Fungrim entry: d25d10

\zeta\!\left(s, a\right) = \sum_{k=0}^{N - 1} \frac{1}{{\left(a + k\right)}^{s}} + \frac{{\left(a + N\right)}^{1 - s}}{s - 1} + \frac{1}{{\left(a + N\right)}^{s}} \left(\frac{1}{2} + \sum_{k=1}^{M} \frac{B_{2 k}}{\left(2 k\right)!} \frac{\left(s\right)_{2 k - 1}}{{\left(a + N\right)}^{2 k - 1}}\right) - \int_{N}^{\infty} \frac{B_{2 M}\!\left(t - \left\lfloor t \right\rfloor\right)}{\left(2 M\right)!} \frac{\left(s\right)_{2 M}}{{\left(a + t\right)}^{s + 2 M}} \, dt

Assumptions:

s \in \mathbb{C} \;\mathbin{\operatorname{and}}\; s \ne 1 \;\mathbin{\operatorname{and}}\; a \in \mathbb{C} \;\mathbin{\operatorname{and}}\; N \in \mathbb{Z}_{\ge 1} \;\mathbin{\operatorname{and}}\; M \in \mathbb{Z}_{\ge 1} \;\mathbin{\operatorname{and}}\; \operatorname{Re}\!\left(a + N\right) > 0 \;\mathbin{\operatorname{and}}\; \operatorname{Re}\!\left(s + 2 M - 1\right) > 0 \;\mathbin{\operatorname{and}}\; \left(a \notin \{0, -1, \ldots\} \;\mathbin{\operatorname{or}}\; \operatorname{Re}(s) < 0 \;\mathbin{\operatorname{or}}\; s = 0\right)

References:

http://dx.doi.org/10.1007/s11075-014-9893-1

TeX:

\zeta\!\left(s, a\right) = \sum_{k=0}^{N - 1} \frac{1}{{\left(a + k\right)}^{s}} + \frac{{\left(a + N\right)}^{1 - s}}{s - 1} + \frac{1}{{\left(a + N\right)}^{s}} \left(\frac{1}{2} + \sum_{k=1}^{M} \frac{B_{2 k}}{\left(2 k\right)!} \frac{\left(s\right)_{2 k - 1}}{{\left(a + N\right)}^{2 k - 1}}\right) - \int_{N}^{\infty} \frac{B_{2 M}\!\left(t - \left\lfloor t \right\rfloor\right)}{\left(2 M\right)!} \frac{\left(s\right)_{2 M}}{{\left(a + t\right)}^{s + 2 M}} \, dt

s \in \mathbb{C} \;\mathbin{\operatorname{and}}\; s \ne 1 \;\mathbin{\operatorname{and}}\; a \in \mathbb{C} \;\mathbin{\operatorname{and}}\; N \in \mathbb{Z}_{\ge 1} \;\mathbin{\operatorname{and}}\; M \in \mathbb{Z}_{\ge 1} \;\mathbin{\operatorname{and}}\; \operatorname{Re}\!\left(a + N\right) > 0 \;\mathbin{\operatorname{and}}\; \operatorname{Re}\!\left(s + 2 M - 1\right) > 0 \;\mathbin{\operatorname{and}}\; \left(a \notin \{0, -1, \ldots\} \;\mathbin{\operatorname{or}}\; \operatorname{Re}(s) < 0 \;\mathbin{\operatorname{or}}\; s = 0\right)

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
`BernoulliB`	$B_{n}$	Bernoulli number
`Factorial`	$n !$	Factorial
`RisingFactorial`	$\left(z\right)_{k}$	Rising factorial
`Integral`	$\int_{a}^{b} f(x) \, dx$	Integral
`BernoulliPolynomial`	$B_{n}\!\left(z\right)$	Bernoulli polynomial
`Infinity`	$\infty$	Positive infinity
`CC`	$\mathbb{C}$	Complex numbers
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n
`Re`	$\operatorname{Re}(z)$	Real part
`ZZLessEqual`	$\mathbb{Z}_{\le n}$	Integers less than or equal to n

Source code for this entry:

Entry(ID("d25d10"),
    Formula(Equal(HurwitzZeta(s, a), Sub(Add(Add(Sum(Div(1, Pow(Add(a, k), s)), For(k, 0, Sub(N, 1))), Div(Pow(Add(a, N), Sub(1, s)), Sub(s, 1))), Mul(Div(1, Pow(Add(a, N), s)), Add(Div(1, 2), Sum(Mul(Div(BernoulliB(Mul(2, k)), Factorial(Mul(2, k))), Div(RisingFactorial(s, Sub(Mul(2, k), 1)), Pow(Add(a, N), Sub(Mul(2, k), 1)))), For(k, 1, M))))), Integral(Mul(Div(BernoulliPolynomial(Mul(2, M), Sub(t, Floor(t))), Factorial(Mul(2, M))), Div(RisingFactorial(s, Mul(2, M)), Pow(Add(a, t), Add(s, Mul(2, M))))), For(t, N, Infinity))))),
    Variables(s, a, N, M),
    Assumptions(And(Element(s, CC), NotEqual(s, 1), Element(a, CC), Element(N, ZZGreaterEqual(1)), Element(M, ZZGreaterEqual(1)), Greater(Re(Add(a, N)), 0), Greater(Re(Sub(Add(s, Mul(2, M)), 1)), 0), Or(NotElement(a, ZZLessEqual(0)), Less(Re(s), 0), Equal(s, 0)))),
    References("http://dx.doi.org/10.1007/s11075-014-9893-1"))

Topics using this entry

Hurwitz zeta function