A question concerning positive type polynomials

A Question concerning Positive Type Polynomials

1. The problem. In [l], the following problem is posed: Let 0<A<1. Find the smallest integer re for which there exist re real numbers ai, a2, ■ ■ • , an such that the polynomial (y2 — 2Ay + T) ' II?-1 Cy+0») has all its coefficients non-negative. If N is such an integer, does there exist some real number a such that (y2 — 2Ay + l) •(y+a)N has only non-negative coefficients.1 In the process of g...

Two new conjectures concerning positive Jacobi polynomials sums

A refinement of a conjecture of Gasper concerning the values of (α, β), −1/2 < β < 0, −1 < α + β < 0, for which the inequalities

Reductibility of Positive Type Polynomials

Congruences concerning Legendre Polynomials

Let p be an odd prime. In the paper, by using the properties of Legendre polynomials we prove some congruences for È p−1 2 k=0 2k k ¡ 2 m −k (mod p 2). In particular, we confirm several conjectures of Z.W. Sun. We also pose 13 conjectures on supercongruences.

Multivariate Polynomials: A Spanning Question

There are many different methods of approximating multivariate functions. We have, for example, the classic methods of polynomials, Fourier series, or tensor products, and more modern methods using wavelets, radial basis functions, multivariate splines, or ridge functions. Many of these are natural generalizations of methods developed for approximating univariate functions. However, functions o...