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Lerch transcendent

In mathematics, the Lerch transcendent, is a special function that generalizes the Hurwitz zeta function and the polylogarithm. It is named after Czech mathematician Mathias Lerch, who published a paper about a similar function in 1887. The Lerch transcendent, is given by:

Φ ( z , s , α ) = ∑ n = 0 ∞ z n ( n + α ) s {\displaystyle \Phi (z,s,\alpha )=\sum _{n=0}^{\infty }{\frac {z^{n}}{(n+\alpha )^{s}}}} .

It only converges for any real number α > 0 {\displaystyle \alpha >0} , where | z | < 1 {\displaystyle |z|<1} , or R ( s ) > 1 {\displaystyle {\mathfrak {R}}(s)>1} , and | z | = 1 {\displaystyle |z|=1} .

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Special cases

The Lerch transcendent is related to and generalizes various special functions.

The Lerch zeta function is given by:

L ( λ , s , α ) = ∑ n = 0 ∞ e 2 π i λ n ( n + α ) s = Φ ( e 2 π i λ , s , α ) {\displaystyle L(\lambda ,s,\alpha )=\sum _{n=0}^{\infty }{\frac {e^{2\pi i\lambda n}}{(n+\alpha )^{s}}}=\Phi (e^{2\pi i\lambda },s,\alpha )}

The Hurwitz zeta function is the special case3

ζ ( s , α ) = ∑ n = 0 ∞ 1 ( n + α ) s = Φ ( 1 , s , α ) {\displaystyle \zeta (s,\alpha )=\sum _{n=0}^{\infty }{\frac {1}{(n+\alpha )^{s}}}=\Phi (1,s,\alpha )}

The polylogarithm is another special case:4

Li s ( z ) = ∑ n = 1 ∞ z n n s = z Φ ( z , s , 1 ) {\displaystyle {\textrm {Li}}_{s}(z)=\sum _{n=1}^{\infty }{\frac {z^{n}}{n^{s}}}=z\Phi (z,s,1)}

The Riemann zeta function is a special case of both of the above:5

ζ ( s ) = ∑ n = 1 ∞ 1 n s = Φ ( 1 , s , 1 ) {\displaystyle \zeta (s)=\sum _{n=1}^{\infty }{\frac {1}{n^{s}}}=\Phi (1,s,1)}

The Dirichlet eta function:6

η ( s ) = ∑ n = 1 ∞ ( − 1 ) n − 1 n s = Φ ( − 1 , s , 1 ) {\displaystyle \eta (s)=\sum _{n=1}^{\infty }{\frac {(-1)^{n-1}}{n^{s}}}=\Phi (-1,s,1)}

The Dirichlet beta function:7

β ( s ) = ∑ k = 0 ∞ ( − 1 ) k ( 2 k + 1 ) s = 2 − s Φ ( − 1 , s , 1 2 ) {\displaystyle \beta (s)=\sum _{k=0}^{\infty }{\frac {(-1)^{k}}{(2k+1)^{s}}}=2^{-s}\Phi (-1,s,{\tfrac {1}{2}})}

The Legendre chi function:8

χ s ( z ) = ∑ k = 0 ∞ z 2 k + 1 ( 2 k + 1 ) s = z 2 s Φ ( z 2 , s , 1 2 ) {\displaystyle \chi _{s}(z)=\sum _{k=0}^{\infty }{\frac {z^{2k+1}}{(2k+1)^{s}}}={\frac {z}{2^{s}}}\Phi (z^{2},s,{\tfrac {1}{2}})}

The inverse tangent integral:9

Ti s ( z ) = ∑ k = 0 ∞ ( − 1 ) k z 2 k + 1 ( 2 k + 1 ) s = z 2 s Φ ( − z 2 , s , 1 2 ) {\displaystyle {\textrm {Ti}}_{s}(z)=\sum _{k=0}^{\infty }{\frac {(-1)^{k}z^{2k+1}}{(2k+1)^{s}}}={\frac {z}{2^{s}}}\Phi (-z^{2},s,{\tfrac {1}{2}})}

The polygamma functions for positive integers n:1011

ψ ( n ) ( α ) = ( − 1 ) n + 1 n ! Φ ( 1 , n + 1 , α ) {\displaystyle \psi ^{(n)}(\alpha )=(-1)^{n+1}n!\Phi (1,n+1,\alpha )}

