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

Chebyshev polynomials