# Fungrim entry: b5049d

$\sum_{n=0}^{\infty} U_{n}\!\left(x\right) {z}^{n} = \frac{1}{1 - 2 x z + {z}^{2}}$
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} U_{n}\!\left(x\right) {z}^{n} = \frac{1}{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
ChebyshevU$U_{n}\!\left(x\right)$ Chebyshev polynomial of the second 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("b5049d"),
Formula(Equal(Sum(Mul(ChebyshevU(n, x), Pow(z, n)), For(n, 0, Infinity)), Div(1, 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

Copyright (C) Fredrik Johansson and contributors. Fungrim is provided under the MIT license. The source code is on GitHub.

2021-03-15 19:12:00.328586 UTC