The Clausen function:12

Cl 2 ( z ) = i e − i z 2 Φ ( e − i z , 2 , 1 ) − i e i z 2 Φ ( e i z , 2 , 1 ) {\displaystyle {\text{Cl}}_{2}(z)={\frac {ie^{-iz}}{2}}\Phi (e^{-iz},2,1)-{\frac {ie^{iz}}{2}}\Phi (e^{iz},2,1)}

Integral representations

The Lerch transcendent has an integral representation:

Φ ( z , s , a ) = 1 Γ ( s ) ∫ 0 ∞ t s − 1 e − a t 1 − z e − t d t {\displaystyle \Phi (z,s,a)={\frac {1}{\Gamma (s)}}\int _{0}^{\infty }{\frac {t^{s-1}e^{-at}}{1-ze^{-t}}}\,dt}

The proof is based on using the integral definition of the gamma function to write

Φ ( z , s , a ) Γ ( s ) = ∑ n = 0 ∞ z n ( n + a ) s ∫ 0 ∞ x s e − x d x x = ∑ n = 0 ∞ ∫ 0 ∞ t s z n e − ( n + a ) t d t t {\displaystyle \Phi (z,s,a)\Gamma (s)=\sum _{n=0}^{\infty }{\frac {z^{n}}{(n+a)^{s}}}\int _{0}^{\infty }x^{s}e^{-x}{\frac {dx}{x}}=\sum _{n=0}^{\infty }\int _{0}^{\infty }t^{s}z^{n}e^{-(n+a)t}{\frac {dt}{t}}}

and then interchanging the sum and integral. The resulting integral representation converges for z ∈ C ∖ [ 1 , ∞ ) , {\displaystyle z\in \mathbb {C} \setminus [1,\infty ),} Re(s) > 0, and Re(a) > 0. This analytically continues Φ ( z , s , a ) {\displaystyle \Phi (z,s,a)} to z outside the unit disk. The integral formula also holds if z = 1, Re(s) > 1, and Re(a) > 0; see Hurwitz zeta function.1314

A contour integral representation is given by

Φ ( z , s , a ) = − Γ ( 1 − s ) 2 π i ∫ C ( − t ) s − 1 e − a t 1 − z e − t d t {\displaystyle \Phi (z,s,a)=-{\frac {\Gamma (1-s)}{2\pi i}}\int _{C}{\frac {(-t)^{s-1}e^{-at}}{1-ze^{-t}}}\,dt}

where C is a Hankel contour counterclockwise around the positive real axis, not enclosing any of the points t = log ⁡ ( z ) + 2 k π i {\displaystyle t=\log(z)+2k\pi i} (for integer k) which are poles of the integrand. The integral assumes Re(a) > 0.15

Other integral representations

A Hermite-like integral representation is given by

Φ ( z , s , a ) = 1 2 a s + ∫ 0 ∞ z t ( a + t ) s d t + 2 a s − 1 ∫ 0 ∞ sin ⁡ ( s arctan ⁡ ( t ) − t a log ⁡ ( z ) ) ( 1 + t 2 ) s / 2 ( e 2 π a t − 1 ) d t {\displaystyle \Phi (z,s,a)={\frac {1}{2a^{s}}}+\int _{0}^{\infty }{\frac {z^{t}}{(a+t)^{s}}}\,dt+{\frac {2}{a^{s-1}}}\int _{0}^{\infty }{\frac {\sin(s\arctan(t)-ta\log(z))}{(1+t^{2})^{s/2}(e^{2\pi at}-1)}}\,dt}

for

ℜ ( a ) > 0 ∧ | z | < 1 {\displaystyle \Re (a)>0\wedge |z|<1}

and

Φ ( z , s , a ) = 1 2 a s + log s − 1 ⁡ ( 1 / z ) z a Γ ( 1 − s , a log ⁡ ( 1 / z ) ) + 2 a s − 1 ∫ 0 ∞ sin ⁡ ( s arctan ⁡ ( t ) − t a log ⁡ ( z ) ) ( 1 + t 2 ) s / 2 ( e 2 π a t − 1 ) d t {\displaystyle \Phi (z,s,a)={\frac {1}{2a^{s}}}+{\frac {\log ^{s-1}(1/z)}{z^{a}}}\Gamma (1-s,a\log(1/z))+{\frac {2}{a^{s-1}}}\int _{0}^{\infty }{\frac {\sin(s\arctan(t)-ta\log(z))}{(1+t^{2})^{s/2}(e^{2\pi at}-1)}}\,dt}

