Paul Erdős (1913–1996) 1. Prologue

at Kerepesi Cemetery in Budapest to pay their last respects to Paul Erdős. If there was one theme suggested by the farewell orations, it was that the world of mathematics had lost a legend, one of its great representatives. On October 21, 1996, in accordance with his last wishes, Paul Erdős ’ ashes were buried in his parents ’ grave at the Jewish cemetery on Kozma street in Budapest. Paul Erdős was one of this century’s greatest and most prolific mathematicians. He is said to have written about 1500 papers, withalmost 500 co-authors. He made fundamental contributions in numerous areas of mathematics. There is a Hungarian saying to the effect that one can forget everything but one’s first love. When considering Erdős and his mathematics, we cannot speak of “first love”, but of “first loves”, and approximation theory was among them. Paul Erdős wrote more than 100 papers that are connected, in one way or another, with the approximation of functions. In these two short reviews, we try to present some of Paul’s fundamental contributions to approximation theory. A list of Paul’s papers in approximation theory is given at the end of this article

Combinatorial Trigonometry with Chebyshev Polynomials

We provide a combinatorial proof of the trigonometric identity cos(nθ) = Tncos(θ),where Tn is the Chebyshev polynomial of the first kind. We also provide combinatorial proofs of other trigonometric identities, including those involving Chebyshev polynomials of the second kind

Multivariate Jacobi and Laguerre polynomials, infinite-dimensional extensions, and their probabilistic connections with multivariate

Multivariate versions of classical orthogonal polynomials such as Jacobi, Hahn, Laguerre, Meixner are reviewed and their connection explored by adopting a probabilistic approach. Hahn and Meixner polynomials are interpreted as posterior mixtures of Jacobi and Laguerre polynomials, respectively. By using known properties of Gamma point processes and related transformations, an infinite-dimensional version of Jacobi polynomials is constructed with respect to the size-biased version of the Poisson-Dirichlet weight measure and to the law of the Gamma point process from which it is derived. 1 Introduction. The Dirichlet distribution Dα on d < ∞ points, where α = (α1,..., αd) ∈ Rd +, is the probability distribution on the (d − 1)−dimensional simplex described by Dα(dx1,..., dxd−1)

Stieltjes Polynomials

> n (x), the Chebyshev polynomial of the second kind, where E n+1 (x) = T n+1 (x), the Chebyshev polynomial of the first kind. For h(x) = (1 \Gamma x 2 ) \Gamma1=2 , P n (x) = T n (x) and E n+1 (x) = (1 \Gamma x 2 )U n\Gamma1 (x). A generalisation are the Bernstein-Szego weight functions h(x) = (1 \Gamma x 2 ) \Sigma1=2 =ae m (x), where ae m is a polynomial of degree m that is