Legendre's Chi-Function

The function defined by

(1)

It is related to the polylogarithm by

			(2)
			(3)

and to the Lerch transcendent by

(4)

It takes the special values

			(5)
			(6)
			(7)
			(8)
			(9)
			(10)

where i is the imaginary unit and  is Catalan's constant (Lewin 1958, p. 19). Other special values include

			(11)
			(12)

where  is the Dirichlet lambda function and  is the Dirichlet beta function.

REFERENCES:

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

Edwards, J. A Treatise on the Integral Calculus, Vol. 2. New York: Chelsea, p. 290, 1955.

Legendre, A. M. Exercices de calcul intégral, tome 1. p. 247, 1811.

Lewin, L. "Legendre's Chi-Function." §1.8 in Dilogarithms and Associated Functions. London: Macdonald, pp. 17-19, 1958.

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

Nielsen, N. "Der Eulersche Dilogarithmus und seine Verallgemeinerungen." Nova Acta Leopoldina, Abh. der Kaiserlich Leopoldinisch-Carolinischen Deutschen Akad. der Naturforsch. 90, 121-212, 1909.