# Fungrim entry: 9d66de

${U}^{(r)}_{n}(x) = \frac{\left(n + 1\right)_{r + 1} \left(n - r + 1\right)_{r}}{\left(2 r + 1\right)!!} \,{}_2F_1\!\left(r + n + 2, r - n, \frac{3}{2} + r, \frac{1 - x}{2}\right)$
Assumptions:$n \in \mathbb{Z} \,\mathbin{\operatorname{and}}\, r \in \mathbb{Z}_{\ge 0} \,\mathbin{\operatorname{and}}\, x \in \mathbb{C} \,\mathbin{\operatorname{and}}\, \left(r \le n \,\mathbin{\operatorname{or}}\, x \ne -1\right)$
TeX:
{U}^{(r)}_{n}(x) = \frac{\left(n + 1\right)_{r + 1} \left(n - r + 1\right)_{r}}{\left(2 r + 1\right)!!} \,{}_2F_1\!\left(r + n + 2, r - n, \frac{3}{2} + r, \frac{1 - x}{2}\right)

n \in \mathbb{Z} \,\mathbin{\operatorname{and}}\, r \in \mathbb{Z}_{\ge 0} \,\mathbin{\operatorname{and}}\, x \in \mathbb{C} \,\mathbin{\operatorname{and}}\, \left(r \le n \,\mathbin{\operatorname{or}}\, x \ne -1\right)
Definitions:
Fungrim symbol Notation Short description
ComplexDerivative$\frac{d}{d z}\, f\!\left(z\right)$ Complex derivative
ChebyshevU$U_{n}\!\left(x\right)$ Chebyshev polynomial of the second kind
RisingFactorial$\left(z\right)_{k}$ Rising factorial
Hypergeometric2F1$\,{}_2F_1\!\left(a, b, c, z\right)$ Gauss hypergeometric function
ZZ$\mathbb{Z}$ Integers
ZZGreaterEqual$\mathbb{Z}_{\ge n}$ Integers greater than or equal to n
CC$\mathbb{C}$ Complex numbers
Source code for this entry:
Entry(ID("9d66de"),
Assumptions(And(Element(n, ZZ), Element(r, ZZGreaterEqual(0)), Element(x, CC), Or(LessEqual(r, n), Unequal(x, -1)))))