Lerchs transcendent är en speciell funktion som generaliserar Hurwitzs zetafunktion och många andra kända funktioner. Funktionen är uppkallad efter Matyáš Lerch . Dess definition är
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}}}.}
En integralrepresentation för Lerchs transcendent ges av
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}
då
ℜ
(
a
)
>
0
∧
ℜ
(
s
)
>
0
∧
z
<
1
∨
ℜ
(
a
)
>
0
∧
ℜ
(
s
)
>
1
∧
z
=
1.
{\displaystyle \Re (a)>0\wedge \Re (s)>0\wedge z<1\vee \Re (a)>0\wedge \Re (s)>1\wedge z=1.}
En annan integralrepresentation ges av
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}
då
ℜ
(
a
)
>
0.
{\displaystyle \Re (a)>0.}
Hurwitzs zetafunktion är ett specialfall:
ζ
(
s
,
α
)
=
L
(
0
,
α
,
s
)
=
Φ
(
1
,
s
,
α
)
.
{\displaystyle \,\zeta (s,\alpha )=L(0,\alpha ,s)=\Phi (1,s,\alpha ).}
Polylogaritmen är också ett specialfall:
Li
s
(
z
)
=
z
Φ
(
z
,
s
,
1
)
.
{\displaystyle \,{\textrm {Li}}_{s}(z)=z\Phi (z,s,1).}
Legendres chifunktion ges av
χ
n
(
z
)
=
2
−
n
z
Φ
(
z
2
,
n
,
1
/
2
)
.
{\displaystyle \,\chi _{n}(z)=2^{-n}z\Phi (z^{2},n,1/2).}
Riemanns zetafunktion ges av
ζ
(
s
)
=
Φ
(
1
,
s
,
1
)
.
{\displaystyle \,\zeta (s)=\Phi (1,s,1).}
Dirichlets etafunktion ges av
η
(
s
)
=
Φ
(
−
1
,
s
,
1
)
.
{\displaystyle \,\eta (s)=\Phi (-1,s,1).}
Andra specialfall ges av
Φ
(
z
,
s
,
1
)
=
L
i
s
(
z
)
z
{\displaystyle \Phi (z,s,1)={\frac {\mathrm {Li} _{s}(z)}{z}}}
Φ
(
z
,
0
,
a
)
=
1
1
−
z
{\displaystyle \Phi (z,0,a)={\frac {1}{1-z}}}
Φ
(
0
,
s
,
a
)
=
(
a
2
)
−
s
2
{\displaystyle \Phi (0,s,a)=\left(a^{2}\right)^{-{\frac {s}{2}}}}
Φ
(
0
,
s
,
a
)
=
a
−
s
{\displaystyle \Phi (0,s,a)=a^{-s}\,}
Φ
(
z
,
1
,
1
)
=
−
log
(
1
−
z
)
z
{\displaystyle \Phi (z,1,1)=-{\frac {\log(1-z)}{z}}}
Φ
(
1
,
s
,
1
2
)
=
(
2
s
−
1
)
ζ
(
s
)
{\displaystyle \Phi (1,s,{\tfrac {1}{2}})=(2^{s}-1)\zeta (s)}
Φ
(
−
1
,
s
,
1
)
=
(
1
−
2
1
−
s
)
ζ
(
s
)
{\displaystyle \Phi (-1,s,1)=(1-2^{1-s})\zeta (s)\,}
Φ
(
0
,
1
,
a
)
=
1
a
2
{\displaystyle \Phi (0,1,a)={\frac {1}{\sqrt {a^{2}}}}}
Flera kända konstanter kan skrivas med hjälp av Lerchs trascendent:
Φ
(
−
1
,
2
,
1
2
)
=
4
G
∂
Φ
∂
s
(
−
1
,
−
1
,
1
)
=
log
(
A
3
2
3
e
4
)
∂
Φ
∂
s
(
−
1
,
−
2
,
1
)
=
7
ζ
(
3
)
4
π
2
∂
Φ
∂
s
(
−
1
,
−
1
,
1
2
)
=
G
π
{\displaystyle {\begin{aligned}&\Phi (-1,2,{\tfrac {1}{2}})&=&\;4\,G\\&{\frac {\partial \Phi }{\partial s}}(-1,-1,1)&=&\;\log \left({\frac {A^{3}}{{\sqrt[{3}]{2}}\,{\sqrt[{4}]{\mathrm {e} }}}}\right)\\&{\frac {\partial \Phi }{\partial s}}(-1,-2,1)&=&\;{\frac {7\,\zeta (3)}{4\,\pi ^{2}}}\\&{\frac {\partial \Phi }{\partial s}}(-1,-1,{\tfrac {1}{2}})&=&\;{\frac {G}{\pi }}\end{aligned}}}
där
G
{\displaystyle G}
Catalans konstant ,
A
{\displaystyle A}
Glaisher–Kinkelins konstant och
ζ
(
3
)
{\displaystyle \zeta (3)}
Apérys konstant .
