Editing Theta function (section)

=== 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>