Editing Theta function (section)

== Partition sequences and Pochhammer products ==

=== Regular partition number sequence ===

The regular partition sequence <math>P(n)</math> itself indicates the number of ways in which a positive [[Integer|integer number]] <math>n</math> can be split into positive integer summands. For the numbers <math>n = 1</math> to <math>n = 5</math>, the associated partition numbers <math>P</math> with all associated number partitions are listed in the following table:

{| class="wikitable"
|+Example values of P(n) and associated number partitions
!n
!P(n)
!paying partitions
|-
|0
|1
|() empty partition/[[empty sum]]
|-
|1
|1
|(1)
|-
|2
|2
|(1+1), (2)
|-
|3
|3
|(1+1+1), (1+2), (3)
|-
|4
|5
|(1+1+1+1), (1+1+2), (2+2), (1+3), (4)
|-
|5
|7
|(1+1+1+1+1), (1+1+1+2), (1+2+2), (1+1+3), (2+3), (1+4), (5)
|}

The generating function of the regular partition number sequence can be represented via Pochhammer product in the following way:
: <math>\sum _{k = 0}^\infty P(k)x^k = \frac{1}{(x;x)_{\infty}} = \theta_{3}(x)^{-1/6}\theta_{4}(x)^{-2/3} \biggl[\frac{\theta_{3}(x)^4 - \theta_{4}(x)^4}{16\,x}\biggr]^{-1/24}</math>

The summandization of the now mentioned [[Pochhammer symbol|Pochhammer product]] is described by the [[Pentagonal number theorem]] in this way:

:<math>(x;x)_{\infty} = 1 + \sum_{n = 1}^{\infty} \bigl[- x^{\text{Fn}(2n-1)} - x^ {\text{Kr}(2n-1)} + x^{\text{Fn}(2n)} + x^{\text{Kr}(2n)}\bigr]</math>

The following basic definitions apply to the [[pentagonal number]]s and the card house numbers:

: <math>\text{Fn}(z) = \tfrac{1}{2}z(3z-1)</math>
: <math>\text{Kr}(z) = \tfrac{1}{2}z(3z+1)</math>

As a further application<ref>[https://www.researchgate.net/publication/235432739_Ramanujan%27s_theta-function_identities_involving_Lambert_series Ramanujan's theta-function identities involving Lambert series]</ref> one obtains a formula for the third power of the Euler product:
:<math>(x;x)^3 = \prod_{n=1}^\infty (1-x^n)^3 = \sum _{m=0}^\infty (-1)^m(2m +1)x^{m(m+1)/2}</math>

=== Strict partition number sequence ===

And the strict partition sequence <math>Q(n)</math> indicates the number of ways in which such a positive integer number <math>n</math> can be splitted into positive integer summands such that each summand appears at most once<ref>{{cite web|accessdate=2022-03-09|title=code golf - Strict partitions of a positive integer|url=https://codegolf.stackexchange.com/questions/71941/strict-partitions-of-a-positive-integer}}</ref> and no summand value occurs repeatedly. Exactly the same sequence<ref>{{cite web|date=2022-03-09|title=A000009 - OEIS|url=https://oeis.org/A000009}}</ref> is also generated if in the partition only odd summands are included, but these odd summands may occur more than once. Both representations for the strict partition number sequence are compared in the following table:

{| class="wikitable"
|+Example values of Q(n) and associated number partitions
!n
!Q(n)
!Number partitions without repeated summands
!Number partitions with only odd addends
|-
|0
|1
|() empty partition/[[empty sum]]
|() empty partition/[[empty sum]]
|-
|1
|1
|(1)
|(1)
|-
|2
|1
|(2)
|(1+1)
|-
|3
|2
|(1+2), (3)
|(1+1+1), (3)
|-
|4
|2
|(1+3), (4)
|(1+1+1+1), (1+3)
|-
|5
|3
|(2+3), (1+4), (5)
|(1+1+1+1+1), (1+1+3), (5)
|-
|6
|4
|(1+2+3), (2+4), (1+5), (6)
|(1+1+1+1+1+1), (1+1+1+3), (3+3), (1+5)
|-
|7
|5
|(1+2+4), (3+4), (2+5), (1+6), (7)
|(1+1+1+1+1+1+1), (1+1+1+1+3), (1+3+3), (1+1+5), (7)
|-
|8
|6
|(1+3+4), (1+2+5), (3+5), (2+6), (1+7), (8)
|(1+1+1+1+1+1+1+1), (1+1+1+1+1+3), (1+1+3+3), (1+1+1+ 5), (3+5), (1+7)
|}

The generating function of the strict partition number sequence can be represented using Pochhammer's product:
: <math>\sum _{k = 0}^\infty Q(k)x^k = \frac{1}{(x;x^2)_{\infty }} = \theta_{3}(x)^{1/6}\theta_{4}(x)^{-1/3} \biggl[\frac{\theta_{3}(x)^4 - \theta_{4}(x)^4}{16\,x}\biggr]^{1/24}</math>

