  Every character on a graded connected Hopf algebra decomposes uniquely as a
product of an even character and an odd character (Aguiar, Bergeron, and
Sottile, math.CO/0310016).
  We obtain explicit formulas for the even and odd parts of the universal
character on the Hopf algebra of quasi-symmetric functions. They can be
described in terms of Legendre's beta function evaluated at half-integers, or
in terms of bivariate Catalan numbers:
  $$ C(m,n)=\frac{(2m)!(2n)!}{m!(m+n)!n!}. $$
  Properties of characters and of quasi-symmetric functions are then used to
derive several interesting identities among bivariate Catalan numbers and in
particular among Catalan numbers and central binomial coefficients.
