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Fungrim entry: 859445

P2n+1 ⁣(z)=(1)n4n(2n+1)(2nn)z2F1 ⁣(n,n+32,32,z2)P_{2 n + 1}\!\left(z\right) = \frac{{\left(-1\right)}^{n}}{{4}^{n}} \left(2 n + 1\right) {2 n \choose n} z \,{}_2F_1\!\left(-n, n + \frac{3}{2}, \frac{3}{2}, {z}^{2}\right)
Assumptions:nZ0  and  zCn \in \mathbb{Z}_{\ge 0} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C}
TeX:
P_{2 n + 1}\!\left(z\right) = \frac{{\left(-1\right)}^{n}}{{4}^{n}} \left(2 n + 1\right) {2 n \choose n} z \,{}_2F_1\!\left(-n, n + \frac{3}{2}, \frac{3}{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("859445"),
    Formula(Equal(LegendrePolynomial(Add(Mul(2, n), 1), z), Mul(Mul(Mul(Mul(Div(Pow(-1, n), Pow(4, n)), Add(Mul(2, n), 1)), Binomial(Mul(2, n), n)), z), Hypergeometric2F1(Neg(n), Add(n, Div(3, 2)), Div(3, 2), Pow(z, 2))))),
    Variables(n, z),
    Assumptions(And(Element(n, ZZGreaterEqual(0)), Element(z, CC))))

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2021-03-15 19:12:00.328586 UTC