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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
x -Differences
Forward differences: … Backward differences: … Central differences in imaginary direction: … In Koekoek et al. (2010) δ x denotes the operator i δ x .
4: 2.9 Difference Equations
§2.9 Difference Equations
or equivalently the second-order homogeneous linear difference equation …in which Δ 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
§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 [ z 0 , z 1 , , z n + 1 ] 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 [ z 0 , z 0 , , z 0 ] 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
Table 18.22.2: Difference equations (18.22.12) for Krawtchouk, Meixner, and Charlier polynomials.
p n ( x ) A ( x ) C ( x ) λ n
§18.22(iii) x -Differences
18.22.25 Δ x C n ( x ; a ) = n a C n 1 ( x ; a ) ,
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 [Uncaptioned image] 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). …