Legendre chi function

In mathematics, the Legendre chi function is defined as

$\chi _{n}(z)=\sum _{k=0}^{\infty }{\frac {z^{2k+1}}{(2k+1)^{n}}}.$

The discrete fourier transform of the Legendre chi function with respect to the order n is the Hurwitz zeta function (Cvijovic).

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

$\chi _{n}(z)=2^{-n}z\,\Phi (z^{2},n,1/2).\,$

References

• Djurdje Cvijović and Jacek Klinowski, "Values of the Legendre chi and Hurwitz zeta functions at rational arguments", Mathematics of Computation 68 (1999), 1623-1630.