勒奇超越函數

勒奇超越函數是一種特殊函數，推廣了赫爾維茨ζ函數和多重對數函數，定義如下

Lerch transcendent

Lerch plot with complex variable

${\displaystyle L(z,s,a)=\sum _{n=0}^{\infty }{\frac {z^{n}}{(a+n)^{s}}}}$

特例

赫爾維茨ζ函數。當勒奇函數中的z=1時，化為赫爾維茨ζ函數：

${\displaystyle L(1,s,a)=\zeta (s,a)}$

多重對數函數,當勒奇函數中a=1,則化為多重對數函數

${\displaystyle L(z,s,1)=Li_{s}(z)}$

勒讓德χ函數可以用勒奇超越函數表示,

${\displaystyle \,\chi _{n}(z)=2^{-n}z\Phi (z^{2},n,1/2).}$

作為赫爾維茨ζ函數的特例，黎曼ζ函數可以表示為

${\displaystyle \,\zeta (s)=\Phi (1,s,1).}$

狄利克雷η函數可以表示為

${\displaystyle \,\eta (s)=\Phi (-1,s,1).}$

積分形式

${\displaystyle L(z,s,a)={\frac {1}{\Gamma (s)}}\int _{0}^{\infty }{\frac {z^{x}}{(a+x)^{s}}}dx}$

級數展開

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

參考文獻

• Apostol, T. M., Lerch's Transcendent, Olver, Frank W. J.; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W. (編), NIST Handbook of Mathematical Functions, Cambridge University Press, 2010, ISBN 978-0521192255, MR2723248.
• Bateman, H.; Erdélyi, A., Higher Transcendental Functions, Vol. I (PDF), New York: McGraw-Hill, 1953 [2015-02-14], （原始內容存檔 (PDF)於2011-08-11）. (See § 1.11, "The function Ψ(z,s,v)", p. 27)
• Gradshteyn, I.S.; Ryzhik, I.M., Tables of Integrals, Series, and Products 4th, New York: Academic Press, 1980, ISBN 0-12-294760-6. (see Chapter 9.55)
• Guillera, Jesus; Sondow, Jonathan, Double integrals and infinite products for some classical constants via analytic continuations of Lerch's transcendent, The Ramanujan Journal, 2008, 16 (3): 247–270, MR 2429900, arXiv:math.NT/0506319 , doi:10.1007/s11139-007-9102-0. (Includes various basic identities in the introduction.)
• Jackson, M., On Lerch's transcendent and the basic bilateral hypergeometric series ₂ψ₂, J. London Math. Soc., 1950, 25 (3): 189–196, MR 0036882, doi:10.1112/jlms/s1-25.3.189.
• Laurinčikas, Antanas; Garunkštis, Ramūnas, The Lerch zeta-function, Dordrecht: Kluwer Academic Publishers, 2002, ISBN 978-1-4020-1014-9, MR 1979048.
• Lerch, Mathias, Note sur la fonction ${\displaystyle \scriptstyle {\mathfrak {K}}(w,x,s)=\sum _{k=0}^{\infty }{e^{2k\pi ix} \over (w+k)^{s}}}$, Acta Mathematica, 1887, 11 (1–4): 19–24, JFM 19.0438.01, MR 1554747, doi:10.1007/BF02612318 （法語）.