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Lerch transcendent
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<h2 id="special-cases">Special cases</h2> <p>The Lerch transcendent is related to and generalizes various special functions. </p><p>The Lerch zeta function is given by: </p> 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 )} <p>The <a href="/facts/Hurwitz_zeta_function/4EH7sKXB">Hurwitz zeta function</a> is the special case<a class="footnote-ref" id="fnref:3" href="#fn:3"><sup>3</sup></a> </p> 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 )} <p>The <a href="/facts/Polylogarithm/CKrkJMKn">polylogarithm</a> is another special case:<a class="footnote-ref" id="fnref:4" href="#fn:4"><sup>4</sup></a> </p> 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)} <p>The <a href="/facts/Riemann_zeta_function/MnfRu7lY">Riemann zeta function</a> is a special case of both of the above:<a class="footnote-ref" id="fnref:5" href="#fn:5"><sup>5</sup></a> </p> 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)} <p>The <a href="/facts/Dirichlet_eta_function/32tUb6a7">Dirichlet eta function</a>:<a class="footnote-ref" id="fnref:6" href="#fn:6"><sup>6</sup></a> </p> 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)} <p>The <a href="/facts/Dirichlet_beta_function/FKwkVguL">Dirichlet beta function</a>:<a class="footnote-ref" id="fnref:7" href="#fn:7"><sup>7</sup></a> </p> 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}})} <p>The <a href="/facts/Legendre_chi_function/QF7pSJNt">Legendre chi function</a>:<a class="footnote-ref" id="fnref:8" href="#fn:8"><sup>8</sup></a> </p> 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}})} <p>The <a href="/facts/Inverse_tangent_integral/CKrkJMKn">inverse tangent integral</a>:<a class="footnote-ref" id="fnref:9" href="#fn:9"><sup>9</sup></a> </p> 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}})} <p>The <a href="/facts/Polygamma_function/IF6W68aL">polygamma functions</a> for positive integers <i>n</i>:<a class="footnote-ref" id="fnref:10" href="#fn:10"><sup>10</sup></a><a class="footnote-ref" id="fnref:11" href="#fn:11"><sup>11</sup></a> </p> 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 )} <p>The <a href="/facts/Clausen_function/1mDkvwcL">Clausen function</a>:<a class="footnote-ref" id="fnref:12" href="#fn:12"><sup>12</sup></a> </p> 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)} <h2 id="integral-representations">Integral representations</h2> <p>The Lerch transcendent has an integral representation: </p> 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} <p>The proof is based on using the integral definition of the <a href="/facts/Gamma_function/SjGQcnyw">gamma function</a> to write </p> 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}}} <p>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(<i>s</i>) > 0, and Re(<i>a</i>) > 0. This <a href="/facts/Analytic_continuation/aBCuEgvt">analytically continues</a> Φ ( z , s , a ) {\displaystyle \Phi (z,s,a)} to <i>z</i> outside the unit disk. The integral formula also holds if <i>z</i> = 1, Re(<i>s</i>) > 1, and Re(<i>a</i>) > 0; see <a href="/facts/Hurwitz_zeta_function/4EH7sKXB">Hurwitz zeta function</a>.<a class="footnote-ref" id="fnref:13" href="#fn:13"><sup>13</sup></a><a class="footnote-ref" id="fnref:14" href="#fn:14"><sup>14</sup></a> </p><p>A <a href="/facts/Contour_integral/4ILprcRr">contour integral</a> representation is given by </p> 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} <p>where <i>C</i> is a <a href="/facts/Hankel_contour/cGD27Hus">Hankel contour</a> 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 <i>k</i>) which are <a href="/facts/Pole_(complex_analysis)/gnA3U8VP">poles</a> of the integrand. The integral assumes Re(<i>a</i>) > 0.<a class="footnote-ref" id="fnref:15" href="#fn:15"><sup>15</sup></a> </p> <h3>Other integral representations</h3> <p>A Hermite-like integral representation is given by </p> 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} <p>for </p> ℜ ( a ) > 0 ∧ | z | < 1 {\displaystyle \Re (a)>0\wedge |z|<1} <p>and </p> 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} <p>for </p> ℜ ( a ) > 0. {\displaystyle \Re (a)>0.