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Fungrim entry: 6cd4a1

P2n ⁣(z)=(1)n4n(2nn)2F1 ⁣(n,n+12,12,z2)P_{2 n}\!\left(z\right) = \frac{{\left(-1\right)}^{n}}{{4}^{n}} {2 n \choose n} \,{}_2F_1\!\left(-n, n + \frac{1}{2}, \frac{1}{2}, {z}^{2}\right)
Assumptions:nZ0andzCn \in \mathbb{Z}_{\ge 0} \,\mathbin{\operatorname{and}}\, z \in \mathbb{C}
TeX:
P_{2 n}\!\left(z\right) = \frac{{\left(-1\right)}^{n}}{{4}^{n}} {2 n \choose n} \,{}_2F_1\!\left(-n, n + \frac{1}{2}, \frac{1}{2}, {z}^{2}\right)

n \in \mathbb{Z}_{\ge 0} \,\mathbin{\operatorname{and}}\, z \in \mathbb{C}
Definitions:
Fungrim symbol Notation Short description
LegendrePolynomialPn ⁣(z)P_{n}\!\left(z\right) Legendre polynomial
Powab{a}^{b} Power
Binomial(nk){n \choose k} Binomial coefficient
Hypergeometric2F12F1 ⁣(a,b,c,z)\,{}_2F_1\!\left(a, b, c, z\right) Gauss hypergeometric function
ZZGreaterEqualZn\mathbb{Z}_{\ge n} Integers greater than or equal to n
CCC\mathbb{C} Complex numbers
Source code for this entry:
Entry(ID("6cd4a1"),
    Formula(Equal(LegendrePolynomial(Mul(2, n), z), Mul(Mul(Div(Pow(-1, n), Pow(4, n)), Binomial(Mul(2, n), n)), Hypergeometric2F1(Neg(n), Add(n, Div(1, 2)), Div(1, 2), Pow(z, 2))))),
    Variables(n, z),
    Assumptions(And(Element(n, ZZGreaterEqual(0)), Element(z, SetMinus(CC)))))

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

2019-10-05 13:11:19.856591 UTC