Fungrim entry: 7a85b7

P_{n}\!\left(z\right) = \frac{1}{{2}^{n}} \sum_{k=0}^{\left\lfloor n / 2 \right\rfloor} {\left(-1\right)}^{k} {n \choose k} {2 n - 2 k \choose n} {z}^{n - 2 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}^{\left\lfloor n / 2 \right\rfloor} {\left(-1\right)}^{k} {n \choose k} {2 n - 2 k \choose n} {z}^{n - 2 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("7a85b7"),
    Formula(Equal(LegendrePolynomial(n, z), Mul(Div(1, Pow(2, n)), Sum(Mul(Mul(Mul(Pow(-1, k), Binomial(n, k)), Binomial(Sub(Mul(2, n), Mul(2, k)), n)), Pow(z, Sub(n, Mul(2, k)))), For(k, 0, Floor(Div(n, 2))))))),
    Variables(n, z),
    Assumptions(And(Element(n, ZZGreaterEqual(0)), Element(z, CC))))

Topics using this entry

Legendre polynomials