} <p>Similar representations include </p> 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,} <p>and </p> 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,} <p>holding for positive <i>z</i> (and more generally wherever the integrals converge). Furthermore, </p> 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,} <p>The last formula is also known as <i>Lipschitz formula</i>. </p> <h2 id="identities">Identities</h2> <p>For λ rational, the summand is a <a href="/facts/Root_of_unity/EqH0ho1Y">root of unity</a>, 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} . </p> 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)} <p>Various identities include: </p> 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}}}} <p>and </p> 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)} <p>and </p> 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).} <h2 id="series-representations">Series representations</h2> <p>A series representation for the Lerch transcendent is given by </p> 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}.} <p>(Note that ( n k ) {\displaystyle {\tbinom {n}{k}}} is a <a href="/facts/Binomial_coefficient/Tsx7eY5h">binomial coefficient</a>.) </p><p>The series is valid for all <i>s</i>, and for complex <i>z</i> with Re(<i>z</i>)<1/2. Note a general resemblance to a similar series representation for the Hurwitz zeta function.<a class="footnote-ref" id="fnref:16" href="#fn:16"><sup>16</sup></a> </p><p>A <a href="/facts/Taylor_series/M0aTuftH">Taylor series</a> in the first parameter was given by <a href="/facts/Arthur_Erd%25C3%25A9lyi/LC2upm0N">Arthur Erdélyi</a>. It may be written as the following series, which is valid for<a class="footnote-ref" id="fnref:17" href="#fn:17"><sup>17</sup></a> </p> | 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]} <p>If <i>n</i> is a positive integer, then </p> 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\},} <p>where ψ ( n ) {\displaystyle \psi (n)} is the <a href="/facts/Digamma_function/nDTNu86e">digamma function</a>. </p><p>A <a href="/facts/Taylor_series/M0aTuftH">Taylor series</a> in the third variable is given by </p> 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),} <p>where ( s ) k {\displaystyle (s)_{k}} is the <a href="/facts/Pochhammer_symbol/sAxnmGoC">Pochhammer symbol</a>. </p><p>Series at <i>a</i> = −<i>n</i> is given by </p> 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} <p>A special case for <i>n</i> = 0 has the following series </p> 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,} <p>where Li s ( z ) {\displaystyle \operatorname {Li} _{s}(z)} is the <a href="/facts/Polylogarithm/CKrkJMKn">polylogarithm</a>. </p><p>An <a href="/facts/Asymptotic_series/4W79BNve">asymptotic series</a> for s → − ∞ {\displaystyle s\rightarrow -\infty } </p> 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}} <p>for | a | < 1 ; ℜ ( s ) < 0 ; z ∉ ( − ∞ , 0 ) {\displaystyle |a|<1;\Re (s)<0;z\notin (-\infty ,0)} and </p> 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}} <p>for | a | < 1 ; ℜ ( s ) < 0 ; z ∉ ( 0 , ∞ ) . {\displaystyle |a|<1;\Re (s)<0;z\notin (0,\infty ).} </p><p>An asymptotic series in the <a href="/facts/Incomplete_gamma_function/vojnYSu4">incomplete gamma function</a> </p> 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}}}} <p>for | a | < 1 ; ℜ ( s ) < 0. {\displaystyle |a|<1;\Re (s)<0.} </p><p>The representation as a generalized hypergeometric function is<a class="footnote-ref" id="fnref:18" href="#fn:18"><sup>18</sup></a> </p> 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).} <h2 id="asymptotic-expansion">Asymptotic expansion</h2> <p>The polylogarithm function L i n ( z ) {\displaystyle \mathrm {Li} _{n}(z)} is defined as </p> 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).} <p>Let </p> 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}}} <p>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 </p> 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})} <p>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 <a href="/facts/Falling_and_rising_factorials/sAxnmGoC">Pochhammer symbol</a>.