Fungrim entry: 685d1a

\sum_{n=0}^{\infty} T_{n}\!\left(x\right) {z}^{n} = \frac{1 - x z}{1 - 2 x z + {z}^{2}}

Assumptions:

x \in \left[-1, 1\right] \;\mathbin{\operatorname{and}}\; z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \left|z\right| < 1

TeX:

\sum_{n=0}^{\infty} T_{n}\!\left(x\right) {z}^{n} = \frac{1 - x z}{1 - 2 x z + {z}^{2}}

x \in \left[-1, 1\right] \;\mathbin{\operatorname{and}}\; z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \left|z\right| < 1

Definitions:

Fungrim symbol	Notation	Short description
`Sum`	$\sum_{n} f(n)$	Sum
`ChebyshevT`	$T_{n}\!\left(x\right)$	Chebyshev polynomial of the first kind
`Pow`	${a}^{b}$	Power
`Infinity`	$\infty$	Positive infinity
`ClosedInterval`	$\left[a, b\right]$	Closed interval
`CC`	$\mathbb{C}$	Complex numbers
`Abs`	$\left\|z\right\|$	Absolute value

Source code for this entry:

Entry(ID("685d1a"),
    Formula(Equal(Sum(Mul(ChebyshevT(n, x), Pow(z, n)), For(n, 0, Infinity)), Div(Sub(1, Mul(x, z)), Add(Sub(1, Mul(Mul(2, x), z)), Pow(z, 2))))),
    Variables(x, z),
    Assumptions(And(Element(x, ClosedInterval(-1, 1)), Element(z, CC), Less(Abs(z), 1))))

Topics using this entry

Chebyshev polynomials