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1: 17.2 Calculus

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17.2.41

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Defines:

${𝒟}_q$: $q$-differential operator (locally)

Symbols:

$q$: complex base and $z$: complex variable

Referenced by:

§17.2(iv), §18.27(i)

Permalink:

http://dlmf.nist.gov/17.2.E41

Encodings:

pMML, png, TeX

See also:

Annotations for §17.2(iv), §17.2 and Ch.17

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17.2.42

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Symbols:

$\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $!$: factorial (as in $n!$), ${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial), ${\left[\genfrac{}{}{0.0pt}{}{n}{m}\right]}_q$: $q$-binomial coefficient (or Gaussian polynomial), $q$: complex base, $j$: nonnegative integer, $n$: nonnegative integer, $z$: complex variable and ${𝒟}_q$: $q$-differential operator

Referenced by:

§17.2(iv)

Permalink:

http://dlmf.nist.gov/17.2.E42

Encodings:

pMML, png, TeX

See also:

Annotations for §17.2(iv), §17.2 and Ch.17

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17.2.43 $${𝒟}_q(f(z)g(z))=g(z)f^{[1]}(z)+f(zq)g^{[1]}(z).$$

ⓘ

Symbols:

$q$: complex base, $z$: complex variable and ${𝒟}_q$: $q$-differential operator

Referenced by:

§17.2(iv)

Permalink:

http://dlmf.nist.gov/17.2.E43

Encodings:

pMML, png, TeX

See also:

Annotations for §17.2(iv), §17.2(iv), §17.2 and Ch.17

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17.2.44 $${𝒟}_q^n(f(z)g(z))=\sum _{j=0}^n{\left[\genfrac{}{}{0.0pt}{}{n}{j}\right]}_q{f}^{[n-j]}(z{q}^j)g^{[j]}(z).$$

ⓘ

Symbols:

${\left[\genfrac{}{}{0.0pt}{}{n}{m}\right]}_q$: $q$-binomial coefficient (or Gaussian polynomial), $q$: complex base, $j$: nonnegative integer, $n$: nonnegative integer, $z$: complex variable and ${𝒟}_q$: $q$-differential operator

Referenced by:

§17.2(iv)

Permalink:

http://dlmf.nist.gov/17.2.E44

Encodings:

pMML, png, TeX

See also:

Annotations for §17.2(iv), §17.2(iv), §17.2 and Ch.17

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2: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions

§1.18 Linear Second Order Differential Operators and Eigenfunction Expansions

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Formally Self-Adjoint and Self-Adjoint Differential Operators: Self-Adjoint Extensions

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3: 17.6 ${}_2{}^{}\varphi _1^{}$ Function

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Iterations of $𝒟$

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17.6.25 $${𝒟}_q^n{}_2{}^{}\varphi _1^{}(\genfrac{}{}{0pt}{}{a,b}{c};q,zd)=\frac{{(a,b;q)}_n{d}^n}{{(c;q)}_n{(1-q)}^n}{}_2{}^{}\varphi _1^{}(\genfrac{}{}{0pt}{}{a{q}^n,b{q}^n}{c{q}^n};q,dz),$$

ⓘ

Symbols:

${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial), ${}_{r+1}{}^{}\varphi _s^{}(a_0,\mathrm{\dots },a_r;b_1,\mathrm{\dots },b_s;q,z)$ or ${}_{r+1}{}^{}\varphi _s^{}(\genfrac{}{}{0pt}{}{a_0,\mathrm{\dots },a_r}{b_1,\mathrm{\dots },b_s};q,z)$: basic hypergeometric (or $q$-hypergeometric) function, ${(a_1,a_2,\mathrm{\dots },a_r;q)}_n$: multiple $q$-Pochhammer symbol, $q$: complex base, $n$: nonnegative integer, $z$: complex variable and ${𝒟}_q$: $q$-differential operator

Permalink:

http://dlmf.nist.gov/17.6.E25

Encodings:

pMML, png, TeX

See also:

Annotations for §17.6(iv), §17.6(iv), §17.6 and Ch.17

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17.6.26 $${𝒟}_q^n\left(\frac{{(z;q)}_{\mathrm{\infty }}}{{(abz/c;q)}_{\mathrm{\infty }}}{}_2{}^{}\varphi _1^{}(\genfrac{}{}{0pt}{}{a,b}{c};q,z)\right)=\frac{{(c/a,c/b;q)}_n}{{(c;q)}_n{(1-q)}^n}{\left(\frac{ab}{c}\right)}^n\frac{{(z{q}^n;q)}_{\mathrm{\infty }}}{{(abz/c;q)}_{\mathrm{\infty }}}{}_2{}^{}\varphi _1^{}(\genfrac{}{}{0pt}{}{a,b}{c{q}^n};q,z{q}^n).$$

