# via divided differences

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## 1—10 of 21 matching pages

##### 1: 3.3 Interpolation

###### §3.3(iii) Divided Differences

…##### 2: 19.36 Methods of Computation

###### §19.36(iii) Via Theta Functions

…##### 3: 18.30 Associated OP’s

##### 4: 3.6 Linear Difference Equations

###### §3.6 Linear Difference Equations

… ► … ►See also Gautschi (1967) and Gil et al. (2007a, Chapter 4) for the computation of recessive solutions via continued fractions. … ►The difference equation … …##### 5: 11.13 Methods of Computation

###### §11.13(v) Difference Equations

►Sequences of values of ${\mathbf{H}}_{\nu}\left(z\right)$ and ${\mathbf{L}}_{\nu}\left(z\right)$, with $z$ fixed, can be computed by application of the inhomogeneous difference equations (11.4.23) and (11.4.25). …##### 6: 18.18 Sums

##### 7: 22.20 Methods of Computation

###### §22.20(i) Via Theta Functions

►A powerful way of computing the twelve Jacobian elliptic functions for real or complex values of both the argument $z$ and the modulus $k$ is to use the definitions in terms of theta functions given in §22.2, obtaining the theta functions via methods described in §20.14. … ►for $n\ge 1$, where the square root is chosen so that $\mathrm{ph}{b}_{n}=\frac{1}{2}(\mathrm{ph}{a}_{n-1}+\mathrm{ph}{b}_{n-1})$, where $\mathrm{ph}{a}_{n-1}$ and $\mathrm{ph}{b}_{n-1}$ are chosen so that their difference is numerically less than $\pi $. … ►From (22.7.1), ${k}_{1}=\frac{1}{19}$ and $x/(1+{k}_{1})=0.19$. … ►If $k={k}^{\prime}=1/\sqrt{2}$, then three iterations of (22.20.1) give $M=\mathrm{0.84721\hspace{0.33em}30848}$, and from (22.20.6) $K=\pi /(2M)=\mathrm{1.85407\hspace{0.33em}46773}$ — in agreement with the value of ${\left(\mathrm{\Gamma}\left(\frac{1}{4}\right)\right)}^{2}/\left(4\sqrt{\pi}\right)$; compare (23.17.3) and (23.22.2). …##### 8: 18.19 Hahn Class: Definitions

*Askey scheme*extends the three families of classical OP’s (Jacobi, Laguerre and Hermite) with eight further families of OP’s for which the role of the differentiation operator $\frac{d}{dx}$ in the case of the classical OP’s is played by a suitable difference operator. … ►

*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.

*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.