Showing papers on "Discrete orthogonal polynomials published in 1977"
TL;DR: In this article, a positive measure on the circumference and almost everywhere on the globe is defined, and the orthogonal polynomials corresponding to the positive measure are chosen.
Abstract: Let be a positive measure on the circumference and let almost everywhere on . Let be the orthogonal polynomials corresponding to , and let be their parameters. Then .Bibliography: 5 titles.
244 citations
TL;DR: In this article, a discrete inner product for which the associated sequence of monic orthogonal polynomials coincides with the sequence of appropriately normalized characteristic polynomorphisms of the left principal submatrices of the Jacobi matrix is constructed.
176 citations
01 Jan 1977
TL;DR: The theory of partitions has long been associated with so-called basic hypergeometric functions or Eulerian series as mentioned in this paper, and it has been shown that the Rogers-Ramanujan identities can be deduced from the connection coefficient problem for the little q-Jacobi polynomials.
Abstract: The theory of partitions has long been associated with so called basic hypergeometric functions or Eulerian series We begin with discussion of some of the lesser known identities of LJ Rogers which have interesting interpretations in the theory of partitions Illustrations are given for the numerous ways partition studies lead to Eulerian series The main portion of our work is primarily an introduction to recent work on orthogonal polynomials defined by basic hypergeometric series and to the applications that can be made of these results to the theory of partitions Perhaps it is most interesting to note that we deduce the Rogers-Ramanujan identities from our solution to the connection coefficient problem for the little q-Jacobi polynomials
100 citations
TL;DR: In this paper, the authors proved Jackson type estimates for the approximation of monotone functions by polynomials, in terms of the modulus of continuity of the function.
Abstract: We prove Jackson type estimates for the approximation of monotone functions by monotone polynomials. The results are given in terms of the modulus of continuity of $f^{(k)} $ , for any $k \geqq 0$. The estimates are of the same order as for the unconstrained approximation by polynomials.
57 citations
TL;DR: It is shown that certain problems involving sparse polynomials with integer coefficients are at least as hard as any problem in NP.
55 citations
TL;DR: In this article, some well-known properties of polynomials orthogonal on the unit circle are used to provide a simple proof of the Schur-Cohn stability criterion.
Abstract: Some well-known properties of polynomials orthogonal on the unit circle are used to provide a simple proof of the Schur-Cohn stability criterion. Some complements are also briefly noted.
49 citations
01 Jan 1977
49 citations
01 Jun 1977
TL;DR: In this paper, it is shown that this problem is equivalent to considering a new class of orthogonal polynomials on the hypercircle, the properties of which are investigated.
Abstract: This paper is concerned with a possible extension of a well-known stabilization technique for one-variable recursive digital filters to the two-dimensional case, as recently conjectured. It is shown that this problem is equivalent to considering a new class of orthogonal polynomials, the two-variable orthogonal polynomials on the hypercircle, the properties of which are investigated. As a result, the zeros of these polynomials are proved not to lie necessarily in an appropriate region compatible with the proposed conjecture, which therefore turns out to be in error.
42 citations
01 Jan 1977
TL;DR: In this article, three-term contiguous relations for 2 F 1's, 3 F 2's and 4 F 3's were derived, and a list of the 3F 2 relations was given, including the recurrence relation for a set of orthogonal polynomials generalizing the classical polynomial.
Abstract: In this paper, we show how three-term contiguous relations for 2 F 1 's, 3 F 2 's, and 4 F 3 's may be derived, and list the 3 F 2 relations. These relations may be a source for many interesting continued fractions, and the 4 F 3 relations include the recurrence relation for a set of orthogonal polynomials generalizing the classical polynomials.
31 citations
TL;DR: In this article, the authors explain combinatorially the occurrence of certain classical sequences of orthogonal polynomials as sequences of rook polynomial, and give some new examples related to general stairstep boards.
Abstract: We explain combinatorially the occurrence of certain classical sequences of orthogonal polynomials as sequences of rook polynomials, and we give some new examples related to general stairstep boards.
29 citations
TL;DR: Bateman's addition formula for Laguerre polynomials of order zero was generalized to the case of order α > 0 in this article, where the result was obtained as a limit case of the addition formula of disk polynomial.
Abstract: Bateman’s addition formula for Laguerre polynomials of order zero is generalized to the case of order $\alpha > 0$. The result is obtained as a limit case of the addition formula for disk polynomials.
TL;DR: In this paper, a class of Markovian models based on a form of "dynamic occupancy problem" originating in statistical mechanics are described and investigated, and a number of interesting formulas are obtained as by-products.
TL;DR: In this article, short proofs of the addition formulas for Gegenbauer polynomials and Jacobi polynomorphisms are given, and the properties of certain special orthogonal polynomial in two, respectively three, variables are used.
Book•
01 Jan 1977
TL;DR: In this paper, an asymptotic formula for the derivatives of orthogonal polynomials on the unit circle is given for the case where the distribution function is locally local.
