Binomial Polynomials

Abstract

The binomial polynomials are defined by the sum representation,

$$\begin{aligned} Q_{n}^{(r)} (z) = \sum _{k=0}^n \genfrac(){0.0pt}{}{n}{k}^{r+1} z^k, \qquad n\in \mathbb{N },\,z\in \mathbb{C }, \end{aligned}$$

where $r$ is a non-negative integer. Using a multivariate complex integral representation, the asymptotical behavior of the sequence $(Q_{n}^{(r)})_{n=1}^{\infty }$ is studied on the whole complex plane as $n\rightarrow \infty$. The proofs are essentially based on a multivariate version of the method of saddle points. Moreover, results on the asymptotic zero distribution for $(Q_{n}^{(r)})_{n=1}^{\infty }$ and for some related polynomials are established by means of the theory of logarithmic potentials with external fields. Classical results on sums of powers of binomial coefficients and for Legendre polynomials are generalized, as well as a specific weighted equilibrium problem on the unit interval is solved.