for

ℜ ( a ) > 0. {\displaystyle \Re (a)>0.}

Similar representations include

Φ ( z , s , a ) = 1 2 a s + ∫ 0 ∞ cos ⁡ ( t log ⁡ z ) sin ⁡ ( s arctan ⁡ t a ) − sin ⁡ ( t log ⁡ z ) cos ⁡ ( s arctan ⁡ t a ) ( a 2 + t 2 ) s 2 tanh ⁡ π t d t , {\displaystyle \Phi (z,s,a)={\frac {1}{2a^{s}}}+\int _{0}^{\infty }{\frac {\cos(t\log z)\sin {\Big (}s\arctan {\tfrac {t}{a}}{\Big )}-\sin(t\log z)\cos {\Big (}s\arctan {\tfrac {t}{a}}{\Big )}}{{\big (}a^{2}+t^{2}{\big )}^{\frac {s}{2}}\tanh \pi t}}\,dt,}

and

Φ ( − z , s , a ) = 1 2 a s + ∫ 0 ∞ cos ⁡ ( t log ⁡ z ) sin ⁡ ( s arctan ⁡ t a ) − sin ⁡ ( t log ⁡ z ) cos ⁡ ( s arctan ⁡ t a ) ( a 2 + t 2 ) s 2 sinh ⁡ π t d t , {\displaystyle \Phi (-z,s,a)={\frac {1}{2a^{s}}}+\int _{0}^{\infty }{\frac {\cos(t\log z)\sin {\Big (}s\arctan {\tfrac {t}{a}}{\Big )}-\sin(t\log z)\cos {\Big (}s\arctan {\tfrac {t}{a}}{\Big )}}{{\big (}a^{2}+t^{2}{\big )}^{\frac {s}{2}}\sinh \pi t}}\,dt,}

holding for positive z (and more generally wherever the integrals converge). Furthermore,

Φ ( e i φ , s , a ) = L ( φ 2 π , s , a ) = 1 a s + 1 2 Γ ( s ) ∫ 0 ∞ t s − 1 e − a t ( e i φ − e − t ) cosh ⁡ t − cos ⁡ φ d t , {\displaystyle \Phi (e^{i\varphi },s,a)=L{\big (}{\tfrac {\varphi }{2\pi }},s,a{\big )}={\frac {1}{a^{s}}}+{\frac {1}{2\Gamma (s)}}\int _{0}^{\infty }{\frac {t^{s-1}e^{-at}{\big (}e^{i\varphi }-e^{-t}{\big )}}{\cosh {t}-\cos {\varphi }}}\,dt,}

The last formula is also known as Lipschitz formula.

Identities

For λ rational, the summand is a root of unity, and thus L ( λ , s , α ) {\displaystyle L(\lambda ,s,\alpha )} may be expressed as a finite sum over the Hurwitz zeta function. Suppose λ = p q {\textstyle \lambda ={\frac {p}{q}}} with p , q ∈ Z {\displaystyle p,q\in \mathbb {Z} } and q > 0 {\displaystyle q>0} . Then z = ω = e 2 π i p q {\displaystyle z=\omega =e^{2\pi i{\frac {p}{q}}}} and ω q = 1 {\displaystyle \omega ^{q}=1} .

Φ ( ω , s , α ) = ∑ n = 0 ∞ ω n ( n + α ) s = ∑ m = 0 q − 1 ∑ n = 0 ∞ ω q n + m ( q n + m + α ) s = ∑ m = 0 q − 1 ω m q − s ζ ( s , m + α q ) {\displaystyle \Phi (\omega ,s,\alpha )=\sum _{n=0}^{\infty }{\frac {\omega ^{n}}{(n+\alpha )^{s}}}=\sum _{m=0}^{q-1}\sum _{n=0}^{\infty }{\frac {\omega ^{qn+m}}{(qn+m+\alpha )^{s}}}=\sum _{m=0}^{q-1}\omega ^{m}q^{-s}\zeta \left(s,{\frac {m+\alpha }{q}}\right)}

Various identities include:

