# Fungrim entry: 9d7c61

$\sum_{n=0}^{\infty} T_{n}\!\left(x\right) \frac{{z}^{n}}{n !} = {e}^{z x} \cosh\!\left(z \sqrt{{x}^{2} - 1}\right)$
Assumptions:$x \in \mathbb{C} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C}$
TeX:
\sum_{n=0}^{\infty} T_{n}\!\left(x\right) \frac{{z}^{n}}{n !} = {e}^{z x} \cosh\!\left(z \sqrt{{x}^{2} - 1}\right)

x \in \mathbb{C} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C}
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
Factorial$n !$ Factorial
Infinity$\infty$ Positive infinity
Exp${e}^{z}$ Exponential function
Sqrt$\sqrt{z}$ Principal square root
CC$\mathbb{C}$ Complex numbers
Source code for this entry:
Entry(ID("9d7c61"),
Formula(Equal(Sum(Mul(ChebyshevT(n, x), Div(Pow(z, n), Factorial(n))), For(n, 0, Infinity)), Mul(Exp(Mul(z, x)), Cosh(Mul(z, Sqrt(Sub(Pow(x, 2), 1))))))),
Variables(x, z),
Assumptions(And(Element(x, CC), Element(z, CC))))

## 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