Fungrim entry: c5dd9b

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

Assumptions:

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

TeX:

P_{n}\!\left(z\right) = \frac{1}{{2}^{n}} \sum_{k=0}^{n} {{n \choose k}}^{2} {\left(z - 1\right)}^{n - k} {\left(z + 1\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
`Pow`	${a}^{b}$	Power
`Sum`	$\sum_{n} f(n)$	Sum
`Binomial`	${n \choose k}$	Binomial coefficient
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("c5dd9b"),
    Formula(Equal(LegendrePolynomial(n, z), Mul(Div(1, Pow(2, n)), Sum(Mul(Mul(Pow(Binomial(n, k), 2), Pow(Sub(z, 1), Sub(n, k))), Pow(Add(z, 1), k)), For(k, 0, n))))),
    Variables(n, z),
    Assumptions(And(Element(n, ZZGreaterEqual(0)), Element(z, CC))))

Topics using this entry

Legendre polynomials