ⓘ

Symbols:

${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial), ${}_{r+1}{}^{}\varphi _s^{}(a_0,\mathrm{\dots },a_r;b_1,\mathrm{\dots },b_s;q,z)$ or ${}_{r+1}{}^{}\varphi _s^{}(\genfrac{}{}{0pt}{}{a_0,\mathrm{\dots },a_r}{b_1,\mathrm{\dots },b_s};q,z)$: basic hypergeometric (or $q$-hypergeometric) function, ${(a_1,a_2,\mathrm{\dots },a_r;q)}_n$: multiple $q$-Pochhammer symbol, $q$: complex base, $n$: nonnegative integer, $z$: complex variable and ${𝒟}_q$: $q$-differential operator

Permalink:

http://dlmf.nist.gov/17.6.E26

Encodings:

pMML, png, TeX

See also:

Annotations for §17.6(iv), §17.6(iv), §17.6 and Ch.17

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17.6.27 $$z(c-abqz){𝒟}_q^2{}_2{}^{}\varphi _1^{}(\genfrac{}{}{0pt}{}{a,b}{c};q,z)+\left(\frac{1-c}{1-q}+\frac{(1-a)(1-b)-(1-abq)}{1-q}z\right){𝒟}_q{}_2{}^{}\varphi _1^{}(\genfrac{}{}{0pt}{}{a,b}{c};q,z)-\frac{(1-a)(1-b)}{{(1-q)}^2}{}_2{}^{}\varphi _1^{}(\genfrac{}{}{0pt}{}{a,b}{c};q,z)=0.$$

ⓘ

Symbols:

${}_{r+1}{}^{}\varphi _s^{}(a_0,\mathrm{\dots },a_r;b_1,\mathrm{\dots },b_s;q,z)$ or ${}_{r+1}{}^{}\varphi _s^{}(\genfrac{}{}{0pt}{}{a_0,\mathrm{\dots },a_r}{b_1,\mathrm{\dots },b_s};q,z)$: basic hypergeometric (or $q$-hypergeometric) function, $q$: complex base, $z$: complex variable and ${𝒟}_q$: $q$-differential operator

Referenced by:

§17.6(iv)

Permalink:

http://dlmf.nist.gov/17.6.E27

Encodings:

pMML, png, TeX

See also:

Annotations for §17.6(iv), §17.6(iv), §17.6 and Ch.17

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4: 16.8 Differential Equations

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16.8.3 $$\left(\mathit{\vartheta }(\mathit{\vartheta }+b_1-1)\mathrm{\cdots }(\mathit{\vartheta }+b_q-1)-z(\mathit{\vartheta }+a_1)\mathrm{\cdots }(\mathit{\vartheta }+a_p)\right)w=0.$$

ⓘ

Symbols:

$p$: nonnegative integer, $q$: nonnegative integer, $z$: complex variable, $a,a_1,\mathrm{\dots },a_p$: real or complex parameters, $b,b_1,\mathrm{\dots },b_q$: real or complex parameters, $\mathit{\vartheta }$: differential operator and $w$: solution

Referenced by:

§16.8(ii), §16.8(ii), §16.8(ii)

Permalink:

http://dlmf.nist.gov/16.8.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §16.8(ii), §16.8 and Ch.16

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16.8.4 $$z^q{𝐷}^{q+1}w+\sum _{j=1}^q{z}^{j-1}({\alpha }_{j}z+{\beta }_j){𝐷}^{j}w+{\alpha }_{0}w=0,$$ $p\le q$,

ⓘ

Symbols:

$p$: nonnegative integer, $q$: nonnegative integer, $z$: complex variable, $𝐷$: differential operator, $w$: solution, ${\alpha }_j$: constant and ${\beta }_j$: constant

Referenced by:

§16.8(ii)

Permalink:

http://dlmf.nist.gov/16.8.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §16.8(ii), §16.8 and Ch.16

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16.8.5 $$z^q(1-z){𝐷}^{q+1}w+\sum _{j=1}^q{z}^{j-1}({\alpha }_{j}z+{\beta }_j){𝐷}^{j}w+{\alpha }_{0}w=0,$$ $p=q+1$,

ⓘ

Symbols:

$p$: nonnegative integer, $q$: nonnegative integer, $z$: complex variable, $𝐷$: differential operator, $w$: solution, ${\alpha }_j$: constant and ${\beta }_j$: constant

