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:
It only converges for any real number , where , or , and .
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
The polylogarithm is another special case:
The Riemann zeta function is a special case of both of the above:
The polygamma functions for positive integers n:
The Clausen function:
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 Re(s) > 0, and Re(a) > 0. 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.
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 (for integer k) which are poles of the integrand. The integral assumes Re(a) > 0.
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). Furthermore,
The last formula is also known as Lipschitz formula.
For û rational, the summand is a root of unity, and thus may be expressed as a finite sum over the Hurwitz zeta function. Suppose with and . Then and .
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.
A Taylor series in the first parameter was given by Arthur Erdélyi. It may be written as the following series, which is valid for
If n is a positive integer, then
where is the digamma function.
A Taylor series in the third variable is given by
where is the Pochhammer symbol.
Series at a = âÂÂn is given by
A special case for n = 0 has the following series
where is the polylogarithm.
An asymptotic series for
for and
for
An asymptotic series in the incomplete gamma function
for
The representation as a generalized hypergeometric function is
The polylogarithm function is defined as
Let
For and , an asymptotic expansion of for large and fixed and is given by
for , where is the Pochhammer symbol.
Let
Let be its Taylor coefficients at . Then for fixed and ,
as .
The Lerch transcendent is implemented as LerchPhi in Maple and Mathematica, and as lerchphi in mpmath and SymPy.