Let α, β ≥ −1/2, and for k = 0, 1, · · ·, pk (α,β) denote the orthonormalized Jacobi polynomial of degree k. We discuss the construction of a matrix H so that there exist positive constants c, c1, depending only on H , α, and β such that

Recently the author derived a Laplace integral representation, a product formula and an addition formula for Jacobi polynomials Ppfi). The results are (1) (2) and (3) * j, ,I " (~0~2 0-f-2 sin2 0 + ir cos #I sin 28)n * ' P$ycos 281) PFycos 282) 2&x + 1) Pgq 1) Pfq 1)-= l/22 qa-p)r(/l+#. . .

In this note we have obtained some novel result on mixed trilateral relations involving extended Jacobi polynomials by group theoretic method which inturn yields the corresponding results involving Hermite, Laguerre and Jacobi polynomials.

In this paper we show how one can obtain simultaneous rational approximants for ζq(1) and ζq(2) with a common denominator by means of Hermite-Padé approximation using multiple little q-Jacobi polynomials and we show that properties of these rational approximants prove that 1, ζq(1), ζq(2) are linearly independent over Q. In particular this implies that ζq(1) and ζq(2) are irrational. Furthermor...

The Littlewood–Paley theory is extended to weighted spaces of distributions on [−1, 1] with Jacobi weights w(t) = (1−t)(1+t) . Almost exponentially localized polynomial elements (needlets) {φξ}, {ψξ} are constructed and, in complete analogy with the classical case on R, it is shown that weighted Triebel–Lizorkin and Besov spaces can be characterized by the size of the needlet coefficients {〈f, ...

Let μ be the Jacobi measure supported on the interval [−1, 1]. Let introduce the Sobolev-type inner product 〈f, g〉 = ∫ 1 −1 f(x)g(x) dμ(x) +Mf(1)g(1) +Nf ′(1)g′(1), where M,N ≥ 0. In this paper we prove that, for certain indices δ, there are functions whose Cesàro means of order δ in the Fourier expansion in terms of the orthonormal polynomials associated with the above Sobolev inner product ar...

Let JTλ be the Jacobi-Trudi matrix corresponding to the partition λ, so det JTλ is the Schur function sλ in the variables x1, x2, . . . . Set x1 = · · · = xn = 1 and all other xi = 0. Then the entries of JTλ become polynomials in n of the form ( n+j−1 j ) . We determine the Smith normal form over the ring Q[n] of this specialization of JTλ . The proof carries over to the specialization xi = q i...

We study ratio asymptotics, that is, existence of the limit of Pnþ1ðzÞ=PnðzÞ (Pn 1⁄4 monic orthogonal polynomial) and the existence of weak limits of pn dm ðpn 1⁄4 Pn=jjPnjjÞ as n-N for orthogonal polynomials on the real line. We show existence of ratio asymptotics at a single z0 with Imðz0Þa0 implies dm is in a Nevai class (i.e., an-a and bn-b where an; bn are the offdiagonal and diagonal Jaco...

We study the Hankel determinant of the generalized Jacobi weight (x − t)x(1 − x) for x ∈ [0, 1] with α, β > 0, t < 0 and γ ∈ R. Based on the ladder operators for the corresponding monic orthogonal polynomials Pn(x), it is shown that the logarithmic derivative of Hankel determinant is characterized by a τ -function for the Painlevé VI system.