# Fungrim entry: fff8ff

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

x \in \mathbb{C} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C}
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
Factorial$n !$ Factorial
Infinity$\infty$ Positive infinity
Exp${e}^{z}$ Exponential function
Sqrt$\sqrt{z}$ Principal square root
Sinc$\operatorname{sinc}(z)$ Sinc function
ConstI$i$ Imaginary unit
CC$\mathbb{C}$ Complex numbers
Source code for this entry:
Entry(ID("fff8ff"),
Formula(Equal(Sum(Mul(ChebyshevU(n, x), Div(Pow(z, n), Factorial(n))), For(n, 0, Infinity)), Mul(Exp(Mul(z, x)), Add(Cosh(Mul(z, Sqrt(Sub(Pow(x, 2), 1)))), Mul(Mul(z, x), Sinc(Mul(Mul(ConstI, 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