differences

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1: 28.27 Addition Theorems

… ►Addition theorems provide important connections between Mathieu functions with different parameters and in different coordinate systems. …

2: 3.6 Linear Difference Equations

§3.6 Linear Difference Equations

… ► … ► … ►The difference equation … …

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

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

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

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§18.22(ii) Difference Equations in $x$

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Table 18.22.2: Difference equations (18.22.12) for Krawtchouk, Meixner, and Charlier polynomials.

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$p_n(x)$	$A(x)$	$C(x)$	${\lambda }_n$
…

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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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Symbols:

$C_n(x;a)$: Charlier polynomial, $\mathrm{\Delta }$: forward difference operator, $n$: nonnegative integer and $x$: real variable

Referenced by:

§18.22(iii)

Permalink:

http://dlmf.nist.gov/18.22.E25

Encodings:

pMML, png, TeX

See also:

Annotations for §18.22(iii), §18.22(iii), §18.22 and Ch.18

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8: Possible Errors in DLMF

… ►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. …

9: Sidebar 9.SB1: Supernumerary Rainbows

… ►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. …

10: 16.25 Methods of Computation

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