Referenced by:

§16.8(ii)

Permalink:

http://dlmf.nist.gov/16.8.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §16.8(ii), §16.8 and Ch.16

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5: 19.4 Derivatives and Differential Equations

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19.4.8 $$(kk_{}^{\prime }{}_{}{}^2{D}_k^2+(1-3k^2)D_k-k)F(\varphi ,k)=\frac{-k\sin \varphi \cos \varphi }{{(1-k^2{\sin }^2\varphi )}^{3/2}},$$

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Symbols:

$\cos z$: cosine function, $F(\varphi ,k)$: Legendre’s incomplete elliptic integral of the first kind, $\sin z$: sine function, $\varphi$: real or complex argument, $k$: real or complex modulus, $k^{\prime }$: complementary modulus and $D_k$: differential operator

Referenced by:

§19.4(ii)

Permalink:

http://dlmf.nist.gov/19.4.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §19.4(ii), §19.4 and Ch.19

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19.4.9 $$(kk_{}^{\prime }{}_{}{}^2{D}_k^2+k_{}^{\prime }{}_{}{}^2{D}_k+k)E(\varphi ,k)=\frac{k\sin \varphi \cos \varphi }{\sqrt{1-k^2{\sin }^2\varphi }}.$$

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Symbols:

$\cos z$: cosine function, $E(\varphi ,k)$: Legendre’s incomplete elliptic integral of the second kind, $\sin z$: sine function, $\varphi$: real or complex argument, $k$: real or complex modulus, $k^{\prime }$: complementary modulus and $D_k$: differential operator

Permalink:

http://dlmf.nist.gov/19.4.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §19.4(ii), §19.4 and Ch.19

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6: 16.21 Differential Equation

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16.21.1 $$\left({(-1)}^{p-m-n}z(\mathit{\vartheta }-a_1+1)\mathrm{\cdots }(\mathit{\vartheta }-a_p+1)-(\mathit{\vartheta }-b_1)\mathrm{\cdots }(\mathit{\vartheta }-b_q)\right)w=0,$$

ⓘ

Symbols:

$p$: nonnegative integer, $q$: nonnegative integer, $z$: complex variable, $a,a_1,\mathrm{\dots },a_p$: real or complex parameters, $b,b_1,\mathrm{\dots },b_q$: real or complex parameters, $w$: solution and $\mathit{\vartheta }$: differential operator

Referenced by:

§16.21, §16.21

Permalink:

http://dlmf.nist.gov/16.21.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §16.21 and Ch.16

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7: 1.16 Distributions

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1.16.30 $$𝐃=(\frac{1}{\mathrm{i}}\frac{\partial }{\partial x_1},\frac{1}{\mathrm{i}}\frac{\partial }{\partial x_2},\mathrm{\dots },\frac{1}{\mathrm{i}}\frac{\partial }{\partial x_n}).$$

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Defines:

$𝐃$: vector differential operator (locally)

Symbols:

$\mathrm{i}$: imaginary unit, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $n$: nonnegative integer, $𝐷f$: distributional derivative and $\left|x\right|$: absolute value of $x$

Referenced by:

(1.16.32), item (ii), item (ii)

Permalink:

http://dlmf.nist.gov/1.16.E30

Encodings:

pMML, png, TeX

Clarification (effective with 1.0.15):

Previously this equation was completely different:

$${𝐷}_{𝜶}={\mathrm{i}}^{-\left|𝜶\right|}{𝐷}^{𝜶}={\left(\frac{1}{\mathrm{i}}\frac{\partial }{\partial x_1}\right)}^{{\alpha }_1}\mathrm{\cdots }{\left(\frac{1}{\mathrm{i}}\frac{\partial }{\partial x_n}\right)}^{{\alpha }_n}.$$

The change was made to clarify the meaning of (1.16.32).

See also:

Annotations for §1.16(vii), §1.16 and Ch.1

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1.16.32 $$P(𝐃)=\sum _{𝜶}{c}_{𝜶}{𝐃}^{\alpha }=\sum _{𝜶}{c}_{𝜶}{\left(\frac{1}{\mathrm{i}}\frac{\partial }{\partial x_1}\right)}^{{\alpha }_1}\mathrm{\dots }{\left(\frac{1}{\mathrm{i}}\frac{\partial }{\partial x_n}\right)}^{{\alpha }_n}.$$

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Symbols:

$\mathrm{i}$: imaginary unit, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $n$: nonnegative integer, $𝐃$: vector differential operator, $P$: polynomial of several variables and $𝐷f$: distributional derivative

Referenced by:

(1.16.30), (1.16.33), (1.16.34), (1.16.36), (1.16.37), item (iii), item (iii)

Permalink:

http://dlmf.nist.gov/1.16.E32

Encodings:

pMML, png, TeX

Clarification (effective with 1.0.15):

Previously this equation was written

$$P(𝐷)=\sum _{𝜶}{c}_{𝜶}{𝐷}_{𝜶}$$

where ${𝐷}_{𝜶}$ was defined in (1.16.30). The change was made to clarify the meaning of this equation.

