𝑥
superscript
subscript
𝑚
𝑛
0
restricted-partitions-P-at-most
𝑚
absent
𝑘
𝒟
𝑛
superscript
𝑥
𝑚
superscript
𝑞
𝑛
superscript
subscript
𝑚
0
𝑘
q-binomial
𝑘
𝑚
𝑞
superscript
𝑞
𝑚
𝑚
1
2
superscript
𝑥
𝑚
superscript
subscript
product
𝑗
1
𝑘
1
𝑥
superscript
𝑞
𝑗
{\displaystyle{\displaystyle(1-x)\sum_{m,n=0}^{\infty}p_{m}\left(\leq k,%
\mathcal{D},n\right)x^{m}q^{n}=\sum_{m=0}^{k}\genfrac{[}{]}{0.0pt}{}{k}{m}_{q}%
q^{m(m+1)/2}x^{m}=\prod_{j=1}^{k}(1+x\,q^{j}),}}
|
x
|
<
1
𝑥
1
{\displaystyle{\displaystyle|x|<1}}
[
n
m
]
q
q-binomial
𝑛
𝑚
𝑞
{\displaystyle{\displaystyle\genfrac{[}{]}{0.0pt}{}{\NVar{n}}{\NVar{m}}_{\NVar%
{q}}}}
p
k
(
≤
m
,
n
)
restricted-partitions-P-at-most
𝑘
absent
𝑚
𝑛
{\displaystyle{\displaystyle p_{\NVar{k}}\left(\leq\NVar{m},\NVar{n}\right)}}
x
𝑥
{\displaystyle{\displaystyle x}}
j
𝑗
{\displaystyle{\displaystyle j}}
k
𝑘
{\displaystyle{\displaystyle k}}
m
𝑚
{\displaystyle{\displaystyle m}}
n
𝑛
{\displaystyle{\displaystyle n}}
q
𝑞
{\displaystyle{\displaystyle q}}
Identifiers
