Fungrim entry: f0569a

P_{n}\!\left(z\right) = \sum_{k=0}^{n} {n \choose k} {n + k \choose k} {\left(\frac{z - 1}{2}\right)}^{k}

Assumptions:

n \in \mathbb{Z}_{\ge 0} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C}

TeX:

P_{n}\!\left(z\right) = \sum_{k=0}^{n} {n \choose k} {n + k \choose k} {\left(\frac{z - 1}{2}\right)}^{k}

n \in \mathbb{Z}_{\ge 0} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C}

Definitions:

Fungrim symbol	Notation	Short description
`LegendrePolynomial`	$P_{n}\!\left(z\right)$	Legendre polynomial
`Sum`	$\sum_{n} f(n)$	Sum
`Binomial`	${n \choose k}$	Binomial coefficient
`Pow`	${a}^{b}$	Power
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("f0569a"),
    Formula(Equal(LegendrePolynomial(n, z), Sum(Mul(Mul(Binomial(n, k), Binomial(Add(n, k), k)), Pow(Div(Sub(z, 1), 2), k)), For(k, 0, n)))),
    Variables(n, z),
    Assumptions(And(Element(n, ZZGreaterEqual(0)), Element(z, CC))))

Topics using this entry

Legendre polynomials