Fungrim entry: e1797b

{U}^{(r)}_{n}(x) = \frac{\sqrt{\pi} \left(n + 1\right)}{2 {\left(x - 1\right)}^{r}} \,{}_3{\textbf F}_2\!\left(1, -n, n + 2, \frac{3}{2}, 1 - 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} \setminus \left\{-1, 1\right\}

References:

http://functions.wolfram.com/Polynomials/ChebyshevU/20/02/01/0002/

TeX:

{U}^{(r)}_{n}(x) = \frac{\sqrt{\pi} \left(n + 1\right)}{2 {\left(x - 1\right)}^{r}} \,{}_3{\textbf F}_2\!\left(1, -n, n + 2, \frac{3}{2}, 1 - 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} \setminus \left\{-1, 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
`Sqrt`	$\sqrt{z}$	Principal square root
`Pi`	$\pi$	The constant pi (3.14...)
`Pow`	${a}^{b}$	Power
`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("e1797b"),
    Formula(Equal(ComplexDerivative(ChebyshevU(n, x), For(x, x, r)), Mul(Div(Mul(Sqrt(Pi), Add(n, 1)), Mul(2, Pow(Sub(x, 1), r))), Hypergeometric3F2Regularized(1, Neg(n), Add(n, 2), Div(3, 2), Sub(1, r), Div(Sub(1, x), 2))))),
    Variables(n, r, x),
    Assumptions(And(Element(n, ZZ), Element(r, ZZGreaterEqual(0)), Element(x, SetMinus(CC, Set(-1, 1))))),
    References("http://functions.wolfram.com/Polynomials/ChebyshevU/20/02/01/0002/"))

Topics using this entry

Chebyshev polynomials