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Legendre's Chi-Function

The function defined by

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

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


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.

© 1996-9 Eric W. Weisstein