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} .
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 case
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:
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:
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:
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:
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