Fungrim entry: 12ce84

{T}^{(r)}_{n}(x) = \frac{\left(n\right)_{r} \left(n - r + 1\right)_{r}}{\left(2 r - 1\right)!!} \,{}_2F_1\!\left(r + n, r - n, \frac{1}{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:

{T}^{(r)}_{n}(x) = \frac{\left(n\right)_{r} \left(n - r + 1\right)_{r}}{\left(2 r - 1\right)!!} \,{}_2F_1\!\left(r + n, r - n, \frac{1}{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
`ChebyshevT`	$T_{n}\!\left(x\right)$	Chebyshev polynomial of the first 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("12ce84"),
    Formula(Equal(ComplexDerivative(ChebyshevT(n, x), For(x, x, r)), Mul(Div(Mul(RisingFactorial(n, r), RisingFactorial(Add(Sub(n, r), 1), r)), DoubleFactorial(Sub(Mul(2, r), 1))), Hypergeometric2F1(Add(r, n), Sub(r, n), Add(Div(1, 2), r), Div(Sub(1, x), 2))))),
    Variables(n, r, x),
    Assumptions(And(Element(n, ZZ), Element(r, ZZGreaterEqual(0)), Element(x, CC), Or(LessEqual(r, n), NotEqual(x, -1)))))

Topics using this entry

Chebyshev polynomials