查看“勒奇超越函数”的源代码

[[File:Lerch transsendent.gif|thumb|Lerch transcendent]]
[[File:Lerch complex plot.gif|thumb|Lerch   plot with complex variable]]
'''勒奇函数'''是一种特殊函数，定义如下

<math>L(z,s,a)=\sum_{n=0}^{\infty}\frac{z^n}{(a+n)^s}</math>


==特例==
;[[赫尔维茨ζ函数]]。当勒奇函数中的z=1时，化为赫尔维茨ζ函数：
<math>L(1,s,a)=\zeta(s,a)</math>
;[[多重对数函数]],当勒奇函数中a=1,则化为多重对数函数
<math>L(z,s,1)=Li_{s}(z)</math>

==积分形式==
<math>L(z,s,a)=\frac{1}{\Gamma(s)}\int_{0}^{\infty}\frac{z^x}{(a+x)^s}dx</math>

==级数展开==
:<math>\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}.</math>


==参考文献==
{{reflist}}
* {{dlmf | id= 25.14 | first= T. M. | last= Apostol | title= Lerch's Transcendent}}.
* {{citation | first1= H. | last1= Bateman | author1-link= Harry Bateman | first2= A. | last2= Erdélyi | author2-link= Arthur Erdélyi | title= Higher Transcendental Functions, Vol. I | year= 1953 | location= New York | publisher= McGraw-Hill | url=http://apps.nrbook.com/bateman/Vol1.pdf}}. (See § 1.11, "The function Ψ(''z'',''s'',''v'')", p.&nbsp;27)
* {{citation | first1= I.S. | last1= Gradshteyn | first2= I.M. | last2= Ryzhik | title= Tables of Integrals, Series, and Products | edition= 4th | location= New York | publisher= Academic Press | year= 1980 | isbn= 0-12-294760-6}}. (see Chapter 9.55)
* {{citation | first1= Jesus | last1= Guillera | first2= Jonathan | last2= Sondow | arxiv= math.NT/0506319 | mr = 2429900 | title= Double integrals and infinite products for some classical constants via analytic continuations of Lerch's transcendent | journal= The Ramanujan Journal | volume= 16 | year= 2008 | pages= 247–270 | issue= 3 | doi= 10.1007/s11139-007-9102-0}}. (Includes various basic identities in the introduction.)
* {{citation | first= M. | last= Jackson | title= On Lerch's transcendent and the basic bilateral hypergeometric series <sub>2</sub>''ψ''<sub>2</sub> | year= 1950 | journal= J. London Math. Soc. | volume= 25 | issue= 3 | pages= 189–196 | doi= 10.1112/jlms/s1-25.3.189 | mr= 0036882}}.
* {{citation | first1= Antanas | last1= Laurinčikas | first2= Ramūnas | last2= Garunkštis | title= The Lerch zeta-function | publisher= Kluwer Academic Publishers | location= Dordrecht | year= 2002 | isbn= 978-1-4020-1014-9 | mr= 1979048}}.
* {{citation | first= Mathias | last= Lerch | authorlink= Mathias Lerch | title= Note sur la fonction <math>\scriptstyle{\mathfrak K}(w,x,s) = \sum_{k=0}^\infty {e^{2k\pi ix} \over (w+k)^s}</math> | language= fr | year= 1887 | journal= Acta Mathematica | volume= 11 | issue= 1–4 | pages= 19–24 | doi= 10.1007/BF02612318 | mr= 1554747 | jfm= 19.0438.01}}.

[[Category:特殊函数]]