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

As such, it resembles the Dirichlet series for the polylogarithm, and, indeed, is trivially expressible as the odd part

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

Identities

Special Values

It takes the special values:

is the imaginary unit and K is Catalan's constant. Other special values include:

is the Dirichlet lambda function is the Dirichlet beta function.

Integral relations

References