<a class="footnote-ref" id="fnref:19" href="#fn:19"><sup>19</sup></a> </p><p>Let </p> 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}}}.} <p>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} , </p> 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),} <p>as ℜ a → ∞ {\displaystyle \Re a\to \infty } .<a class="footnote-ref" id="fnref:20" href="#fn:20"><sup>20</sup></a> </p> <h2 id="software">Software</h2> <p>The Lerch transcendent is implemented as LerchPhi in <a href="http://www.maplesoft.com/support/help/Maple/view.aspx?path=LerchPhi">Maple</a> and <a href="https://functions.wolfram.com/ZetaFunctionsandPolylogarithms/LerchPhi/">Mathematica</a>, and as lerchphi in <a href="http://mpmath.org/doc/current/functions/zeta.html#lerch-transcendent">mpmath</a> and <a href="https://docs.sympy.org/latest/modules/functions/special.html#sympy.functions.special.zeta_functions.lerchphi">SymPy</a>. </p> <ul><li>Apostol, T. M. (2010), <a href="http://dlmf.nist.gov/25.14">"Lerch's Transcendent"</a>, in <a href="/facts/Frank_W._J._Olver/nQYQVRHc">Olver, Frank W. J.</a>; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W. (eds.), <i><a href="/facts/Digital_Library_of_Mathematical_Functions/ymNQBbgK">NIST Handbook of Mathematical Functions</a></i>, Cambridge University Press, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-521-19225-5, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=2723248">2723248</a>..</li> <li><a href="/facts/Harry_Bateman/9tRXIjI2">Bateman, H.</a>; <a href="/facts/Arthur_Erd%25C3%25A9lyi/LC2upm0N">Erdélyi, A.</a> (1953), <a href="http://apps.nrbook.com/bateman/Vol1.pdf"><i>Higher Transcendental Functions, Vol. I</i></a> (PDF), New York: McGraw-Hill. (See § 1.11, "The function Ψ(<i>z</i>,<i>s</i>,<i>v</i>)", p. 27)</li> <li><a href="/facts/Izrail_Solomonovich_Gradshteyn/SKShStsu">Gradshteyn, Izrail Solomonovich</a>; <a href="/facts/Iosif_Moiseevich_Ryzhik/SKShStsu">Ryzhik, Iosif Moiseevich</a>; <a href="/facts/Yuri_Veniaminovich_Geronimus/SKShStsu">Geronimus, Yuri Veniaminovich</a>; <a href="/facts/Michail_Yulyevich_Tseytlin/SKShStsu">Tseytlin, Michail Yulyevich</a>; Jeffrey, Alan (2015) [October 2014]. "9.55.". In Zwillinger, Daniel; <a href="/facts/Victor_Hugo_Moll/p09LtKPa">Moll, Victor Hugo</a> (eds.). <a href="/facts/Gradshteyn_and_Ryzhik/SKShStsu"><i>Table of Integrals, Series, and Products</i></a>. Translated by Scripta Technica, Inc. (8 ed.). Academic Press. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-12-384933-5. <a href="/facts/LCCN_(identifier)/5hb5Cw2V">LCCN</a> <a href="https://lccn.loc.gov/2014010276">2014010276</a>.</li> <li>Guillera, Jesus; Sondow, Jonathan (2008), "Double integrals and infinite products for some classical constants via analytic continuations of Lerch's transcendent", <i>The Ramanujan Journal</i>, 16 (3): 247–270, <a href="/facts/ArXiv_(identifier)/H6EtgnBe">arXiv</a>:<a href="https://arxiv.org/abs/math.NT/0506319">math.NT/0506319</a>, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1007%2Fs11139-007-9102-0">10.1007/s11139-007-9102-0</a>, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=2429900">2429900</a>, <a href="/facts/S2CID_(identifier)/ldJsHa2Y">S2CID</a> <a href="https://api.semanticscholar.org/CorpusID:119131640">119131640</a>. (Includes various basic identities in the introduction.)</li> <li>Jackson, M. (1950), "On Lerch's transcendent and the basic bilateral hypergeometric series 2<i>ψ</i>2", <i>J. London Math. Soc.</i>, 25 (3): 189–196, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1112%2Fjlms%2Fs1-25.3.189">10.1112/jlms/s1-25.3.189</a>, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0036882">0036882</a>.</li> <li>Johansson, F.; Blagouchine, Ia. (2019), "Computing Stieltjes constants using complex integration", <i>Mathematics of Computation</i>, 88 (318): 1829–1850, <a href="/facts/ArXiv_(identifier)/H6EtgnBe">arXiv</a>:<a href="https://arxiv.org/abs/1804.01679">1804.01679</a>, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1090%2Fmcom%2F3401">10.1090/mcom/3401</a>, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=3925487">3925487</a>, <a href="/facts/S2CID_(identifier)/ldJsHa2Y">S2CID</a> <a href="https://api.semanticscholar.org/CorpusID:4619883">4619883</a>.