Lerchs transcendent satisfierar ett stort antal identiteter, såsom
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}}}}
och
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)}
och
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).}
Då Re(z )<1/2 kan Lerchs transcendent skrivas som
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}.}
(Notera att
(
n
k
)
{\displaystyle {\tbinom {n}{k}}}
binomialkoefficient .)
Om s är ett positivt heltal är
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\},}
då
ψ
(
n
)
{\displaystyle \psi (n)}
digammafunktionen .
En Taylorserie i tredje variabeln ges av
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),}
där
(
s
)
k
{\displaystyle (s)_{k}}
Pochhammersymbolen .
En serie med ofullständiga gammafunktionen är
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
|
a
|
<
1
,
R
e
s
<
0.
{\displaystyle \Phi (z,s,a)={\frac {1}{2\,a^{s}}}+{\frac {1}{z^{a}}}\sum _{k=1}^{\infty }{\frac {\mathrm {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 {\mathrm {e} ^{2\,\pi \,i\,k\,a}\,\Gamma (1-s,a\,(2\,\pi \,i\,k-\log z))}{(2\,\pi \,i\,k-\log z)^{1-s}}}\quad |a|<1,\;\mathrm {Re} \,s<0.}
Bateman, H. ; Erdélyi, A. (1953), Higher Transcendental Functions, Vol. I , http://apps.nrbook.com/bateman/Vol1.pdf z ,s ,v )", p. 27)Gradshteyn, I.S.; Ryzhik, I.M. (1980), Tables of Integrals, Series, and Products ISBN 0-12-294760-6 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, doi :10.1007/s11139-007-9102-0 Jackson, M. (1950), ”On Lerch's transcendent and the basic bilateral hypergeometric series 2 ψ 2 ”, J. London Math. Soc. 25 (3): 189–196, doi :10.1112/jlms/s1-25.3.189 Laurinčikas, Antanas; Garunkštis, Ramūnas (2002), The Lerch zeta-function ISBN 978-1-4020-1014-9 Lerch, Matyáš (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 11 (1–4): 19–24, doi :10.1007/BF02612318 
![{\displaystyle {\begin{aligned}&\Phi (-1,2,{\tfrac {1}{2}})&=&\;4\,G\\&{\frac {\partial \Phi }{\partial s}}(-1,-1,1)&=&\;\log \left({\frac {A^{3}}{{\sqrt[{3}]{2}}\,{\sqrt[{4}]{\mathrm {e} }}}}\right)\\&{\frac {\partial \Phi }{\partial s}}(-1,-2,1)&=&\;{\frac {7\,\zeta (3)}{4\,\pi ^{2}}}\\&{\frac {\partial \Phi }{\partial s}}(-1,-1,{\tfrac {1}{2}})&=&\;{\frac {G}{\pi }}\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/604103b7e47a2180ac4304d0b2d034077e64001f)
![{\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\},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6f97bb2052f7804ca5b0247792e7a427e3f9cee9)