=== Overpartition number sequence ===

The [[Maclaurin series]] for the reciprocal of the function {{math|''{{not a typo|ϑ}}<sub>01</sub>''}} has the numbers of [[Partition function (number theory)|over partition sequence]] as coefficients with a positive sign:<ref>{{cite journal | last=Mahlburg | first=Karl | title=The overpartition function modulo small powers of 2 | journal=Discrete Mathematics | volume=286 | issue=3 | date=2004 | doi=10.1016/j.disc.2004.03.014 | pages=263–267}}</ref>

: <math>\frac{1}{\theta_{4}(x)} = \prod_{n=1}^{\infty} \frac{1 + x^{n}}{1 - x^{n }} = \sum_{k=0}^{\infty} \overline{P}(k)x^{k}</math>
: <math>\frac{1}{\theta_{4}(x)} = 1+2x+4x^2+8x^3+14x^4+24x^5+40x^6+64x^7+100x^ 8+154x^9+232x^{10} + \dots</math>

If, for a given number <math>k</math>, all partitions are set up in such a way that the summand size never increases, and all those summands that do not have a summand of the same size to the left of themselves can be marked for each partition of this type, then it will be the resulting number<ref>{{cite journal|language=en|title=Elsevier Enhanced Reader|journal=Discrete Mathematics |date=28 April 2009 |volume=309 |issue=8 |pages=2528–2532 |doi=10.1016/j.disc.2008.05.007 |last1=Kim |first1=Byungchan |doi-access=free }}<!-- auto-translated by Module:CS1 translator --></ref> of the marked partitions depending on <math>k</math> by the overpartition function <math>\overline{P}(k)</math> .

First example:

: <math>\overline{P}(4) = 14</math>

These 14 possibilities of partition markings exist for the sum 4:

{| class="wikitable"
|(4), ('''4'''), (3+1), ('''3'''+1), (3+'''1'''), ('''3'''+'''1'''), (2+2), ('''2'''+2), (2+1+1), ('''2'''+1+1), (2+'''1'''+1), ('''2'''+'''1'''+1), (1+1+1+1), ('''1'''+1+1+1)
|}

Second example:

: <math>\overline{P}(5) = 24</math>

These 24 possibilities of partition markings exist for the sum 5:

{| class="wikitable"
|(5), ('''5'''), (4+1), ('''4'''+1), (4+'''1'''), ('''4'''+'''1'''), (3+2), ('''3'''+2), (3+'''2'''), ('''3'''+'''2'''), (3+1+1), ('''3'''+1+1), (3+'''1'''+1), ('''3'''+'''1'''+1), (2+2+1), ('''2'''+2+1), (2+2+'''1'''), ('''2'''+2+'''1'''),
(2+1+1+1), ('''2'''+1+1+1), (2+'''1'''+1+1), ('''2'''+'''1'''+1+1), (1+1+1+1+1), ('''1'''+1+1+1+1)
|}

=== Relations of the partition number sequences to each other ===
In the Online Encyclopedia of Integer Sequences (OEIS), the sequence of regular partition numbers <math>P(n)</math> is under the code A000041, the sequence of strict partitions is <math> Q(n)</math> under the code A000009 and the sequence of superpartitions <math>\overline{P}(n)</math> under the code A015128. All parent partitions from index <math>n = 1</math> are even.

The sequence of superpartitions <math>\overline{P}(n)</math> can be written with the regular partition sequence P<ref>{{cite web|date=2022-03-11|author=Eric W. Weisstein|language=en|title=Partition Function P|url=https://mathworld.wolfram.com/PartitionFunctionP.html}}</ref> and the strict partition sequence Q<ref>{{cite web|date=2022-03-11|author=Eric W. Weisstein|language=en|title=Partition Function Q|url=https://mathworld.wolfram.com/PartitionFunctionQ.html}}</ref> can be generated like this:
: <math>\overline{P}(n) = \sum_{k=0}^{n} P(n - k)Q(k)</math>
In the following table of sequences of numbers, this formula should be used as an example:
{| class="wikitable"
!n
!P(n)
!Q(n)
!<math>\overline{P}(n)</math>
|-
|0
|1
|1
|1 = 1*1
|-
|1
|1
|1
|2 = 1 * 1 + 1 * 1
|-
|2
|2
|1
|4 = 2 * 1 + 1 * 1 + 1 * 1
|-
|3
|3
|2
|8 = 3 * 1 + 2 * 1 + 1 * 1 + 1 * 2
|-
|4
|5
|2
|14 = 5 * 1 + 3 * 1 + 2 * 1 + 1 * 2 + 1 * 2
|-
|5
|7
|3
|24 = 7 * 1 + 5 * 1 + 3 * 1 + 2 * 2 + 1 * 2 + 1 * 3
|}
Related to this property, the following combination of two series of sums can also be set up via the function {{math|{{not a typo|ϑ}}<sub>01</sub>}}:

:<math>\theta_{4}(x) = \biggl[\sum_{k = 0}^{\infty} P(k) x^k \biggr]^{-1} \biggl[\sum_{k = 0}^{\infty} Q(k) x^k \biggr]^{-1}</math>