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.
[1] The Lerch transcendent, is given by: It only converges for any real number
[2] The Lerch transcendent is related to and generalizes various special functions.
The Lerch zeta function is given by: The Hurwitz zeta function is the special case[3] The polylogarithm is another special case:[3] The Riemann zeta function is a special case of both of the above:[3] The Dirichlet eta function:[3] The Dirichlet beta function:[3] The Legendre chi function:[3] The inverse tangent integral:[4] The polygamma functions for positive integers n:[5][6] The Clausen function:[7] The Lerch transcendent has an integral representation: The proof is based on using the integral definition of the Gamma function to write and then interchanging the sum and integral.
The resulting integral representation converges for
This analytically continues
to z outside the unit disk.
The integral formula also holds if z = 1, Re(s) > 1, and Re(a) > 0; see Hurwitz zeta function.
[8][9] A contour integral representation is given by where C is a Hankel contour counterclockwise around the positive real axis, not enclosing any of the points
t = log ( z ) + 2 k π i
(for integer k) which are poles of the integrand.
[10] A Hermite-like integral representation is given by for and for Similar representations include and holding for positive z (and more generally wherever the integrals converge).
For λ rational, the summand is a root of unity, and thus
( λ , s , α )
may be expressed as a finite sum over the Hurwitz zeta function.
Various identities include: and and A series representation for the Lerch transcendent is given by (Note that
is a binomial coefficient.)
The series is valid for all s, and for complex z with Re(z)<1/2.
Note a general resemblance to a similar series representation for the Hurwitz zeta function.
[11] A Taylor series in the first parameter was given by Arthur Erdélyi.
It may be written as the following series, which is valid for[12] If n is a positive integer, then where
A Taylor series in the third variable is given by where
is the Pochhammer symbol.
An asymptotic series for
An asymptotic series in the incomplete gamma function for
The representation as a generalized hypergeometric function is[13] The polylogarithm function
, an asymptotic expansion of
be its Taylor coefficients at
[15] The Lerch transcendent is implemented as LerchPhi in Maple and Mathematica, and as lerchphi in mpmath and SymPy.