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 \alpha > 0, where |z| < 1, or, and |z| = 1.

Special cases

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 Dirichlet eta function: The Dirichlet beta function: The Legendre chi function: The inverse tangent integral: The polygamma functions for positive integers n: The Clausen function:

Integral representations

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 \Phi(z,s,a) 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.

Other integral representations

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.

Identities

For λ rational, the summand is a root of unity, and thus may be expressed as a finite sum over the Hurwitz zeta function. Suppose with p, q \in \Z and q > 0. Then and. Various identities include: and and

Series representations

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 \psi(n) is the digamma function. A Taylor series in the third variable is given by where (s)_{k} is the Pochhammer symbol. Series at a = −n 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

Asymptotic expansion

The polylogarithm function is defined as Let For and, an asymptotic expansion of \Phi(z,s,a) for large a and fixed s and z is given by for , where is the Pochhammer symbol. Let Let C_{n}(z,a) be its Taylor coefficients at x=0. Then for fixed and \Re s > 0, as.

Software

The Lerch transcendent is implemented as LerchPhi in Maple and Mathematica, and as lerchphi in mpmath and SymPy.