In mathematics, the Lerch zeta function, sometimes called the HurwitzLerch zeta function, 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 the function in 1887.[1]

Definition

The Lerch zeta function is given by

A related function, the Lerch transcendent, is given by

.

The transcendent only converges for any real number , where:

, or

, and .[2]

The two are related, as

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 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.[3][4]

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.[5]

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.

Special cases

The Lerch zeta function and Lerch transcendent generalize various special functions.

The Hurwitz zeta function is the special case[6]

The polylogarithm is another special case:[6]

The Riemann zeta function is a special case of both of the above:[6]

Other special cases include:

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 and . 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.[7]

A Taylor series in the first parameter was given by Arthur Erdélyi. It may be written as the following series, which is valid for[8]

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 = −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[9]

Asymptotic expansion

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.[10]

Let

Let be its Taylor coefficients at . Then for fixed and ,

as .[11]

Software

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

References

  1. Lerch, Mathias (1887), "Note sur la fonction ", Acta Mathematica (in French), 11 (1–4): 19–24, doi:10.1007/BF02612318, JFM 19.0438.01, MR 1554747, S2CID 121885446
  2. https://arxiv.org/pdf/math/0506319.pdf
  3. Bateman & Erdélyi 1953, p. 27
  4. Guillera & Sondow 2008, Lemma 2.1 and 2.2
  5. Bateman & Erdélyi 1953, p. 28
  6. 1 2 3 4 5 6 Guillera & Sondow 2008, p. 248–249
  7. "The Analytic Continuation of the Lerch Transcendent and the Riemann Zeta Function". 27 April 2020. Retrieved 28 April 2020.
  8. B. R. Johnson (1974). "Generalized Lerch zeta function". Pacific J. Math. 53 (1): 189–193. doi:10.2140/pjm.1974.53.189.
  9. Gottschalk, J. E.; Maslen, E. N. (1988). "Reduction formulae for generalized hypergeometric functions of one variable". J. Phys. A. 21 (9): 1983–1998. Bibcode:1988JPhA...21.1983G. doi:10.1088/0305-4470/21/9/015.
  10. Ferreira, Chelo; López, José L. (October 2004). "Asymptotic expansions of the Hurwitz–Lerch zeta function". Journal of Mathematical Analysis and Applications. 298 (1): 210–224. doi:10.1016/j.jmaa.2004.05.040.
  11. Cai, Xing Shi; López, José L. (10 June 2019). "A note on the asymptotic expansion of the Lerch's transcendent". Integral Transforms and Special Functions. 30 (10): 844–855. arXiv:1806.01122. doi:10.1080/10652469.2019.1627530. S2CID 119619877.
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