Fungrim entry: d0d91a

\sum_{n=0}^{\infty} F_{n} \frac{{z}^{n}}{n !} = \frac{2}{\sqrt{5}} {e}^{z / 2} \sinh\!\left(\frac{\sqrt{5}}{2} z\right)

Assumptions:

z \in \mathbb{C}

TeX:

\sum_{n=0}^{\infty} F_{n} \frac{{z}^{n}}{n !} = \frac{2}{\sqrt{5}} {e}^{z / 2} \sinh\!\left(\frac{\sqrt{5}}{2} z\right)

z \in \mathbb{C}

Definitions:

Fungrim symbol	Notation	Short description
`Sum`	$\sum_{n} f(n)$	Sum
`Fibonacci`	$F_{n}$	Fibonacci number
`Pow`	${a}^{b}$	Power
`Factorial`	$n !$	Factorial
`Infinity`	$\infty$	Positive infinity
`Sqrt`	$\sqrt{z}$	Principal square root
`Exp`	${e}^{z}$	Exponential function
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("d0d91a"),
    Formula(Equal(Sum(Mul(Fibonacci(n), Div(Pow(z, n), Factorial(n))), For(n, 0, Infinity)), Mul(Mul(Div(2, Sqrt(5)), Exp(Div(z, 2))), Sinh(Mul(Div(Sqrt(5), 2), z))))),
    Variables(z),
    Assumptions(Element(z, CC)))

Topics using this entry

Fibonacci numbers

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