Orthogonal Laurent polynomials and strong moment theory: a survey

PC-fractions and orthogonal Laurent polynomials for log-normal distributions

The Strong Chebyshev Distribution and Orthogonal Laurent Polynomials

Introduction Numerous elementary and higher transcendental functions of complex analysis possess representations in terms of continued fractions of the Stieltjes type. (An abundance of examples are given in the textbooks by WALL [1948] and PERRON [t957]; for some recent examples and applications, see MAEHLY [1960], CONTE and FRIED [t96t], MILLER, MOLMUD and MEECHAM [t962], BLANCH [1964], PATRY and KELLER [1964], W¥1~N [1964].) Similar representations also exist for the remainder terms of the asymptotic expansions of functions defined by certain Laplacian integrals (HESRICI [t963, 1965]). Such representations are of considerable computational value, both because of the ease with which a continued fraction can be evaluated recursively, and because of the fact that continued fraction representations frequently are convergent (and computationally efficient) in wider regions of the complex plane than, say, a Taylor series. In actual computation it is necessary to approximate the value of a continued fraction by some approximant of finite order. This paper makes an a t tempt to s tudy the error committed in this approximation. Both a posteriori bounds depending on the knowledge of a finite number of approximants -and a priori bounds depending only on the elements of the continued fraction or on the coefficients of its corresponding power series are presented. A solution of the problem of finding a posteriori bounds which in a sense is best possible is given in section 2. Several different a priori bounds are given in section 3. Although these bounds are not shown to be best possible, they are shown to have the correct order of magnitude in some important special cases. The tack of a generally accepted terminology and even of clear definitions in the analytic theory of continued fractions made it necessary to recollect, in a unified notation, some basic facts of the theory in section t.

A three-complex-parameter class of orthogonal Laurent polynomials on the unit circle associated with basic hypergeometric or q q -hypergeometric functions is considered. To be precise, we consider the orthogonality properties of the sequence of polynomials { 2 Φ 1 ( q − n , q b + 1 ; q − c + b − n ; q , q − c + d − 1 z ) } n = 0 ∞ \{\,_2\Phi _1(q^{-n},q^{b+1};q^{-c+b-n}; q, q^{-c+d-1}z)\}_{n=0}^{\infty } , where 0 > q > 1 0 > q > 1 and the complex parameters b b , c c and d d are such that b ≠ − 1 , − 2 , … b \neq -1, -2, \ldots , c − b + 1 ≠ − 1 , − 2 , … c-b+1 \neq -1, -2, \ldots , R e ( d ) > 0 \mathcal {R}e(d) > 0 and R e ( c − d + 2 ) > 0 \mathcal {R}e(c-d+2) > 0 . Explicit expressions for the recurrence coefficients, moments, orthogonality and also asymptotic properties are given. By a special choice of the parameters, results regarding a class of Szegő polynomials are also derived.

Orthogonal polynomials, special functions and mathematical physics

The extended Krylov subspace method and orthogonal Laurent polynomials

The obvious way to compute the continued fraction of a real number α > 1 is to compute a very accurate numerical approximation of α, and then to iterate the well-known truncate-and-invert step which computes the next partial quotient a = bαc and the next complete quotient α′ = 1/(α − a). This method is called the basic method. In the course of this process precision is lost, and one has to take precautions to stop before the partial quotients become incorrect. Lehmer [6] gives a safe stopping-criterion, and a trick to reduce the amount of multi-length arithmetic. Schonhage [12] describes an algorithm for computing the greatest common divisor of u and v and the related continued fraction expansion of u/v in O(n log n log log n) steps if both u and v do not exceed 2. A disadvantage of this approach is that if we wish to extend the list of partial quotients computed from an initial approximation of α, we have to compute a more accurate initial approximation of α, compute the new complete quotient using this new approximation and the partial quotients already computed from the old approximation, and then extend the list of partial quotients using that new complete quotient (we notice that Shiu in [13, p.1312] incorrectly states that all the previous calculations have to be repeated, but the partial quotients computed so far don’t have to be recomputed). Bombieri and Van der Poorten [1], and Shiu [13] have recently proposed a remedy for this problem. They give a formula for computing a rational approximation of the next complete convergent from the first n partial quotients. From that complete convergent about n new partial quotients can be computed. So each step gives an approximate doubling of the number of partial quotients. To start the method, a few partial quotients have to be computed with the basic or indirect method. In [1] this method is proposed for algebraic numbers (which are zeros of polynomials) of degree ≥ 3, whereas Shiu also applies it to more general numbers, namely to transcendental numbers that can be defined as the zero of a function for which the logarithmic derivative at a rational point can be computed with arbitrary precision. This includes numbers like π, log π, and log 2. For each of thirteen different numbers, Shiu computes 10000 partial quotients. Their frequency distributions are compared with the one which almost all numbers should obey, according to the Khintchine– Levy theory [3, 7]. No significant deviations from this theory are reported. Shiu calls his method the direct method. Curiously, Shiu does not refer to what we would call the polynomial method for algebraic numbers [2, 5, 11] of degree ≥ 3, which computes the partial quotients of α using only the coefficients of its defining polynomial. Moreover, Shiu gives neither implementational

