Fungrim entry: 60c6da

\zeta\!\left(s, a\right) = \frac{1}{s - 1} + \sum_{n=0}^{\infty} \frac{{\left(-1\right)}^{n}}{n !} \gamma_{n}\!\left(a\right) {\left(s - 1\right)}^{n}

Assumptions:

s \in \mathbb{C} \;\mathbin{\operatorname{and}}\; a \in \mathbb{C} \;\mathbin{\operatorname{and}}\; a \notin \{0, -1, \ldots\}

TeX:

\zeta\!\left(s, a\right) = \frac{1}{s - 1} + \sum_{n=0}^{\infty} \frac{{\left(-1\right)}^{n}}{n !} \gamma_{n}\!\left(a\right) {\left(s - 1\right)}^{n}

s \in \mathbb{C} \;\mathbin{\operatorname{and}}\; a \in \mathbb{C} \;\mathbin{\operatorname{and}}\; a \notin \{0, -1, \ldots\}

Definitions:

Fungrim symbol	Notation	Short description
`HurwitzZeta`	$\zeta\!\left(s, a\right)$	Hurwitz zeta function
`Sum`	$\sum_{n} f(n)$	Sum
`Pow`	${a}^{b}$	Power
`Factorial`	$n !$	Factorial
`StieltjesGamma`	$\gamma_{n}\!\left(a\right)$	Stieltjes constant
`Infinity`	$\infty$	Positive infinity
`CC`	$\mathbb{C}$	Complex numbers
`ZZLessEqual`	$\mathbb{Z}_{\le n}$	Integers less than or equal to n

Source code for this entry:

Entry(ID("60c6da"),
    Formula(Equal(HurwitzZeta(s, a), Add(Div(1, Sub(s, 1)), Sum(Mul(Mul(Div(Pow(-1, n), Factorial(n)), StieltjesGamma(n, a)), Pow(Sub(s, 1), n)), For(n, 0, Infinity))))),
    Variables(s, a),
    Assumptions(And(Element(s, CC), Element(a, CC), NotElement(a, ZZLessEqual(0)))))

Fungrim entry: 60c6da

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