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In mathematics, the Legendre chi function (named after Adrien-Marie Legendre) is a special function whose Taylor series is also a Dirichlet series, given by

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

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

It takes the special values:

where is the imaginary unit and K is Catalan's constant.^[1] Other special values include:

where is the Dirichlet lambda function and is the Dirichlet beta function.^[1]

• Weisstein, Eric W. "Legendre's Chi Function". MathWorld.
• Djurdje Cvijović, Jacek Klinowski (1999). "Values of the Legendre chi and Hurwitz zeta functions at rational arguments". Mathematics of Computation. 68 (228): 1623–1630. doi:10.1090/S0025-5718-99-01091-1.
• Djurdje Cvijović (2007). "Integral representations of the Legendre chi function". Journal of Mathematical Analysis and Applications. 332 (2): 1056–1062. arXiv:0911.4731. doi:10.1016/j.jmaa.2006.10.083. S2CID 115155704.