Fungrim entry: 599417

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

Assumptions:

q \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \left|q\right| < 1

TeX:

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

q \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \left|q\right| < 1

Definitions:

Fungrim symbol	Notation	Short description
`Sum`	$\sum_{n} f(n)$	Sum
`PartitionsP`	$p(n)$	Integer partition function
`Pow`	${a}^{b}$	Power
`Infinity`	$\infty$	Positive infinity
`EulerQSeries`	$\phi(q)$	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)), For(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.

2021-03-15 19:12:00.328586 UTC