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Fungrim entry: d25d10

ζ ⁣(s,a)=k=0N11(a+k)s+(a+N)1ss1+1(a+N)s(12+k=1MB2k(2k)!(s)2k1(a+N)2k1)NB2M ⁣(tt)(2M)!(s)2M(a+t)s+2Mdt\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:sC  and  s1  and  aC  and  NZ1  and  MZ1  and  Re ⁣(a+N)>0  and  Re ⁣(s+2M1)>0  and  (a{0,1,}  or  Re(s)<0  or  s=0)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
HurwitzZetaζ ⁣(s,a)\zeta\!\left(s, a\right) Hurwitz zeta function
Sumnf(n)\sum_{n} f(n) Sum
Powab{a}^{b} Power
BernoulliBBnB_{n} Bernoulli number
Factorialn!n ! Factorial
RisingFactorial(z)k\left(z\right)_{k} Rising factorial
Integralabf(x)dx\int_{a}^{b} f(x) \, dx Integral
BernoulliPolynomialBn ⁣(z)B_{n}\!\left(z\right) Bernoulli polynomial
Infinity\infty Positive infinity
CCC\mathbb{C} Complex numbers
ZZGreaterEqualZn\mathbb{Z}_{\ge n} Integers greater than or equal to n
ReRe(z)\operatorname{Re}(z) Real part
ZZLessEqualZn\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"))

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2021-03-15 19:12:00.328586 UTC