Legendre's Chi-Function

The function defined by

$\begin{displaymath} \chi_\nu(z)=\sum_{k=0}^\infty {z^{2k+1}\over(2k+1)^\nu} \end{displaymath}$

for integral $\nu=2$ , 3, .... It is related to the Polylogarithm by

$\displaystyle \chi_\nu(z)$	$\textstyle =$	$\displaystyle {\textstyle{1\over 2}}[\mathop{\rm Li}\nolimits _\nu(z)-\mathop{\rm Li}\nolimits _\nu(-z)]$
	$\textstyle =$	$\displaystyle \mathop{\rm Li}\nolimits _\nu(z)-2^{-\nu}\mathop{\rm Li}\nolimits _\nu(z^2).$

See also Polylogarithm

References

Cvijovic, D. and Klinowski, J. ``Closed-Form Summation of Some Trigonometric Series.'' Math. Comput. 64, 205-210, 1995.

Lewin, L. Polylogarithms and Associated Functions. Amsterdam, Netherlands: North-Holland, pp. 282-283, 1981.