Abstract: An asymptotic formula is found for the derivatives of orthogonal polynomials on the unit circle. The condition on the distribution function is essentially local and the result is stronger than those known before.
Journal Article•
TL;DR: In this paper, it was shown that problems involving /-free numbers and polynomials of degree r are almost inevitably easy to resolve when /^r, while the successive work of Erdös [3] and the author ([6], [7], and [8]) has fully covered the question of the representation of /free numbers by fixed polynomorphisms of degree n with integer arguments when / = r − 1.
Abstract: A füll history and background of this subject having been furnished by both I and two earlier memoirs of similar title (Hooley, [6] and [7]), the briefest of introductions serves to usher in the preliminary Statement of the results we shall obtain. Suffice it then to repeat that problems involving /-free numbers and polynomials of degree r are almost inevitably easy to resolve when /^r, while the successive work of Erdös [3] and the author ([6], [7], and [8]) has fully covered the question of the representation of /-free numbers by fixed polynomials of degree r with integer arguments when / = r — 1 . Nevertheless, this work did not settle the general Situation relating to the case / = r — l , since, for instance, it shed no light whatsoever here on either the important problem regarding polynomials with prime arguments or the conjugate problem involving the representation of large numbers äs the sum of an rth power and an (r— l)-th power-free number. The latter are the problems, briefly alluded to in I, that will now occupy our attention.
TL;DR: In this paper, the sharpness of a theorem concerning zero-free parabolic regions for certain sequences of polynomials satisfying a three-term recurrence relation was established, and it was shown that this is also the case for certain Pade approximants.
Abstract: In this paper, we establish the sharpness of a theorem concerning zero-free parabolic regions for certain sequences of polynomials satisfying a three-term recurrence relation. Similarly, we establish the sharpness of a zero-free sectorial region for certain sequences of Pade approximants toe z .
TL;DR: Several recursion-theoretic facts and an improvement on the exponential Diophantine representation are applied to refine the exponential case of a result of Davis [1972] regarding the difficulty of determining the number of zeros of a polynomial.
Abstract: ?0. The negative solution of Hilbert's Tenth Problem brought with it a number of unsolvable Diophantine problems. Moreover, by actually providing a Diophantine characterization of recursive enumerability, the proof of the negative solution opened the door to the techniques of recursion theory. In this note, we wish to apply several recursion-theoretic facts and an improvement on the exponential Diophantine representation to refine the exponential case of a result of Davis [1972] regarding the difficulty of determining the number of zeros of a polynomial. P, Q, etc. will denote polynomials or exponential polynomials-exactly which will be clear from the context. Let # (P) denote the number of distinct nonnegative zeros of P. Further, let C = {0, 1, * * *, N4o} be the set of possible values of # (P). For A C C, we define A * to be
TL;DR: In this article, a new basis for discrete analytic polynomials is presented for which the series converges absolutely to a discrete analytic function in the upper right quarter lattice whenever
Abstract: A new basis for discrete analytic polynomials is presented for which the series converges absolutely to a discrete analytic function in the upper right quarter lattice whenever
TL;DR: In this paper, the authors discuss some properties of matrix polynomials and a computational procedure for finding the matrix roots of such polynomial coefficients and their relationship to spectral factorization.
01 Jan 1977
01 Jan 1977
TL;DR: In this paper, the general theory of orthogonal polynomials with respect to a functional defined by its moments is used to derive old and new results on continued fractions, Pade approximants, and the ǫ-algorithm.
Abstract: The general theory of orthogonal polynomials with respect to a functional defined by its moments is used to derive old and new results on continued fractions, Pade approximants, and the ɛ-algorithm. Orthogonal polynomials seem to be the mathematical basis on which Pade approximants and related matters are to be studied.
TL;DR: In this article, an operator T that maps a set of orthogonal polynomials {P n (x )} n = o ∞ to another set of O(n) orthogonals is presented.
01 Jan 1977
TL;DR: In this article, a family of reproducing kernels for the q-Jacobi polynomials 4(n"J)(X) = 241(q-f, q -l+f3; qa; q, qx).
Abstract: We derive a family of reproducing kernels for the q-Jacobi polynomials 4(n"J)(X) = 241(q-f, q -l+f3; qa; q, qx). This is achieved by proving that the polynomials 4'fl)(x) satisfy a discrete Fredholm integral equation of the second kind with a positive symmetric kernel, then applying Mercer's theorem.
TL;DR: The augmented monomial symmetric function in n variables corresponding to the partition p E 9(f) is defined by David and Kendall [I] as a lexicon.
TL;DR: In this paper, conditions for the reducibility of polynomials with holomorphic coefficients and locally simple roots were studied and the role of the arithmetical properties of the degree of the polynomial was clarified.
Abstract: In this paper, we study conditions for the reducibility of polynomials with holomorphic coefficients and locally simple roots. Results are obtained which sharpen the theorems of Gorin and Lin on the reducibility of such polynomials under certain restrictions on the index of the discriminant and the domain in which the coefficients are defined; the role of the arithmetical properties of the degree of the polynomial is clarified.Bibliography: 5 titles.