Φ ( z , s , a ) = z n Φ ( z , s , a + n ) + ∑ k = 0 n − 1 z k ( k + a ) s {\displaystyle \Phi (z,s,a)=z^{n}\Phi (z,s,a+n)+\sum _{k=0}^{n-1}{\frac {z^{k}}{(k+a)^{s}}}}

and

Φ ( z , s − 1 , a ) = ( a + z ∂ ∂ z ) Φ ( z , s , a ) {\displaystyle \Phi (z,s-1,a)=\left(a+z{\frac {\partial }{\partial z}}\right)\Phi (z,s,a)}

and

Φ ( z , s + 1 , a ) = − 1 s ∂ ∂ a Φ ( z , s , a ) . {\displaystyle \Phi (z,s+1,a)=-{\frac {1}{s}}{\frac {\partial }{\partial a}}\Phi (z,s,a).}

Series representations

A series representation for the Lerch transcendent is given by

Φ ( z , s , q ) = 1 1 − z ∑ n = 0 ∞ ( − z 1 − z ) n ∑ k = 0 n ( − 1 ) k ( n k ) ( q + k ) − s . {\displaystyle \Phi (z,s,q)={\frac {1}{1-z}}\sum _{n=0}^{\infty }\left({\frac {-z}{1-z}}\right)^{n}\sum _{k=0}^{n}(-1)^{k}{\binom {n}{k}}(q+k)^{-s}.}

(Note that ( n k ) {\displaystyle {\tbinom {n}{k}}} is a binomial coefficient.)

The series is valid for all s, and for complex z with Re(z)<1/2. Note a general resemblance to a similar series representation for the Hurwitz zeta function.16

A Taylor series in the first parameter was given by Arthur Erdélyi. It may be written as the following series, which is valid for17

| log ⁡ ( z ) | < 2 π ; s ≠ 1 , 2 , 3 , … ; a ≠ 0 , − 1 , − 2 , … {\displaystyle \left|\log(z)\right|<2\pi ;s\neq 1,2,3,\dots ;a\neq 0,-1,-2,\dots } Φ ( z , s , a ) = z − a [ Γ ( 1 − s ) ( − log ⁡ ( z ) ) s − 1 + ∑ k = 0 ∞ ζ ( s − k , a ) log k ⁡ ( z ) k ! ] {\displaystyle \Phi (z,s,a)=z^{-a}\left[\Gamma (1-s)\left(-\log(z)\right)^{s-1}+\sum _{k=0}^{\infty }\zeta (s-k,a){\frac {\log ^{k}(z)}{k!}}\right]}

If n is a positive integer, then

Φ ( z , n , a ) = z − a { ∑ k = 0 k ≠ n − 1 ∞ ζ ( n − k , a ) log k ⁡ ( z ) k ! + [ ψ ( n ) − ψ ( a ) − log ⁡ ( − log ⁡ ( z ) ) ] log n − 1 ⁡ ( z ) ( n − 1 ) ! } , {\displaystyle \Phi (z,n,a)=z^{-a}\left\{\sum _{{k=0} \atop k\neq n-1}^{\infty }\zeta (n-k,a){\frac {\log ^{k}(z)}{k!}}+\left[\psi (n)-\psi (a)-\log(-\log(z))\right]{\frac {\log ^{n-1}(z)}{(n-1)!}}\right\},}

where ψ ( n ) {\displaystyle \psi (n)} is the digamma function.

A Taylor series in the third variable is given by

Φ ( z , s , a + x ) = ∑ k = 0 ∞ Φ ( z , s + k , a ) ( s ) k ( − x ) k k ! ; | x | < ℜ ( a ) , {\displaystyle \Phi (z,s,a+x)=\sum _{k=0}^{\infty }\Phi (z,s+k,a)(s)_{k}{\frac {(-x)^{k}}{k!}};|x|<\Re (a),}

where ( s ) k {\displaystyle (s)_{k}} is the Pochhammer symbol.

