Integral representations of the Legendre chi function

Abstract

We, by making use of elementary arguments, deduce integral representations of the Legendre chi function χ s ( z ) valid for | z | < 1 and Re s > 1 . Our earlier established results on the integral representations for the Riemann zeta function ζ ( 2 n + 1 ) and the Dirichlet beta function β ( 2 n ) , n ∈ N , are a direct consequence of these representations.