Three-term Recurrence Formula Research Articles

Orthogonal polynomials on several intervals via a polynomial mapping

Starting from a sequence { p n ( x ; μ 0 ) } \{ {p_n}(x;\,{\mu _0})\} of orthogonal polynomials with an orthogonality measure μ 0 {\mu _0} supported on E 0 ⊂ [ − 1 , 1 ] {E_0} \subset [ - 1,\,1] , we construct a new sequence { p n ( x ; μ ) } \{ {p_n}(x;\,\mu )\} of orthogonal polynomials on E = T − 1 ( E 0 ) E = {T^{ - 1}}({E_0}) ( T T is a polynomial of degree N N ) with an orthogonality measure μ \mu that is related to μ 0 {\mu _0} . If E 0 = [ − 1 , 1 ] {E_0} = [ - 1,\,1] , then E = T − 1 ( [ − 1 , 1 ] ) E = {T^{ - 1}}([ - 1,\,1]) will in general consist of N N intervals. We give explicit formulas relating { p n ( x ; μ ) } \{ {p_n}(x;\,\mu )\} and { p n ( x ; μ 0 ) } \{ {p_n}(x;\,{\mu _0})\} and show how the recurrence coefficients in the three-term recurrence formulas for these orthogonal polynomials are related. If one chooses T T to be a Chebyshev polynomial of the first kind, then one gets sieved orthogonal polynomials.

Representations and bounds for zeros of orthogonal polynomials and eigenvalues of sign-symmetric tri-diagonal matrices

We consider a sequence ¦ p n ¦ of orthogonal polynomials defined by a three-term recurrence formula. Representations and bounds are derived for the endpoints of the smallest interval containing the (real and distinct) zeros of P n in terms of the parameters in the recurrence relation. These results are brought to light by viewing (−1) n P n as the characteristic polynomial of a sign-symmetric tri-diagonal matrix of order n. Our findings are subsequently used to obtain new proofs for a number of bounds on the endpoints of the true and limit intervals of orthogonality for the sequence ¦ p n ¦.

Scattering Theory and Polynomials Orthogonal on the Real Line

The techniques of scattering theory are used to study polynomials orthogonal on a segment of the real line. Instead of applying these techniques to the usual three-term recurrence formula, we derive a set of two two-term recurrence formulas satisfied by these polynomials. One of the advantages of these new recurrence formulas is that the Jost function is related, in the limit as $n \to \infty$, to the solution of one of the recurrence formulas with the boundary conditions given at $n = 0$. In this paper we investigate the properties of the Jost function and the spectral function assuming the coefficients in the recurrence formulas converge at a particular rate.

Orthogonal Polynomials Whose Distribution Functions Have Finite Point Spectra

A counterexample is given to an assertion by K. M. Case that if the coefficients in the three-term recurrence formula for orthogonal polynomials converge as fast as $n^{ - 2} $, the corresponding distribution function has only finitely many discrete points in its spectrum. Some positive results concerning this situation are also given, and continuity of the distribution function is investigated.

Scattering theory and polynomials orthogonal on the real line

The techniques of scattering theory are used to study polynomials orthogonal on a segment of the real line. Instead of applying these techniques to the usual three-term recurrence formula, we derive a set of two two-term recurrence formulas satisfied by these polynomials. One of the advantages of these new recurrence formulas is that the Jost function is related, in the limit as n → ∞ n\, \to \,\infty , to the solution of one of the recurrence formulas with the boundary conditions given at n = 0 n\, = \,0 . In this paper we investigate the properties of the Jost function and the spectral function assuming the coefficients in the recurrence formulas converge at a particular rate.

General integral transform of the exponential integrals

General integral transform of the exponential integralsEn is considered and will be denoted asB(k)n(τ). Different expressions and the equations satisfied byB(k)n are developed. Two-term recurrence formula forB(k)n(0) and three-term recurrence formula forB(k)n(τ); τ≠0 will be established for a givenk≥1 andn=2,3, ...,N. The computational algorithms based on these formulae are also constructed for the casesk=1,2,3, andn≥2. Finally the numerical results fork=2,3 andn=2(1)25 are presented to 15-digit accuracy

On the A-transform of the exponential integrals

In this paper, a technique of recursive analysis is developed for the integral transform A of the exponential integral functionsE n which is denoted as λ n (τ). The main result of this analysis enables us to establish a two-term recurrence formula for λ n (0) and a three-term recurrence formula for λ n (τ); τ≠0. A computational algorithm based on these formulae is also constructed and its numerical results forn=2(1)25 are presented to 15-digit accuracy.

Converting Interpolation Series into Chebyshev Series by Recurrence Formulas

Interpolation series (divided difference, Gregory-Newton, Gauss, Stirling, Bessel) are converted into Chebyshev (or Jacobi) series by applying a previously derived general five-term recurrence formula [3]. It employs the coefficients in three-term linear recurrence formulas (same kind as for orthogonal polynomials) which have been found for the mth degree nonorthogonal polynomial coefficients of the differences used in the interpolation series. In the Gauss, Stirling and Bessel series, the coefficients in the recurrence formulas vary with the parity of m. The basic five-term recurrence formula is applicable also to: (1) inter- and intraconversion of power series in $ax + b$, divided difference and equal-interval interpolation series (including subtabulation), and Chebyshev series, (2) obtaining Chebyshev series for solutions of difference equations, (3) the derivation of formulas for Chebyshev coefficients in terms of differences, and (4) the conversion of interpolation series into Chebyshev series, for more than one variable.

