  It is shown that several of Brafman's generating functions for the Gegenbauer
polynomials are algebraic functions of their arguments, if the Gegenbauer
parameter differs from an integer by one-fourth or one-sixth. Two examples are
given, which come from recently derived expressions for associated Legendre
functions with octahedral or tetrahedral monodromy. It is also shown that if
the Gegenbauer parameter is restricted as stated, the Poisson kernel for the
Gegenbauer polynomials can be expressed in terms of complete elliptic
integrals. An example is given.
