Fungrim entry: 65c610

{e}^{x + y} = \sum_{k=0}^{\infty} \sum_{n=0}^{\infty} {n + k \choose k} \frac{{x}^{k} {y}^{n}}{\left(n + k\right)!}

Assumptions:

x \in \mathbb{C} \;\mathbin{\operatorname{and}}\; y \in \mathbb{C}

TeX:

{e}^{x + y} = \sum_{k=0}^{\infty} \sum_{n=0}^{\infty} {n + k \choose k} \frac{{x}^{k} {y}^{n}}{\left(n + k\right)!}

x \in \mathbb{C} \;\mathbin{\operatorname{and}}\; y \in \mathbb{C}

Definitions:

Fungrim symbol	Notation	Short description
`Exp`	${e}^{z}$	Exponential function
`Sum`	$\sum_{n} f(n)$	Sum
`Binomial`	${n \choose k}$	Binomial coefficient
`Pow`	${a}^{b}$	Power
`Factorial`	$n !$	Factorial
`Infinity`	$\infty$	Positive infinity
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("65c610"),
    Formula(Equal(Exp(Add(x, y)), Sum(Sum(Mul(Binomial(Add(n, k), k), Div(Mul(Pow(x, k), Pow(y, n)), Factorial(Add(n, k)))), For(n, 0, Infinity)), For(k, 0, Infinity)))),
    Variables(x, y),
    Assumptions(And(Element(x, CC), Element(y, CC))))

Topics using this entry

Factorials and binomial coefficients