# 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

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