We show that the function ((x-dm)(x-dM))1/4xα/2e-x/2Lk(α)(x) is almost equioscillating with the amplitude 2/π provided k and α are large enough. Here Lk(α)(x) is the orthonormal Laguerre polynomial of degree k and dm, dM are some approximations for the extreme zeros. As a corollary we obtain a very explicit, uniform in k and α, sharp upper bound on the Laguerre polynomials.
- Laguerre polynomials
- Orthogonal polynomials