Convolutions of Orthonormal Polynomials

In this paper, we determine all pairs of orthogonal polynomial sequences $\{ p_n (x)\} $ and $\{ q_n (x)\} $, such that their convolution, \[ Q_n (x,y) = \sum_{k = 0}^n {p_k (x)q_{n - k} (y)} ,\quad n \geqq 0.\] defines $\{ Q_n (x,y)\} $ as an orthogonal polynomial sequence in x for all y. All such triples are determined explicitly in terms of their three-term recurrence formulas. Generating functions and “explicit” representation formulas are obtained. The resulting sequences are found to consist of a class of orthogonal polynomials characterized by J. Meixner (which class includes the Laguerre and Hermite polynomials) together with a new class of orthogonal polynomials which includes the orthogonal q-polynomials of Al-Salam and Carlitz. Explicit orthogonality relations are found for one new special case of this latter class.

Converting interpolation series into Chebyshev series by recurrence formulas

Interpolation series (divided difference, Gregory-Newton, Gauss, Stirling, Bessel) are converted into Chebyshev (or Jacobi) series by applying a previously derived general five-term recurrence formula [3]. It employs the coefficients in three-term linear recurrence formulas (same kind as for orthogonal polynomials) which have been found for the mth degree nonorthogonal polynomial coefficients of the differences used in the interpolation series. In the Gauss, Stirling and Bessel series, the coefficients in the recurrence formulas vary with the parity of m. The basic five-term recurrence formula is applicable also to: (1) inter- and intraconversion of power series in a x + b ax + b , divided difference and equal-interval interpolation series (including subtabulation), and Chebyshev series, (2) obtaining Chebyshev series for solutions of difference equations, (3) the derivation of formulas for Chebyshev coefficients in terms of differences, and (4) the conversion of interpolation series into Chebyshev series, for more than one variable.

A recurrence scheme for converting from one orthogonal expansion into another

A generalization of a scheme of Hamming for converting a polynomial P n (x) into a Chebyshev series is combined with a recurrence scheme of Clenshaw for summing any finite series whose terms satisfy a three-term recurrence formula. An application to any two orthogonal expansions P n ( x ) = ∑ n m =0 a m q m ( x ) = ∑ n m =0 A m Q m ( x ) enables one to obtain A m directly from a m , m = 0(1) n , by a five-term recurrence scheme.

Miniaturized tables of Bessel functions. II

In a previous study, we discussed the expansion of two-parameter functions in a double series of Chebyshev polynomials, and, in particular, we presented coefficients for the evaluation of the modified Bessel function ( 2 z / π ) 1 / 2 e z K v ( z ) {(2z/\pi )^{1/2}}{e^z}{K_v}(z) to 20 decimals for all z ≧ 5 z \geqq 5 and all v , 0 ≦ v ≦ 1 v,0 \leqq v \leqq 1 . In the present study, we give similar coefficients for the evaluation of g e − z z − μ I v ( z ) g{e^{ - z}}{z^{ - \mu }}{I_v}(z) to at least 20 decimals where I v ( z ) {I_v}(z) is the modified Bessel function of the first kind and g and μ \mu are certain constants which depend on the range of the parameter and variable for four different situations. The ranges are ( 1 ) 0 > z ≦ 8 , 0 ≦ v ≦ 4 ; ( 2 ) 0 > z ≦ 8 , 4 ≦ v ≦ 8 ; ( 3 ) z ≧ 8 , − 1 ≦ v ≦ 0 ; ( 4 ) z ≧ 8 , 0 ≦ v ≦ 1 (1)\;0 > z \leqq 8,0 \leqq v \leqq 4;(2)\;0 > z \leqq 8,4 \leqq v \leqq 8;(3)\;z \geqq 8, - 1 \leqq v \leqq 0;(4)\;z \geqq 8,0 \leqq v \leqq 1 .

Miniaturized tables of Bessel functions

In this report, we discuss the representation of bivariate functions in double series of Chebyshev polynomials. For an application, we tabulate coefficients which are accurate to 20 decimals for the evaluation of ( 2 z / π ) 1 / 2 e z K v ( z ) {(2z/\pi )^{1/2}}{e^z}{K_v}(z) for all z ≧ 5 z \geqq 5 and all v , 0 ≦ v ≦ 1 v,0 \leqq v \leqq 1 . Since K v ( z ) {K_v}(z) is an even function in v and satisfies a three-term recurrence formula in v which is stable when used in the forward direction, we can readily evaluate K v ( z ) {K_v}(z) for all z ≧ 5 z \geqq 5 and all v ≧ 0 v \geqq 0 . Only 205 coefficients are required to achieve an accuracy of about 20 decimals for the z and v ranges described. Extension of these ideas for the evaluation of all Bessel functions and other important bivariate functions is under way.

A Characterization and a Class of Distribution Functions for the Stieltjes-Wigert Polynomials

In his classic memoir on the moment problem that bears his name, Stieltjes [2] exhibited1as an example of an indeterminate (Stieltjes) moment sequence.Stieltjes also obtained the corresponding S-fraction and thus implicitly obtained the three-term recurrence formula satisfied by the corresponding orthogonal polynomials.