Special cases of this theorem have been proved by Erdös-Grünwald' and Webster2 (the cases a = 1/2 and a = 3/2) . If there is no danger of confusion we shall omit the upper index n in lk`, n ~ (x) . PROOF OF THE THEOREM . It clearly suffices to consider the lk(x) with -1 =< xk < 0 . From the differential equation of the ultraspherical polynomials' we obtain (") lk(xk) = {«~xk) _ axk z zP„ (xk) 1...

Let μ be a probability measure on the real line with finite moments of all orders. Apply the Gram-Schmidt orthogonalization process to the system {1, x, x, . . . , xn, . . . } to get orthogonal polynomials Pn(x), n ≥ 0, which have leading coefficient 1 and satisfy (x − αn)Pn(x) = Pn+1(x) + ωnPn−1(x). In general it is almost impossible to use this process to compute the explicit form of these po...

We consider a filtration of the symmetric function space given by Λ (k) t , the linear span of Hall-Littlewood polynomials indexed by partitions whose first part is not larger than k. We introduce symmetric functions called the k-Schur functions, providing an analog for the Schur functions in the subspaces Λ (k) t . We prove several properties for the k-Schur functions including that they form ...

Generalized Hall–Littlewood polynomials (Macdonald spherical functions) and generalized Kostka–Foulkes polynomials (q-weight multiplicities) arise in many places in combinatorics, representation theory, geometry, and mathematical physics. This paper attempts to organize the different definitions of these objects and prove the fundamental combinatorial results from “scratch”, in a presentation w...

We consider the space Pn of orthogonal polynomials of degree n on the unit disc for a general radially symmetricweight function.We show that there exists a single orthogonal polynomialwhose rotations through the angles j n+1 , j = 0, 1, . . . , n forms an orthonormal basis for Pn, and compute all such polynomials explicitly. This generalises the orthonormal basis of Logan and Shepp for the Lege...

We construct an n-dimensional polytope whose boundary complex is compressed and whose face numbers for any pulling triangulation are the coefficients of the powers of (x − 1)/2 in the n-th Legendre polynomial. We show that the non-central Delannoy numbers count all faces in the lexicographic pulling triangulation that contain a point in a given open generalized orthant. We thus provide a geomet...

Many interesting properties of polynomials are closely related to the geometry of their Newton polytopes. We analyze the coercivity on R of multivariate polynomials f ∈ R[x] in terms of their Newton polytopes. In fact, we introduce the broad class of so-called gem regular polynomials and characterize their coercivity via conditions imposed on the vertex set of their Newton polytopes. These cond...

Biological cells can be treated as an inhomogeneous particle. In addition to biomaterials, inhomogeneous particles are also important in more traditional colloidal science. By using two energy methods that are based on Legendre polynomials and Green’s function, respectively, we investigate the interaction between biological cells or colloidal particles in the presence of an external electric fi...

Although general methods led me to a complete solution, I soon saw that the result is obtained faster when the general procedure is left, and when one follows the path suggested by the particular problem at hand. S. Bernstein first lines of [6] Abstract. One establishes inequalities for the coefficients of orthogonal polynomials Φn(z) = z n + ξnz n−1 + · · · + Φn(0), n = 0, 1, . . . which are o...

Abstract We describe an expansion of Legendre polynomials, analogous to the Taylor expansion, to approximate arbitrary functions. We show that the polynomial coefficients in Legendre expansion, thus, the whole series, converge to zero much more rapidly compared to the Taylor expansion of the same order. Furthermore, using numerical analysis with sixth-order polynomial expansion, we demonstrate ...