Legendre chi function

In mathematics, the Legendre chi function is a special function whose Taylor series is also a Dirichlet series, given by

Failed to parse (Missing texvc executable; please see math/README to configure.): \chi_\nu(z) = \sum_{k=0}^\infty \frac{z^{2k+1}}{(2k+1)^\nu}.

As such, it resembles the Dirichlet series for the polylogarithm, and, indeed, is trivially expressible in terms of the polylogarithm as

Failed to parse (Missing texvc executable; please see math/README to configure.): \chi_\nu(z) = \frac{1}{2}\left[\operatorname{Li}_\nu(z) - \operatorname{Li}_\nu(-z)\right]

The Legendre chi function appears as the discrete fourier transform, with respect to the order ν, of the Hurwitz zeta function, and also of the Euler polynomials, with the explicit relationships given in those articles.

The Legendre chi function is a special case of the Lerch transcendent, and is given by

Failed to parse (Missing texvc executable; please see math/README to configure.): \chi_n(z)=2^{-n}z\,\Phi (z^2,n,1/2).\,

References

• Eric W. Weisstein, Legendre's Chi Function at MathWorld.
• Djurdje Cvijović and Jacek Klinowski, "Values of the Legendre chi and Hurwitz zeta functions at rational arguments", Mathematics of Computation 68 (1999), 1623-1630.
•  Djurdje Cvijović (2006). "Integral representations of the Legendre chi function". Elsevier. Retrieved on December 15, 2006.

fr:Fonction chi de Legendre