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

Orthogonal Laurent polynomials and strong moment theory: a survey

The Strong Chebyshev Distribution and Orthogonal Laurent Polynomials

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.

The extended Krylov subspace method and orthogonal Laurent polynomials

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.

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

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.

Orthogonal polynomials and some q-beta integrals of Ramanujan

Sums of lognormal random variables occur in many wireless communications problems, in part, because a shadowing environment is well modeled by a lognormal distribution. A closed-form expression does not exist for the lognormal characteristic function; as a result no closed-form expression can be derived for the sum distribution. Many approximations exist that are based on approximating a sum of lognormal random variables as another lognormal random variable. A new method for calculating an approximation to a lognormal sum distribution, based on curve fitting on lognormal probability paper is introduced in this paper. Accurate, simple closed-form approximations to distributions and densities of lognormal sum random variables are presented.

Co-channel interference is a major factor in limiting the capacity and link quality in cellular communications. As the co-channel interference is modeled by lognormal distribution, sum of the co-channel interferences of neighboring cells is represented by the sum of lognormal Random Variables (RVs) which has no closed-form expression. Assuming independent, identically distributed (iid) RVs, the sum of lognormal RVs has been approximated by another log-normally distributed RV in the literature. In practice, the co-channel interference RVs have identical standard deviations (SDs) and different means. In this paper, first a new method based on curve fitting is proposed to approximate the sum of two log-normally distributed RV's with identical SDs and different means. Then a recursive method using the surface fitting is developed for approximating the sum of more than two lognormal RVs. Results show that the proposed method can approximate the first and the second moments of the resulting RV very well.

On matrix Hurwitz type polynomials and their interrelations to Stieltjes positive definite sequences and orthogonal matrix polynomials

We establish a combinatorial realization of continued fractions as quotients of cardinalities of sets. These sets are sets of perfect matchings of certain graphs, the snake graphs, that appear naturally in the theory of cluster algebras. To a continued fraction $[a_{1},a_{2},\ldots ,a_{n}]$ we associate a snake graph ${\mathcal{G}}[a_{1},a_{2},\ldots ,a_{n}]$ such that the continued fraction is the quotient of the number of perfect matchings of ${\mathcal{G}}[a_{1},a_{2},\ldots ,a_{n}]$ and ${\mathcal{G}}[a_{2},\ldots ,a_{n}]$. We also show that snake graphs are in bijection with continued fractions. We then apply this connection between cluster algebras and continued fractions in two directions. First we use results from snake graph calculus to obtain new identities for the continuants of continued fractions. Then we apply the machinery of continued fractions to cluster algebras and obtain explicit direct formulas for quotients of elements of the cluster algebra as continued fractions of Laurent polynomials in the initial variables. Building on this formula, and using classical methods for infinite periodic continued fractions, we also study the asymptotic behavior of quotients of elements of the cluster algebra.

Sharp Estimates for Jacobi Matrices and Chain Sequences

Contiguous relations, basic hypergeometric functions, and orthogonal polynomials. II. Associated big q-Jacobi polynomials

These notes are an exposition of work of Francon–Viennot, Flajolet and Viennot dealing with combinatorial aspects of the theory of continued fractions. We have added here some of the missing details, some of the proofs that were not given in the original works, and hopefully have corrected some of the errors without creating new ones.