# differences

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## 1—10 of 176 matching pages

##### 1: 28.27 Addition Theorems

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►Addition theorems provide important connections between Mathieu functions with different parameters and in different coordinate systems.
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##### 2: 3.6 Linear Difference Equations

##### 3: 18.1 Notation

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###### $x$-Differences

►Forward differences: … ►Backward differences: … ►Central differences in imaginary direction: … ►In Koekoek et al. (2010) ${\delta}_{x}$ denotes the operator $\mathrm{i}{\delta}_{x}$.##### 4: 2.9 Difference Equations

###### §2.9 Difference Equations

… ►or equivalently the second-order homogeneous linear difference equation …in which $\mathrm{\Delta}$ is the forward difference operator (§3.6(i)). … ►For analogous results for difference equations of the form … ►##### 5: Simon Ruijsenaars

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►His main research interests cover integrable systems, special functions, analytic difference equations, classical and quantum mechanics, and the relations between these areas.
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##### 6: 3.3 Interpolation

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###### §3.3(iii) Divided Differences

… ►Explicitly, the*divided difference of order $n$*is given by … ►This represents the Lagrange interpolation polynomial in terms of divided differences: …Newton’s formula has the advantage of allowing easy updating: incorporation of a new point ${z}_{n+1}$ requires only addition of the term with $\left[{z}_{0},{z}_{1},\mathrm{\dots},{z}_{n+1}\right]f$ to (3.3.38), plus the computation of this divided difference. …For example, for $k+1$ coincident points the limiting form is given by $\left[{z}_{0},{z}_{0},\mathrm{\dots},{z}_{0}\right]f={f}^{(k)}({z}_{0})/k!$. …##### 7: 18.22 Hahn Class: Recurrence Relations and Differences

###### §18.22 Hahn Class: Recurrence Relations and Differences

… ►###### §18.22(ii) Difference Equations in $x$

… ►${p}_{n}(x)$ | $A(x)$ | $C(x)$ | ${\lambda}_{n}$ |
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###### §18.22(iii) $x$-Differences

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18.22.25
$${\mathrm{\Delta}}_{x}{C}_{n}(x;a)=-\frac{n}{a}{C}_{n-1}(x;a),$$

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##### 8: Possible Errors in DLMF

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►One source of confusion, rather than actual errors, are some new functions which differ from those in Abramowitz and Stegun (1964) by scaling, shifts or constraints on the domain; see the Info box (click or hover over the icon) for links to defining formula.
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##### 9: Sidebar 9.SB1: Supernumerary Rainbows

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►The faint line below the main colored arc is a ‘supernumerary rainbow’, produced by the interference of different sun-rays traversing a raindrop and emerging in the same direction.
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##### 10: 16.25 Methods of Computation

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►There is, however, an added feature in the numerical solution of differential equations and difference equations (recurrence relations).
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