Analytic theory of continued fractions III : L. Jacobsen (ed.), Lecture Notes in Mathematics 1406. Springer, Berlin, 1989. 142 pp., DM 25, ISBN 3-540-51830-4

We show in this paper how the Laplace transform θ* of the duration θ of an excursion by the occupation process {Λt} of an M/M/∞ system above a given threshold can be obtained by means of continued fraction analysis. The representation of θ* by a continued fraction is established and the [m−1/m] Padé approximants are computed by means of well known orthogonal polynomials, namely associated Charlier polynomials. It turns out that the continued fraction considered is an S fraction and as a consequence the Stieltjes transform of some spectral measure. Then, using classic asymptotic expansion properties of hypergeometric functions, the representation of the Laplace transform θ* by means of Kummer's function is obtained. This allows us to recover an earlier result obtained via complex analysis and the use of the strong Markov property satisfied by the occupation process {Λt}. The continued fraction representation enables us to further characterize the distribution of the random variable θ.

Sequences of orthogonal Laurent polynomials, bi-orthogonality and quadrature formulas on the unit circle

Contiguous relations, basic hypergeometric functions, and orthogonal polynomials, I

The effectiveness of dual integral equations for handling mixed boundary value problems has established them as an important tool for applied mathematicians. Their many applications in mathematical physics have prompted extensive research over the last 25 years, and many researchers have made significant contributions to the methodology of solving and to the applications of dual integral equations. However, until now, much of this work has been available only in the form of research papers scattered throughout different journals.In Advances in Dual Integral Equations, the authors systematically present some of the recent developments in dual integral equations involving various special functions as kernel. They examine dual integral equations with Bessel, Legendre, and trigonometric functions as kernel plus dual integral equations involving inverse Mellin transforms. These can be particularly useful in studying certain mixed boundary value problems involving homogeneous media in continuum mechanics. However, when dealing with problems involving non-homogenous media, the corresponding equations may have different kernels. This application prompts the authors to conclude with a discussion of hybrid dual integral equations-mixed kernels with generalized associated Legendre functions and mixed kernels involving Bessel functions.Researchers in the theory of elasticity, fluid dynamics, and mathematical physics will find Advances in Dual Integral Equations a concise, one-stop resource for recent work addressing special functions as kernel.

The Berlekamp—Massey algorithm (further, the BMA) is interpreted as an algorithm for constructing Pade approximations to the Laurent series over an arbitrary field with singularity at infinity. It is shown that the BMA is an iterative procedure for constructing the sequence of polynomials orthogonal to the corresponding space of polynomials with respect to the inner product determined by the given series. The BMA is used to expand the exponential in continued fractions and calculate its Pade approximations.

ZnO tetrapod materials for functional applications

We continue with analyzing quadrature formulas of high degree of precision for computing the Fourier coefficients in expansions of functions with respect to a system of orthogonal polynomials, started recently by Bojanov and Petrova [Quadrature formulae for Fourier coefficients, J. Comput. Appl. Math. 231 (2009), 378–391] and we extend their results. Construction of new Gaussian quadrature formulas for the Fourier coefficients of a function, based on the values of the function and its derivatives, is considered. We prove the existence and uniqueness of Kronrod extensions with multiple nodes of standard Gaussian quadrature formulas with multiple nodes for several weight functions, in order to construct some new generalizations of quadrature formulas for the Fourier coefficients. For the quadrature formulas for the Fourier coefficients based on the zeros of the corresponding orthogonal polynomials we construct Kronrod extensions with multiple nodes and highest algebraic degree of precision. For this very desirable kind of extension there do not exist any results in the theory of standard quadrature formulas.