Uniform inequalities for Gegenbauer polynomials (Q1912675)

The usual asymptotic representations of the Gegenbauer (ultraspherical) polynomials do not yield bounds on their absolute values which hold equally on the interval \(-1\leq x\leq 1\). But in the Legendre case (index \(\lambda= {1\over 2}\)) and more generally in the case of \(0\leq \lambda\leq 1\) such estimates exist. In this paper such bounds are given explicitly for arbitrary \(\lambda>0\).