</li> <li>Laurinčikas, Antanas; Garunkštis, Ramūnas (2002), <i>The Lerch zeta-function</i>, Dordrecht: Kluwer Academic Publishers, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-1-4020-1014-9, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=1979048">1979048</a>.</li></ul> <h2 id="external-links">External links</h2> <ul><li>Aksenov, Sergej V.; Jentschura, Ulrich D. (2002), <a href="http://aksenov.freeshell.org/lerchphi.html"><i>C and Mathematica Programs for Calculation of Lerch's Transcendent</i></a>.</li> <li>Ramunas Garunkstis, <i><a href="http://www.mif.vu.lt/~garunkstis">Home Page</a></i> (2005) <i>(Provides numerous references and preprints.)</i></li> <li>Garunkstis, Ramunas (2004). <a href="http://www.mif.vu.lt/~garunkstis/preprintai/approx.pdf">"Approximation of the Lerch Zeta Function"</a> (PDF). <i>Lithuanian Mathematical Journal</i>. 44 (2): 140–144. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1023%2FB%3ALIMA.0000033779.41365.a5">10.1023/B:LIMA.0000033779.41365.a5</a>. <a href="/facts/S2CID_(identifier)/ldJsHa2Y">S2CID</a> <a href="https://api.semanticscholar.org/CorpusID:123059665">123059665</a>.</li> <li>Kanemitsu, S.; Tanigawa, Y.; Tsukada, H. (2015). <a href="https://hal.archives-ouvertes.fr/hal-02220916">"A generalization of Bochner's formula"</a>. Kanemitsu, S.; Tanigawa, Y.; Tsukada, H. (2004). <a href="https://doi.org/10.46298%2Fhrj.2004.150">"A generalization of Bochner's formula"</a>. <i>Hardy-Ramanujan Journal</i>. 27. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.46298%2Fhrj.2004.150">10.46298/hrj.2004.150</a>.</li> <li><a href="/facts/Eric_W._Weisstein/FbMuVDep">Weisstein, Eric W.</a> <a href="https://mathworld.wolfram.com/LerchTranscendent.html">"Lerch Transcendent"</a>. <i><a href="/facts/MathWorld/yc1jodbq">MathWorld</a></i>.</li> <li><a href="/facts/Frank_W._J._Olver/nQYQVRHc">Olver, Frank W. J.</a>; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W., eds. (2010), <a href="http://dlmf.nist.gov/25.14">"Lerch's Transcendent"</a>, <i><a href="/facts/Digital_Library_of_Mathematical_Functions/ymNQBbgK">NIST Handbook of Mathematical Functions</a></i>, Cambridge University Press, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-521-19225-5, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=2723248">2723248</a>.</li></ul> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>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 <a href="/wiki/Mathias_Lerch" target="_blank">/wiki/Mathias_Lerch</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> <li id="fn:2"><p>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 <a href="https://arxiv.org/abs/math.NT/0506319" target="_blank">https://arxiv.org/abs/math.NT/0506319</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li> <li id="fn:3"><p>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 <a href="https://arxiv.org/abs/math.NT/0506319" target="_blank">https://arxiv.org/abs/math.NT/0506319</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li> <li id="fn:4"><p>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 <a href="https://arxiv.org/abs/math.NT/0506319" target="_blank">https://arxiv.org/abs/math.NT/0506319</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li> <li id="fn:5"><p>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 <a href="https://arxiv.org/abs/math.NT/0506319" target="_blank">https://arxiv.org/abs/math.NT/0506319</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></p></li> <li id="fn:6"><p>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 <a href="https://arxiv.org/abs/math.NT/0506319" target="_blank">https://arxiv.org/abs/math.NT/0506319</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></p></li> <li id="fn:7"><p>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 <a href="https://arxiv.org/abs/math.NT/0506319" target="_blank">https://arxiv.org/abs/math.NT/0506319</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></p></li> <li