See also:

Annotations for §1.16(vii), §1.16 and Ch.1

►Here $𝜶$ ranges over a finite set of multi-indices, $P(𝐱)$ is a multivariate polynomial, and $P(𝐃)$ is a partial differential operator. … ►

1.16.36 $$⟨ℱ\left(P(𝐃)u\right),\varphi ⟩=⟨P_{-}ℱ\left(u\right),\varphi ⟩=⟨ℱ\left(u\right),P_{-}\varphi ⟩,$$

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Symbols:

$⟨\mathrm{\Lambda },\varphi ⟩$: action of distribution on test function, $ℱ\left(u\right)$: Fourier transform of a tempered distribution, $\varphi (x_1,x_2,\mathrm{\dots },x_n)$: test function, $𝐃$: vector differential operator, $P$: polynomial of several variables and $𝐷f$: distributional derivative

Referenced by:

item (iv), §1.16(vii), §1.16(viii), item (iv)

Permalink:

http://dlmf.nist.gov/1.16.E36

Encodings:

pMML, png, TeX

Notational Change (effective with 1.0.15):

Previously this equation was written as

$$ℱ(P(𝐷)u)=P(-𝐱)ℱ(u).$$

It has been rewritten using the notation of (1.16.32) and (1.16.35).

See also:

Annotations for §1.16(vii), §1.16 and Ch.1

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1.16.37 $$⟨ℱ\left(Pu\right),\varphi ⟩=⟨P(𝐃)ℱ\left(u\right),\varphi ⟩,$$

ⓘ

Symbols:

$⟨\mathrm{\Lambda },\varphi ⟩$: action of distribution on test function, $ℱ\left(u\right)$: Fourier transform of a tempered distribution, $\varphi (x_1,x_2,\mathrm{\dots },x_n)$: test function, $𝐃$: vector differential operator, $P$: polynomial of several variables and $𝐷f$: distributional derivative

Referenced by:

item (iv), §1.16(vii), item (iv)

Permalink:

http://dlmf.nist.gov/1.16.E37

Encodings:

pMML, png, TeX

Notational Change (effective with 1.0.15):

Previously this equation was written as

$$ℱ(Pu)=P(𝐷)ℱ(u).$$

It has been rewritten using the notation of (1.16.32) and (1.16.35).

See also:

Annotations for §1.16(vii), §1.16 and Ch.1

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8: 18.38 Mathematical Applications

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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

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16.19.5 $$\mathit{\vartheta }{G}_{p,q}^{m,n}(z;\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q})=G_{p,q}^{m,n}(z;\genfrac{}{}{0pt}{}{a_1-1,a_2,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q})+(a_1-1)G_{p,q}^{m,n}(z;\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q}),$$

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Symbols:

$G_{p,q}^{m,n}(z;\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q})$ or $G_{p,q}^{m,n}(z;𝐚;𝐛)$: Meijer $G$-function, $p$: nonnegative integer, $q$: nonnegative integer, $z$: complex variable, $a,a_1,\mathrm{\dots },a_p$: real or complex parameters, $b,b_1,\mathrm{\dots },b_q$: real or complex parameters and $\mathit{\vartheta }$: differential operator

Permalink:

http://dlmf.nist.gov/16.19.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §16.19 and Ch.16

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10: 1.5 Calculus of Two or More Variables

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1.5.3 $$\frac{\partial f}{\partial x}=D_{x}f=f_x=\underset{h\to 0}{lim}\frac{f(x+h,y)-f(x,y)}{h},$$

ⓘ

Defines:

$\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ and $D_x$: differential operator (locally)

Permalink:

http://dlmf.nist.gov/1.5.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §1.5(i), §1.5 and Ch.1

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1.5.4 $$\frac{\partial f}{\partial y}=D_{y}f=f_y=\underset{h\to 0}{lim}\frac{f(x,y+h)-f(x,y)}{h}.$$

ⓘ

Symbols:

$\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ and $D_x$: differential operator

Permalink:

http://dlmf.nist.gov/1.5.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §1.5(i), §1.5 and Ch.1

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