Legendre chi function
Legendre chi function 10 languages Article Talk Read Edit View history Tools From Wikipedia, the free encyclopedia In mathematics , the Legendre chi function is a special function whose Taylor series is also a Dirichlet series , given by � � ( � ) = ∑ � = 0 ∞ � 2 � + 1 ( 2 � + 1 ) � . As such, it resembles the Dirichlet series for the polylogarithm , and, indeed, is trivially expressible in terms of the polylogarithm as � � ( � ) = 1 2 [ Li � ( � ) − Li � ( − � ) ] . 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 � � ( � ) = 2 − � � Φ ( � 2 , � , 1 / 2 ) . Identities [ edit ] � 2 ( � ) + � 2 ( 1 / � ) = � 2 4 − � � 2 ln | � | . � � � � 2 ...