# differential operators

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##### 1: 17.2 Calculus
17.2.41 $\mathcal{D}_{q}f(z)=\begin{cases}\dfrac{f(z)-f(zq)}{(1-q)z},&z\neq 0,\\ f^{\prime}(0),&z=0,\end{cases}$
17.2.42 $f^{[n]}(z)=\mathcal{D}_{q}^{n}f(z)=\begin{cases}z^{-n}(1-q)^{-n}\sum_{j=0}^{n}% q^{-nj+\genfrac{(}{)}{0.0pt}{}{j+1}{2}}(-1)^{j}\genfrac{[}{]}{0.0pt}{}{n}{j}_{% q}f(zq^{j}),&z\neq 0,\\ \dfrac{f^{(n)}(0)\left(q;q\right)_{n}}{n!(1-q)^{n}},&z=0.\end{cases}$
17.2.44 $\mathcal{D}_{q}^{n}(f(z)g(z))=\sum_{j=0}^{n}\genfrac{[}{]}{0.0pt}{}{n}{j}_{q}f% ^{[n-j]}(zq^{j})g^{[j]}(z).$
##### 3: 17.6 ${{}_{2}\phi_{1}}$ Function
###### Iterations of $\mathcal{D}$
17.6.25 $\mathcal{D}_{q}^{n}{{}_{2}\phi_{1}}\left({a,b\atop c};q,zd\right)=\frac{\left(% a,b;q\right)_{n}d^{n}}{\left(c;q\right)_{n}(1-q)^{n}}{{}_{2}\phi_{1}}\left({aq% ^{n},bq^{n}\atop cq^{n}};q,dz\right),$
17.6.26 $\mathcal{D}_{q}^{n}\left(\frac{\left(z;q\right)_{\infty}}{\left(abz/c;q\right)% _{\infty}}{{}_{2}\phi_{1}}\left({a,b\atop c};q,z\right)\right)=\frac{\left(c/a% ,c/b;q\right)_{n}}{\left(c;q\right)_{n}(1-q)^{n}}\left(\frac{ab}{c}\right)^{n}% \frac{\left(zq^{n};q\right)_{\infty}}{\left(abz/c;q\right)_{\infty}}{{}_{2}% \phi_{1}}\left({a,b\atop cq^{n}};q,zq^{n}\right).$
17.6.27 $z(c-abqz)\mathcal{D}_{q}^{2}{{}_{2}\phi_{1}}\left({a,b\atop c};q,z\right)+% \left(\frac{1-c}{1-q}+\frac{(1-a)(1-b)-(1-abq)}{1-q}z\right)\mathcal{D}_{q}{{}% _{2}\phi_{1}}\left({a,b\atop c};q,z\right)-\frac{(1-a)(1-b)}{(1-q)^{2}}{{}_{2}% \phi_{1}}\left({a,b\atop c};q,z\right)=0.$
##### 4: 16.8 Differential Equations
16.8.3 $\left(\vartheta(\vartheta+b_{1}-1)\cdots(\vartheta+b_{q}-1)-z(\vartheta+a_{1})% \cdots(\vartheta+a_{p})\right)w=0.$
##### 5: 19.4 Derivatives and Differential Equations
Then
19.4.8 $(k{k^{\prime}}^{2}D_{k}^{2}+(1-3k^{2})D_{k}-k)F\left(\phi,k\right)=\frac{-k% \sin\phi\cos\phi}{(1-k^{2}{\sin}^{2}\phi)^{3/2}},$
19.4.9 $(k{k^{\prime}}^{2}D_{k}^{2}+{k^{\prime}}^{2}D_{k}+k)E\left(\phi,k\right)=\frac% {k\sin\phi\cos\phi}{\sqrt{1-k^{2}{\sin}^{2}\phi}}.$
##### 6: 16.21 Differential Equation
16.21.1 $\left((-1)^{p-m-n}z(\vartheta-a_{1}+1)\cdots(\vartheta-a_{p}+1)-(\vartheta-b_{% 1})\cdots(\vartheta-b_{q})\right)w=0,$
##### 7: 1.16 Distributions
1.16.30 $\mathbf{D}=\left(\frac{1}{\mathrm{i}}\frac{\partial}{\partial x_{1}},\frac{1}{% \mathrm{i}}\frac{\partial}{\partial x_{2}},\ldots,\frac{1}{\mathrm{i}}\frac{% \partial}{\partial x_{n}}\right).$
1.16.32 $P(\mathbf{D})=\sum_{\boldsymbol{{\alpha}}}c_{\boldsymbol{{\alpha}}}\mathbf{D}^% {\alpha}=\sum_{\boldsymbol{{\alpha}}}c_{\boldsymbol{{\alpha}}}\left(\frac{1}{% \mathrm{i}}\frac{\partial}{\partial x_{1}}\right)^{\alpha_{1}}\dots\left(\frac% {1}{\mathrm{i}}\frac{\partial}{\partial x_{n}}\right)^{\alpha_{n}}.$
Here $\boldsymbol{{\alpha}}$ ranges over a finite set of multi-indices, $P(\mathbf{x})$ is a multivariate polynomial, and $P(\mathbf{D})$ is a partial differential operator. …
1.16.36 $\left\langle\mathscr{F}\left(P(\mathbf{D})u\right),\phi\right\rangle=\left% \langle P_{-}\mathscr{F}\left(u\right),\phi\right\rangle=\left\langle\mathscr{% F}\left(u\right),P_{-}\phi\right\rangle,$
1.16.37 $\left\langle\mathscr{F}\left(Pu\right),\phi\right\rangle=\left\langle P(% \mathbf{D})\mathscr{F}\left(u\right),\phi\right\rangle,$
##### 8: 18.38 Mathematical Applications
###### Quadrature “Extended” to Pseudo-Spectral (DVR) Representations of Operators in One and Many Dimensions
The basic ideas of Gaussian quadrature, and their extensions to non-classical weight functions, and the computation of the corresponding quadrature abscissas and weights, have led to discrete variable representations, or DVRs, of Sturm–Liouville and other differential operators. …
##### 9: 16.19 Identities
16.19.5 $\vartheta{G^{m,n}_{p,q}}\left(z;{a_{1},\dots,a_{p}\atop b_{1},\dots,b_{q}}% \right)={G^{m,n}_{p,q}}\left(z;{a_{1}-1,a_{2},\dots,a_{p}\atop b_{1},\dots,b_{% q}}\right)+(a_{1}-1){G^{m,n}_{p,q}}\left(z;{a_{1},\dots,a_{p}\atop b_{1},\dots% ,b_{q}}\right),$
##### 10: 1.5 Calculus of Two or More Variables
1.5.3 $\frac{\partial f}{\partial x}=D_{x}f=f_{x}=\lim_{h\to 0}\frac{f(x+h,y)-f(x,y)}% {h},$
1.5.4 $\frac{\partial f}{\partial y}=D_{y}f=f_{y}=\lim_{h\to 0}\frac{f(x,y+h)-f(x,y)}% {h}.$