Series at a = −n is given by

Φ ( z , s , a ) = ∑ k = 0 n z k ( a + k ) s + z n ∑ m = 0 ∞ ( 1 − m − s ) m Li s + m ⁡ ( z ) ( a + n ) m m ! ;   a → − n {\displaystyle \Phi (z,s,a)=\sum _{k=0}^{n}{\frac {z^{k}}{(a+k)^{s}}}+z^{n}\sum _{m=0}^{\infty }(1-m-s)_{m}\operatorname {Li} _{s+m}(z){\frac {(a+n)^{m}}{m!}};\ a\rightarrow -n}

A special case for n = 0 has the following series

Φ ( z , s , a ) = 1 a s + ∑ m = 0 ∞ ( 1 − m − s ) m Li s + m ⁡ ( z ) a m m ! ; | a | < 1 , {\displaystyle \Phi (z,s,a)={\frac {1}{a^{s}}}+\sum _{m=0}^{\infty }(1-m-s)_{m}\operatorname {Li} _{s+m}(z){\frac {a^{m}}{m!}};|a|<1,}

where Li s ⁡ ( z ) {\displaystyle \operatorname {Li} _{s}(z)} is the polylogarithm.

An asymptotic series for s → − ∞ {\displaystyle s\rightarrow -\infty }

Φ ( z , s , a ) = z − a Γ ( 1 − s ) ∑ k = − ∞ ∞ [ 2 k π i − log ⁡ ( z ) ] s − 1 e 2 k π a i {\displaystyle \Phi (z,s,a)=z^{-a}\Gamma (1-s)\sum _{k=-\infty }^{\infty }[2k\pi i-\log(z)]^{s-1}e^{2k\pi ai}}

for | a | < 1 ; ℜ ( s ) < 0 ; z ∉ ( − ∞ , 0 ) {\displaystyle |a|<1;\Re (s)<0;z\notin (-\infty ,0)} and

Φ ( − z , s , a ) = z − a Γ ( 1 − s ) ∑ k = − ∞ ∞ [ ( 2 k + 1 ) π i − log ⁡ ( z ) ] s − 1 e ( 2 k + 1 ) π a i {\displaystyle \Phi (-z,s,a)=z^{-a}\Gamma (1-s)\sum _{k=-\infty }^{\infty }[(2k+1)\pi i-\log(z)]^{s-1}e^{(2k+1)\pi ai}}

for | a | < 1 ; ℜ ( s ) < 0 ; z ∉ ( 0 , ∞ ) . {\displaystyle |a|<1;\Re (s)<0;z\notin (0,\infty ).}

An asymptotic series in the incomplete gamma function

Φ ( z , s , a ) = 1 2 a s + 1 z a ∑ k = 1 ∞ e − 2 π i ( k − 1 ) a Γ ( 1 − s , a ( − 2 π i ( k − 1 ) − log ⁡ ( z ) ) ) ( − 2 π i ( k − 1 ) − log ⁡ ( z ) ) 1 − s + e 2 π i k a Γ ( 1 − s , a ( 2 π i k − log ⁡ ( z ) ) ) ( 2 π i k − log ⁡ ( z ) ) 1 − s {\displaystyle \Phi (z,s,a)={\frac {1}{2a^{s}}}+{\frac {1}{z^{a}}}\sum _{k=1}^{\infty }{\frac {e^{-2\pi i(k-1)a}\Gamma (1-s,a(-2\pi i(k-1)-\log(z)))}{(-2\pi i(k-1)-\log(z))^{1-s}}}+{\frac {e^{2\pi ika}\Gamma (1-s,a(2\pi ik-\log(z)))}{(2\pi ik-\log(z))^{1-s}}}}

for | a | < 1 ; ℜ ( s ) < 0. {\displaystyle |a|<1;\Re (s)<0.}

The representation as a generalized hypergeometric function is18

Φ ( z , s , α ) = 1 α s s + 1 F s ( 1 , α , α , α , ⋯ 1 + α , 1 + α , 1 + α , ⋯ ∣ z ) . {\displaystyle \Phi (z,s,\alpha )={\frac {1}{\alpha ^{s}}}{}_{s+1}F_{s}\left({\begin{array}{c}1,\alpha ,\alpha ,\alpha ,\cdots \\1+\alpha ,1+\alpha ,1+\alpha ,\cdots \\\end{array}}\mid z\right).}

Asymptotic expansion

The polylogarithm function L i n ( z ) {\displaystyle \mathrm {Li} _{n}(z)} is defined as

L i 0 ( z ) = z 1 − z , L i − n ( z ) = z d d z L i 1 − n ( z ) . {\displaystyle \mathrm {Li} _{0}(z)={\frac {z}{1-z}},\qquad \mathrm {Li} _{-n}(z)=z{\frac {d}{dz}}\mathrm {Li} _{1-n}(z).}

