  Many classes of orthogonal polynomials satisfy a specific linearization
property giving rise to a polynomial hypergroup structure, which offers an
elegant and fruitful link to Fourier analysis, harmonic analysis and functional
analysis. From the opposite point of view, this allows regarding certain Banach
algebras as $L^1$-algebras, associated with underlying orthogonal polynomials.
The individual behavior strongly depends on these underlying polynomials. We
study the little $q$-Legendre polynomials, which are orthogonal with respect to
a discrete measure. We will show that their $L^1$-algebras have the property
that every element can be approximated by linear combinations of idempotents.
This particularly implies that these $L^1$-algebras are weakly amenable (i. e.,
every bounded derivation into the dual module is an inner derivation), which is
known to be shared by any $L^1$-algebra of a locally compact group; in the
polynomial hypergroup context, weak amenability is rarely satisfied and of
particular interest because it corresponds to a certain property of the
derivatives of the underlying polynomials and their (Fourier) expansions w.r.t.
the polynomial basis. To our knowledge, the little $q$-Legendre polynomials
yield the first example of a polynomial hypergroup whose $L^1$-algebra is
weakly amenable and right character amenable but fails to be amenable. As a
crucial tool, we establish certain uniform boundedness properties of the
characters. Our strategy relies on the Fourier transformation on hypergroups,
the Plancherel isomorphism, continued fractions, character estimations and
asymptotic behavior.
