Gold 35

Lerch zeta function

Gold ID

35

Link

https://sigir21.wmflabs.org/wiki/Lerch_zeta_function#math.85.57

Formula

$\Phi (z,s,a)={\frac {1}{1-z}}{\frac {1}{a^{s}}}+\sum _{n=1}^{N-1}{\frac {(-1)^{n}\mathrm {Li} _{-n}(z)}{n!}}{\frac {(s)_{n}}{a^{n+s}}}+O(a^{-N-s})$

TeX Source

\Phi(z,s,a) = \frac{1}{1-z} \frac{1}{a^{s}} + \sum_{n=1}^{N-1} \frac{(-1)^{n} \mathrm{Li}_{-n}(z)}{n!} \frac{(s)_{n}}{a^{n+s}} +O(a^{-N-s})

Translation Results
Semantic LaTeX	Mathematica Translation	Maple Translations
		-

Semantic LaTeX

Translation

\Phi(z , s , a) = \frac{1}{1-z} \frac{1}{a^{s}} + \sum_{n=1}^{N-1} \frac{(- 1)^{n} L \iunit_{-n}(z)}{n!} \frac{\Pochhammersym{s}{n}}{a^{n+s}} + O(a^{-N-s})

Expected (Gold Entry)

\Phi(z , s , a) = \frac{1}{1-z} \frac{1}{a^{s}} + \sum_{n=1}^{N-1} \frac{(-1)^{n} \polylog{-n}@{z}}{n!} \frac{\Pochhammersym{s}{n}}{a^{n+s}} + \bigO{a^{-N-s}}

Mathematica

Translation

\[CapitalPhi][z , s , a] == Divide[1,1 - z]*Divide[1,(a)^(s)]+ Sum[Divide[(- 1)^(n)* L*Subscript[I, - n]*(z),(n)!]*Divide[Pochhammer[s, n],(a)^(n + s)], {n, 1, N - 1}, GenerateConditions->None]+ \[CapitalOmicron][(a)^(- N - s)]

Expected (Gold Entry)

\[CapitalPhi][z, s, a] == Divide[1,1 - z]*Divide[1,(a)^(s)]+ Sum[Divide[(- 1)^(n)* PolyLog[-n, z],(n)!]*Divide[Pochhammer[s, n],(a)^(n + s)], {n, 1, N - 1}]+ O[a]^(- N - s)

Maple

Translation

Phi(z , s , a) = (1)/(1 - z)*(1)/((a)^(s))+ sum(((- 1)^(n)* L*I[- n]*(z))/(factorial(n))*(pochhammer(s, n))/((a)^(n + s)), n = 1..N - 1)+ Omicron((a)^(- N - s))

Expected (Gold Entry)