Fungrim entry: b6b014

{U}^{(r)}_{n}(1) = \frac{\left(n + 1\right)_{r + 1} \left(n - r + 1\right)_{r}}{\left(2 r + 1\right)!!}

Assumptions:

n \in \mathbb{Z} \;\mathbin{\operatorname{and}}\; r \in \mathbb{Z}_{\ge 0}

TeX:

{U}^{(r)}_{n}(1) = \frac{\left(n + 1\right)_{r + 1} \left(n - r + 1\right)_{r}}{\left(2 r + 1\right)!!}

n \in \mathbb{Z} \;\mathbin{\operatorname{and}}\; r \in \mathbb{Z}_{\ge 0}

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
`ZZ`	$\mathbb{Z}$	Integers
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n

Source code for this entry:

Entry(ID("b6b014"),
    Formula(Equal(ComplexDerivative(ChebyshevU(n, x), For(x, 1, r)), Div(Mul(RisingFactorial(Add(n, 1), Add(r, 1)), RisingFactorial(Add(Sub(n, r), 1), r)), DoubleFactorial(Add(Mul(2, r), 1))))),
    Variables(n, r),
    Assumptions(And(Element(n, ZZ), Element(r, ZZGreaterEqual(0)))))

Topics using this entry

Chebyshev polynomials