It is well known that the classical families of orthogonal polynomials are characterized as eigenfunctions of a second order linear differential/difference operator. In this paper we present a study of classical orthogonal polynomials in a more general context by using the differential (or difference) calculus and Operator Theory. In such a way we obtain a unified representation of them. Furthe...

This paper considers a problem of Chebyshev approximation by interpolating rationals. Examples are given which show that best approximations may not exist. Sufficient conditions for existence are established, some of which can easily be checked in practice. Illustrative examples are also presented.

Let {φk}k=0, n < m, be a family of polynomials orthogonal with respect to the positive semi-definite bilinear form

We here give the complete data for the remaining cases deg q = 5, 6, 7, which supplement the paper ([1] Th. Stoll, Decomposition of perturbed Chebyshev polynomials, submitted). This is not supposed to be included in the paper.

The results of the computer hunt for the primes of the form q = m + 1 up to 10 are reported. The number of sign changes of the difference πq(x) − Cq 2 ∫ x 2 du √ u log(u) and the error term for this difference is investigated. The analogs of the Brun’s constant and the Skewes number are calculated. An analog of the B conjecture of Hardy–Littlewood is formulated. It is argued that there is no Ch...

Louis W. Shapiro gave a combinatorial proof of a bilinear generating function for Chebyshev polynomials equivalent to the formula 1 1− ax − x2 ∗ 1 1− bx − x2 = 1− x2 1− abx − (2 + a2 + b2)x2 − abx3 + x4 , where ∗ denotes the Hadamard product. In a similar way, by considering tilings of a 2× n rectangle with 1× 1 and 1× 2 bricks in the top row, and 1× 1 and 1× n bricks in the bottom row, we find...

Caution: These polynomials are referred to as w0, w1, w2, ... in Sections 5-8 of [1]. However, beginning with Section 9 of [1], Hazewinkel uses the notations w1, w2, w3, ... for some different polynomials (the so-called big Witt polynomials, defined by formula (9.25) in [1]), which are not the same as our polynomials w1, w2, w3, ... (though they are related to them: in fact, the polynomial wk t...

which, in general, is a strictly proper rational matrix (see [1, 5] and references therein). The computation of (sE−A)−1 can be carried out by using the Cramer rule, which requires the evaluation of n2 determinants of (n− 1)× (n− 1) polynomial matrices. Clearly, this is not a practical procedure for large n. We will describe an extension of the classical Leverrier-Faddeev algorithm using famili...

Using the combinatorial formula for the transformed Macdonald polynomials of Haglund, Haiman, and Loehr, we investigate the combinatorics of the symmetry relation H̃μ(x; q, t) = H̃μ∗(x; t, q). We provide a purely combinatorial proof of the relation in the case of Hall-Littlewood polynomials (q = 0) when μ is a partition with at most three rows, and for the coefficients of the square-free monomial...

We obtain definite integrals for products of associated Legendre functions with Bessel functions, associated Legendre functions, and Chebyshev polynomials of the first kind using orthogonality and integral transforms.