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Lerch zeta function

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In mathematics, the Lerch zeta function, sometimes called the Hurwitz–Lerch 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

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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

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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

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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

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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

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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

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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

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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

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

References

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  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. ^ a b c d e f 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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