Parity results for certain partition functions and identities similar to theta function identities
Authors: Blecksmith Richard , Brillhart John , Gerst Irving 
Published: June 2010

In this paper we give a collection of parity results for partition functions of the form
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="product Underscript n element-of upper S Endscripts left-parenthesis 1 minus x Superscript n Baseline right-parenthesis Superscript negative 1 Baseline identical-to sigma-summation Underscript negative normal infinity Overscript normal infinity Endscripts x Superscript e left-parenthesis n right-parenthesis Baseline left-parenthesis mod 2 right-parenthesis">

∏

n
∈
S

(
1
−

x
n

)

−
1

≡

∑

−
∞

∞

x

e
(
n
)

(
mod

2
)

\prod _{n \in S} (1 - x^n)^{-1} \equiv \sum _{-\infty }^\infty x^{e(n)} \pmod 2

\]
and
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="product Underscript n element-of upper S Endscripts left-parenthesis 1 minus x Superscript n Baseline right-parenthesis Superscript negative 1 Baseline identical-to sigma-summation Underscript negative normal infinity Overscript normal infinity Endscripts left-parenthesis x Superscript e left-parenthesis n right-parenthesis Baseline plus x Superscript f left-parenthesis n right-parenthesis Baseline right-parenthesis left-parenthesis mod 2 right-parenthesis">

∏

n
∈
S

(
1
−

x
n

)

−
1

≡

∑

−
∞

∞

(

x

e
(
n
)

+

x

f
(
n
)

)

(
mod

2
)

\prod _{n \in S} (1 - x^n)^{-1} \equiv \sum _{-\infty }^\infty (x^{e(n)} + x^{f(n)})\pmod 2

\]
for various sets of positive integers S, which are specified with respect to a modulus, and quadratic polynomials

e
(
n
)

e(n)

and

f
(
n
)

f(n)

. Several identities similar to theta function identities, such as
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="product Underscript StartLayout 1st Row n equals 1 2nd Row n not-identical-to plus-or-minus left-parenthesis 4 comma 6 comma 8 comma 10 right-parenthesis left-parenthesis mod 32 right-parenthesis EndLayout Overscript normal infinity Endscripts left-parenthesis 1 minus x Superscript n Baseline right-parenthesis equals 1 plus sigma-summation Underscript n equals 1 Overscript normal infinity Endscripts left-parenthesis negative 1 right-parenthesis Superscript n Baseline left-parenthesis x Superscript n squared Baseline plus x Superscript 2 n squared Baseline right-parenthesis comma">

∏

n
=
1

n

≢

±
(
4
,
6
,
8
,
10
)

(
mod

32
)

∞

(
1
−

x
n

)
=
1
+

∑

n
=
1

∞

(
−
1

)
n

(

x

n
2

+

x

2

n
2

)
,

\prod _{\substack {n = 1\\n \nequiv \pm (4,6,8,10)\pmod {32}}}^\infty (1 - x^n) = 1 + \sum _{n = 1}^\infty (-1)^n (x^{n^2} + x^{2 n^2}),

\]
and its associated congruence
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="product Underscript StartLayout 1st Row n equals 1 2nd Row n not-identical-to 0 comma plus-or-minus 2 comma plus-or-minus 12 comma plus-or-minus 14 comma 16 left-parenthesis mod 32 right-parenthesis EndLayout Overscript normal infinity Endscripts left-parenthesis 1 minus x Superscript n Baseline right-parenthesis Superscript negative 1 Baseline identical-to 1 plus sigma-summation Underscript n equals 1 Overscript normal infinity Endscripts left-parenthesis x Superscript n squared Baseline plus x Superscript 2 n squared Baseline right-parenthesis left-parenthesis mod 2 right-parenthesis comma">

∏

n
=
1

n

≢

0
,
±
2
,
±
12
,
±
14
,
16

(
mod

32
)

∞

(
1
−

x
n

)

−
1

≡
1
+

∑

n
=
1

∞

(

x

n
2

+

x

2

n
2

)

(
mod

2
)
,

\prod _{\substack {n = 1\\n \nequiv 0, \pm 2, \pm 12, \pm 14,16 \pmod {32}}}^\infty (1 - x^n)^{-1} \equiv 1 + \sum _{n = 1}^\infty (x^{n^2} + x^{2 n^2}) \pmod 2,

\]
are also proved.