id="fn:8"><p>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 <a href="https://arxiv.org/abs/math.NT/0506319" target="_blank">https://arxiv.org/abs/math.NT/0506319</a> <a href="#fnref:8" class="footnote-back-ref">↩</a></p></li> <li id="fn:9"><p>Weisstein, Eric W. "Inverse Tangent Integral". mathworld.wolfram.com. Retrieved 2024-10-13. <a href="https://mathworld.wolfram.com/InverseTangentIntegral.html" target="_blank">https://mathworld.wolfram.com/InverseTangentIntegral.html</a> <a href="#fnref:9" class="footnote-back-ref">↩</a></p></li> <li id="fn:10"><p>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. <a href="#fnref:10" class="footnote-back-ref">↩</a></p></li> <li id="fn:11"><p>Weisstein, Eric W. "Polygamma Function". mathworld.wolfram.com. Retrieved 2024-10-14. <a href="https://mathworld.wolfram.com/PolygammaFunction.html" target="_blank">https://mathworld.wolfram.com/PolygammaFunction.html</a> <a href="#fnref:11" class="footnote-back-ref">↩</a></p></li> <li id="fn:12"><p>Weisstein, Eric W. "Clausen Function". mathworld.wolfram.com. Retrieved 2024-10-14. <a href="https://mathworld.wolfram.com/ClausenFunction.html" target="_blank">https://mathworld.wolfram.com/ClausenFunction.html</a> <a href="#fnref:12" class="footnote-back-ref">↩</a></p></li> <li id="fn:13"><p>Bateman & Erdélyi 1953, p. 27 - Bateman, H.; Erdélyi, A. (1953), Higher Transcendental Functions, Vol. I (PDF), New York: McGraw-Hill <a href="http://apps.nrbook.com/bateman/Vol1.pdf" target="_blank">http://apps.nrbook.com/bateman/Vol1.pdf</a> <a href="#fnref:13" class="footnote-back-ref">↩</a></p></li> <li id="fn:14"><p>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 <a href="https://arxiv.org/abs/math.NT/0506319" target="_blank">https://arxiv.org/abs/math.NT/0506319</a> <a href="#fnref:14" class="footnote-back-ref">↩</a></p></li> <li id="fn:15"><p>Bateman & Erdélyi 1953, p. 28 - Bateman, H.; Erdélyi, A. (1953), Higher Transcendental Functions, Vol. I (PDF), New York: McGraw-Hill <a href="http://apps.nrbook.com/bateman/Vol1.pdf" target="_blank">http://apps.nrbook.com/bateman/Vol1.pdf</a> <a href="#fnref:15" class="footnote-back-ref">↩</a></p></li> <li id="fn:16"><p>"The Analytic Continuation of the Lerch Transcendent and the Riemann Zeta Function". 27 April 2020. Retrieved 28 April 2020. <a href="https://www.physicsforums.com/insights/the-analytic-continuation-of-the-lerch-and-the-zeta-functions/" target="_blank">https://www.physicsforums.com/insights/the-analytic-continuation-of-the-lerch-and-the-zeta-functions/</a> <a href="#fnref:16" class="footnote-back-ref">↩</a></p></li> <li id="fn:17"><p>B. R. Johnson (1974). "Generalized Lerch zeta function". Pacific J. Math. 53 (1): 189–193. doi:10.2140/pjm.1974.53.189. <a href="https://doi.org/10.2140%2Fpjm.1974.53.189" target="_blank">https://doi.org/10.2140%2Fpjm.1974.53.189</a> <a href="#fnref:17" class="footnote-back-ref">↩</a></p></li> <li id="fn:18"><p>Gottschalk, J. E.; Maslen, E. N. (1988). "Reduction formulae for generalized hypergeometric functions of one variable". J. Phys. A. 21 (9): 1983–1998. Bibcode:1988JPhA...21.1983G. doi:10.1088/0305-4470/21/9/015. <a href="/wiki/Hypergeometric_function" target="_blank">/wiki/Hypergeometric_function</a> <a href="#fnref:18" class="footnote-back-ref">↩</a></p></li> <li id="fn:19"><p>Ferreira, Chelo; López, José L. (October 2004). "Asymptotic expansions of the Hurwitz–Lerch zeta function". Journal of Mathematical Analysis and Applications. 298 (1): 210–224. doi:10.1016/j.jmaa.2004.05.040. <a href="https://doi.org/10.1016%2Fj.jmaa.2004.05.040" target="_blank">https://doi.org/10.1016%2Fj.jmaa.2004.05.040</a> <a href="#fnref:19" class="footnote-back-ref">↩</a></p></li> <li id="fn:20"><p>Cai, Xing Shi; López, José L. (10 June 2019). "A note on the asymptotic expansion of the Lerch's transcendent". Integral Transforms and Special Functions. 30 (10): 844–855. arXiv:1806.01122. doi:10.1080/10652469.2019.1627530. S2CID 119619877. <a href="/wiki/ArXiv_(identifier)" target="_blank">/wiki/ArXiv_(identifier)</a> <a href="#fnref:20" class="footnote-back-ref">↩</a></p></li> </ol>