Let

Ω a ≡ { C ∖ [ 1 , ∞ ) if  ℜ a > 0 , z ∈ C , | z | < 1 if  ℜ a ≤ 0. {\displaystyle \Omega _{a}\equiv {\begin{cases}\mathbb {C} \setminus [1,\infty )&{\text{if }}\Re a>0,\\{z\in \mathbb {C} ,|z|<1}&{\text{if }}\Re a\leq 0.\end{cases}}}

For | A r g ( a ) | < π , s ∈ C {\displaystyle |\mathrm {Arg} (a)|<\pi ,s\in \mathbb {C} } and z ∈ Ω a {\displaystyle z\in \Omega _{a}} , an asymptotic expansion of Φ ( z , s , a ) {\displaystyle \Phi (z,s,a)} for large a {\displaystyle a} and fixed s {\displaystyle s} and z {\displaystyle z} is given by

Φ ( z , s , a ) = 1 1 − z 1 a s + ∑ n = 1 N − 1 ( − 1 ) n L i − n ( z ) n ! ( s ) n a n + s + O ( a − N − s ) {\displaystyle \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})}

for N ∈ N {\displaystyle N\in \mathbb {N} } , where ( s ) n = s ( s + 1 ) ⋯ ( s + n − 1 ) {\displaystyle (s)_{n}=s(s+1)\cdots (s+n-1)} is the Pochhammer symbol.19

Let

f ( z , x , a ) ≡ 1 − ( z e − x ) 1 − a 1 − z e − x . {\displaystyle f(z,x,a)\equiv {\frac {1-(ze^{-x})^{1-a}}{1-ze^{-x}}}.}

Let C n ( z , a ) {\displaystyle C_{n}(z,a)} be its Taylor coefficients at x = 0 {\displaystyle x=0} . Then for fixed N ∈ N , ℜ a > 1 {\displaystyle N\in \mathbb {N} ,\Re a>1} and ℜ s > 0 {\displaystyle \Re s>0} ,

Φ ( z , s , a ) − L i s ( z ) z a = ∑ n = 0 N − 1 C n ( z , a ) ( s ) n a n + s + O ( ( ℜ a ) 1 − N − s + a z − ℜ a ) , {\displaystyle \Phi (z,s,a)-{\frac {\mathrm {Li} _{s}(z)}{z^{a}}}=\sum _{n=0}^{N-1}C_{n}(z,a){\frac {(s)_{n}}{a^{n+s}}}+O\left((\Re a)^{1-N-s}+az^{-\Re a}\right),}

as ℜ a → ∞ {\displaystyle \Re a\to \infty } .20

Software

The Lerch transcendent is implemented as LerchPhi in Maple and Mathematica, and as lerchphi in mpmath and SymPy.

References

  1. Lerch, Mathias (1887), "Note sur la fonction K ( w , x , s ) = ∑ k = 0 ∞ e 2 k π i x ( w + k ) s {\displaystyle \scriptstyle {\mathfrak {K}}(w,x,s)=\sum _{k=0}^{\infty }{e^{2k\pi ix} \over (w+k)^{s}}} ", Acta Mathematica (in French), 11 (1–4): 19–24, doi:10.1007/BF02612318, JFM 19.0438.01, MR 1554747, S2CID 121885446 /wiki/Mathias_Lerch ↩

  2. Guillera & Sondow 2008. - Guillera, Jesus; Sondow, Jonathan (2008), "Double integrals and infinite products for some classical constants via analytic continuations of Lerch's transcendent", The Ramanujan Journal, 16 (3): 247–270, arXiv:math.NT/0506319, doi:10.1007/s11139-007-9102-0, MR 2429900, S2CID 119131640 https://arxiv.org/abs/math.NT/0506319 ↩

  3. Guillera & Sondow 2008, p. 248–249 - Guillera, Jesus; Sondow, Jonathan (2008), "Double integrals and infinite products for some classical constants via analytic continuations of Lerch's transcendent", The Ramanujan Journal, 16 (3): 247–270, arXiv:math.NT/0506319, doi:10.1007/s11139-007-9102-0, MR 2429900, S2CID 119131640 https://arxiv.org/abs/math.NT/0506319 ↩

