# difference operators

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##### 1: 18.1 Notation
###### $x$-Differences
Forward differences: … Backward differences: … Central differences in imaginary direction: … In Koekoek et al. (2010) $\delta_{x}$ denotes the operator $\mathrm{i}\delta_{x}$.
##### 2: 18.2 General Orthogonal Polynomials
###### §18.2(ii) $x$-DifferenceOperators
If the orthogonality discrete set $X$ is $\{0,1,\dots,N\}$ or $\{0,1,2,\dots\}$, then the role of the differentiation operator $\ifrac{\mathrm{d}}{\mathrm{d}x}$ in the case of classical OP’s (§18.3) is played by $\Delta_{x}$, the forward-difference operator, or by $\nabla_{x}$, the backward-difference operator; compare §18.1(i). … If the orthogonality interval is $(-\infty,\infty)$ or $(0,\infty)$, then the role of $\ifrac{\mathrm{d}}{\mathrm{d}x}$ can be played by $\delta_{x}$, the central-difference operator in the imaginary direction (§18.1(i)). …
##### 3: 18.20 Hahn Class: Explicit Representations
For comments on the use of the forward-difference operator $\Delta_{x}$, the backward-difference operator $\nabla_{x}$, and the central-difference operator $\delta_{x}$, see §18.2(ii). …
##### 4: 18.26 Wilson Class: Continued
For comments on the use of the forward-difference operator $\Delta_{x}$, the backward-difference operator $\nabla_{x}$, and the central-difference operator $\delta_{x}$, see §18.2(ii). …
18.26.16 $\frac{\Delta_{y}\left(R_{n}\left(y(y+\gamma+\delta+1);\alpha,\beta,\gamma,% \delta\right)\right)}{\Delta_{y}\left(y(y+\gamma+\delta+1)\right)}=\frac{n(n+% \alpha+\beta+1)}{(\alpha+1)(\beta+\delta+1)(\gamma+1)}\*R_{n-1}\left(y(y+% \gamma+\delta+2);\alpha+1,\beta+1,\gamma+1,\delta\right).$
18.26.17 $\frac{\Delta_{y}\left(R_{n}\left(y(y+\gamma+\delta+1);\gamma,\delta,N\right)% \right)}{\Delta_{y}\left(y(y+\gamma+\delta+1)\right)}=-\frac{n}{(\gamma+1)N}\*% R_{n-1}\left(y(y+\gamma+\delta+2);\gamma+1,\delta,N-1\right).$
##### 5: 3.9 Acceleration of Convergence
3.9.2 $S=\sum_{k=0}^{\infty}(-1)^{k}2^{-k-1}\Delta^{k}a_{0},$
Here $\Delta$ is the forward difference operator:
3.9.3 $\Delta^{k}a_{0}=\Delta^{k-1}a_{1}-\Delta^{k-1}a_{0},$ $k=1,2,\dotsc$.
3.9.4 $\Delta^{k}a_{0}=\sum_{m=0}^{k}(-1)^{m}\genfrac{(}{)}{0.0pt}{}{k}{m}a_{k-m}.$
##### 6: Mathematical Introduction
 $\mathbb{C}$ complex plane (excluding infinity). … forward difference operator: $\Delta f(x)=f(x+1)-f(x)$. backward difference operator: $\nabla f(x)=f(x)-f(x-1)$. (See also del operator in the Notations section.) …
##### 7: 3.6 Linear Difference Equations
3.6.2 $a_{n}\Delta^{2}w_{n-1}+(2a_{n}-b_{n})\Delta w_{n-1}+(a_{n}-b_{n}+c_{n})w_{n-1}% =d_{n},$
where $\Delta w_{n-1}=w_{n}-w_{n-1}$, $\Delta^{2}w_{n-1}=\Delta w_{n}-\Delta w_{n-1}$, and $n\in\mathbb{Z}$. …
##### 8: 18.22 Hahn Class: Recurrence Relations and Differences
18.22.21 $\Delta_{x}K_{n}\left(x;p,N\right)=-\frac{n}{pN}K_{n-1}\left(x;p,N-1\right),$
##### 9: 18.25 Wilson Class: Definitions
For the Wilson class OP’s $p_{n}(x)$ with $x=\lambda(y)$: if the $y$-orthogonality set is $\{0,1,\dots,N\}$, then the role of the differentiation operator $\ifrac{\mathrm{d}}{\mathrm{d}x}$ in the Jacobi, Laguerre, and Hermite cases is played by the operator $\Delta_{y}$ followed by division by $\Delta_{y}(\lambda(y))$, or by the operator $\nabla_{y}$ followed by division by $\nabla_{y}(\lambda(y))$. Alternatively if the $y$-orthogonality interval is $(0,\infty)$, then the role of $\ifrac{\mathrm{d}}{\mathrm{d}x}$ is played by the operator $\delta_{y}$ followed by division by $\delta_{y}(\lambda(y))$. …
##### 10: 2.9 Difference Equations
2.9.2 $\Delta^{2}w(n)+(2+f(n))\Delta w(n)+(1+f(n)+g(n))w(n)=0,$ $n=0,1,2,\dots$,
in which $\Delta$ is the forward difference operator3.6(i)). …