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11: 3.10 Continued Fractions

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Backward Recurrence Algorithm

… ►($\nabla$ is the backward difference operator.) …

12: 18.20 Hahn Class: Explicit Representations

… ►For comments on the use of the forward-difference operator ${\mathrm{\Delta }}_x$, the backward-difference operator ${\nabla }_x$, and the central-difference operator ${\delta }_x$, see §18.2(ii). … ►

18.20.1 $$p_n(x)=\frac{1}{{\kappa }_n{w}_x}{\nabla }_x^n\left(w_x\prod _{\mathrm{\ell }=0}^{n-1}F(x+\mathrm{\ell })\right),$$ $x\in X$.

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Symbols:

${\nabla }_x$: backward difference, $\in$: element of, $p_n(x)$: polynomial of degree $n$ , $w_x$: weights, $X$: subset of $ℝ$ , $\mathrm{\ell }$: nonnegative integer, $n$: nonnegative integer, $x$: real variable, $F(x)$: coefficient and ${\kappa }_n$: coefficient

A&S Ref:

22.17.2

Referenced by:

§18.20(i), Table 18.20.1

Permalink:

http://dlmf.nist.gov/18.20.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §18.20(i), §18.20(i), §18.20 and Ch.18

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13: Mathematical Introduction

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$ℂ$	complex plane (excluding infinity).
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$\nabla$ (or ${\nabla }_x$)	backward difference operator: $\nabla f(x)=f(x)-f(x-1)$. (See also del operator in the Notations section.)
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14: 6.20 Approximations

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Luke (1969b, p. 25) gives a Chebyshev expansion near infinity for the confluent hypergeometric $U$-function (§13.2(i)) from which Chebyshev expansions near infinity for $E_1\left(z\right)$, $\mathrm{f}\left(z\right)$, and $\mathrm{g}\left(z\right)$ follow by using (6.11.2) and (6.11.3). Luke also includes a recursion scheme for computing the coefficients in the expansions of the $U$ functions. If the scheme can be used in backward direction.

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15: 15.19 Methods of Computation

… ►For example, in the half-plane $\mathrm{\Re }z\le \frac{1}{2}$ we can use (15.12.2) or (15.12.3) to compute $F(a,b;c+N+1;z)$ and $F(a,b;c+N;z)$, where $N$ is a large positive integer, and then apply (15.5.18) in the backward direction. …

16: 18.19 Hahn Class: Definitions

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Hahn class (or linear lattice class). These are OP’s $p_n(x)$ where the role of $\frac{d}{dx}$ is played by ${\mathrm{\Delta }}_x$ or ${\nabla }_x$ or ${\delta }_x$ (see §18.1(i) for the definition of these operators). The Hahn class consists of four discrete and two continuous families.

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Wilson class (or quadratic lattice class). These are OP’s $p_n(x)=p_n(\lambda (y))$ ($p_n(x)$ of degree $n$ in $x$, $\lambda (y)$ quadratic in $y$) where the role of the differentiation operator is played by $\frac{{\mathrm{\Delta }}_y}{{\mathrm{\Delta }}_y(\lambda (y))}$ or $\frac{{\nabla }_y}{{\nabla }_y(\lambda (y))}$ or $\frac{{\delta }_y}{{\delta }_y(\lambda (y))}$. The Wilson class consists of two discrete and two continuous families.

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17: 18.25 Wilson Class: Definitions

… ►For the Wilson class OP’s $p_n(x)$ with $x=\lambda (y)$: if the $y$-orthogonality set is $\{0,1,\mathrm{\dots },N\}$, then the role of the differentiation operator $d/dx$ in the Jacobi, Laguerre, and Hermite cases is played by the operator ${\mathrm{\Delta }}_y$ followed by division by ${\mathrm{\Delta }}_y(\lambda (y))$, or by the operator ${\nabla }_y$ followed by division by ${\nabla }_y(\lambda (y))$. …

18: Bibliography O

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Note on backward recurrence algorithms. Math. Comp. 26 (120), pp. 941–947.

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External Links:

FullText, MathReview (Walter Gautschi), ZentralBlatt

Referenced by:

§10.74(iv), §3.6(v).

Encodings:

BibTeX

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19: 18.26 Wilson Class: Continued

… ►For comments on the use of the forward-difference operator ${\mathrm{\Delta }}_x$, the backward-difference operator ${\nabla }_x$, and the central-difference operator ${\delta }_x$, see §18.2(ii). …

20: 18.2 General Orthogonal Polynomials

… ►If the orthogonality discrete set $X$ is $\{0,1,\mathrm{\dots },N\}$ or $\{0,1,2,\mathrm{\dots }\}$, then the role of the differentiation operator $d/dx$ in the case of classical OP’s (§18.3) is played by ${\mathrm{\Delta }}_x$, the forward-difference operator, or by ${\nabla }_x$, the backward-difference operator; compare §18.1(i). …