  4. Guillera & Sondow 2008, p. 248–249 - Guillera, Jesus; Sondow, Jonathan (2008), "Double integrals and infinite products for some classical constants via analytic continuations of Lerch's transcendent", The Ramanujan Journal, 16 (3): 247–270, arXiv:math.NT/0506319, doi:10.1007/s11139-007-9102-0, MR 2429900, S2CID 119131640 https://arxiv.org/abs/math.NT/0506319 ↩

  5. Guillera & Sondow 2008, p. 248–249 - Guillera, Jesus; Sondow, Jonathan (2008), "Double integrals and infinite products for some classical constants via analytic continuations of Lerch's transcendent", The Ramanujan Journal, 16 (3): 247–270, arXiv:math.NT/0506319, doi:10.1007/s11139-007-9102-0, MR 2429900, S2CID 119131640 https://arxiv.org/abs/math.NT/0506319 ↩

  6. Guillera & Sondow 2008, p. 248–249 - Guillera, Jesus; Sondow, Jonathan (2008), "Double integrals and infinite products for some classical constants via analytic continuations of Lerch's transcendent", The Ramanujan Journal, 16 (3): 247–270, arXiv:math.NT/0506319, doi:10.1007/s11139-007-9102-0, MR 2429900, S2CID 119131640 https://arxiv.org/abs/math.NT/0506319 ↩

  7. Guillera & Sondow 2008, p. 248–249 - Guillera, Jesus; Sondow, Jonathan (2008), "Double integrals and infinite products for some classical constants via analytic continuations of Lerch's transcendent", The Ramanujan Journal, 16 (3): 247–270, arXiv:math.NT/0506319, doi:10.1007/s11139-007-9102-0, MR 2429900, S2CID 119131640 https://arxiv.org/abs/math.NT/0506319 ↩

  8. Guillera & Sondow 2008, p. 248–249 - Guillera, Jesus; Sondow, Jonathan (2008), "Double integrals and infinite products for some classical constants via analytic continuations of Lerch's transcendent", The Ramanujan Journal, 16 (3): 247–270, arXiv:math.NT/0506319, doi:10.1007/s11139-007-9102-0, MR 2429900, S2CID 119131640 https://arxiv.org/abs/math.NT/0506319 ↩

  9. Weisstein, Eric W. "Inverse Tangent Integral". mathworld.wolfram.com. Retrieved 2024-10-13. https://mathworld.wolfram.com/InverseTangentIntegral.html ↩

  10. The polygamma function has the series representation ψ ( m ) ( z ) = ( − 1 ) m + 1 m ! ∑ k = 0 ∞ 1 ( z + k ) m + 1 {\displaystyle \psi ^{(m)}(z)=(-1)^{m+1}\,m!\sum _{k=0}^{\infty }{\frac {1}{(z+k)^{m+1}}}} which holds for integer values of m > 0 and any complex z not equal to a negative integer. ↩

  11. Weisstein, Eric W. "Polygamma Function". mathworld.wolfram.com. Retrieved 2024-10-14. https://mathworld.wolfram.com/PolygammaFunction.html ↩

  12. Weisstein, Eric W. "Clausen Function". mathworld.wolfram.com. Retrieved 2024-10-14. https://mathworld.wolfram.com/ClausenFunction.html ↩

  13. Bateman & Erdélyi 1953, p. 27 - Bateman, H.; Erdélyi, A. (1953), Higher Transcendental Functions, Vol. I (PDF), New York: McGraw-Hill http://apps.nrbook.com/bateman/Vol1.pdf ↩

  14. Guillera & Sondow 2008, Lemma 2.1 and 2.2 - Guillera, Jesus; Sondow, Jonathan (2008), "Double integrals and infinite products for some classical constants via analytic continuations of Lerch's transcendent", The Ramanujan Journal, 16 (3): 247–270, arXiv:math.NT/0506319, doi:10.1007/s11139-007-9102-0, MR 2429900, S2CID 119131640 https://arxiv.org/abs/math.NT/0506319 ↩

  15. Bateman & Erdélyi 1953, p. 28 - Bateman, H.; Erdélyi, A. (1953), Higher Transcendental Functions, Vol. I (PDF), New York: McGraw-Hill http://apps.nrbook.com/bateman/Vol1.pdf ↩

  16. "The Analytic Continuation of the Lerch Transcendent and the Riemann Zeta Function". 27 April 2020. Retrieved 28 April 2020. https://www.physicsforums.com/insights/the-analytic-continuation-of-the-lerch-and-the-zeta-functions/ ↩

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