  From Koornwinder's interpretation of big $q$-Legendre polynomials as
spherical elements on the quantum $SU(2)$ group an addition formula is derived
for the big $q$-Legendre polynomial. The formula involves Al-Salam--Carlitz
polynomials, little $q$-Jacobi polynomials and dual $q$-Krawtchouk polynomials.
For the little $q$-ultraspherical polynomials a product formula in terms of a
big $q$-Legendre polynomial follows by $q$-integration. The addition and
product formula for the Legendre polynomials are obtained when $q$ tends to 1.
