Fungrim entry: 599417

\sum_{n=0}^{\infty} p\!\left(n\right) {q}^{n} = \frac{1}{\phi\!\left(q\right)}

Assumptions:

q \in \mathbb{C} \,\mathbin{\operatorname{and}}\, \left|q\right| \lt 1

TeX:

\sum_{n=0}^{\infty} p\!\left(n\right) {q}^{n} = \frac{1}{\phi\!\left(q\right)}

q \in \mathbb{C} \,\mathbin{\operatorname{and}}\, \left|q\right| \lt 1

Definitions:

Fungrim symbol	Notation	Short description
`PartitionsP`	$p\!\left(n\right)$	Integer partition function
`Pow`	${a}^{b}$	Power
`Infinity`	$\infty$	Positive infinity
`EulerQSeries`	$\phi\!\left(q\right)$	Euler's q-series
`CC`	$\mathbb{C}$	Complex numbers
`Abs`	$\left\|z\right\|$	Absolute value

Source code for this entry:

Entry(ID("599417"),
    Formula(Equal(Sum(Mul(PartitionsP(n), Pow(q, n)), Tuple(n, 0, Infinity)), Div(1, EulerQSeries(q)))),
    Variables(q),
    Assumptions(And(Element(q, CC), Less(Abs(q), 1))))

Topics using this entry

Partition function

Copyright (C) Fredrik Johansson and contributors. Fungrim is provided under the MIT license. The source code is on GitHub.

2019-06-18 07:49:59.356594 UTC