Formula:DLMF:25.11:E10

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ζ ( s , a ) = n = 0 Γ ( n + s ) n ! Γ ( s ) ζ ( n + s ) ( 1 - a ) n Hurwitz-zeta s a superscript subscript n 0 Euler-Gamma n s n Euler-Gamma s Riemann-zeta n s superscript 1 a n {\displaystyle\mathop{\zeta\/}\nolimits\!\left(s,a\right)=\sum_{n=0}^{\infty}% \frac{\mathop{\Gamma\/}\nolimits\!\left(n+s\right)}{n!\mathop{\Gamma\/}% \nolimits\!\left(s\right)}\mathop{\zeta\/}\nolimits\!\left(n+s\right)(1-a)^{n}}


Constraint(s)


s 1 𝑠 1 {\displaystyle s\neq 1} & | a - 1 | < 1 𝑎 1 1 {\displaystyle|a-1|<1}


Proof


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Use Taylor's theorem and
a ζ ( s , a ) = - s ζ ( s + 1 , a ) partial-derivative a Hurwitz-zeta s a s Hurwitz-zeta s 1 a {\displaystyle\frac{\partial}{\partial a}\mathop{\zeta\/}\nolimits\!\left(s,a% \right)=-s\mathop{\zeta\/}\nolimits\!\left(s+1,a\right)} .



Symbols List


a b fragments a fragments b {\displaystyle\frac{\mathopen{\partial}\hskip 0.0pta}{\mathopen{\partial}% \hskip 0.0ptb}}  : partial derivative : http://dlmf.nist.gov/1.5#E3

ζ 𝜁 {\displaystyle\zeta}  : Hurwitz zeta function : http://dlmf.nist.gov/25.11#E1
Γ Γ {\displaystyle\Gamma}  : Euler's gamma function : http://dlmf.nist.gov/5.2#E1
ζ 𝜁 {\displaystyle\zeta}  : Riemann zeta function : http://dlmf.nist.gov/25.2#E1

Bibliography


Equation (17), Section 25.11 of DLMF.

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