delta operator

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11: 2.9 Difference Equations

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2.9.2 $${\mathrm{\Delta }}^{2}w(n)+(2+f(n))\mathrm{\Delta }w(n)+(1+f(n)+g(n))w(n)=0,$$ $n=0,1,2,\mathrm{\dots }$,

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Symbols:

$\mathrm{\Delta }$: forward difference operator, $f(n)$: function, $g(n)$: function, $n$: nonnegative integer and $w(n)$: solution

Permalink:

http://dlmf.nist.gov/2.9.E2

Encodings:

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See also:

Annotations for §2.9(i), §2.9 and Ch.2

►in which $\mathrm{\Delta }$ is the forward difference operator (§3.6(i)). …

12: 18.19 Hahn Class: Definitions

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Hahn class (or linear lattice class). These are OP’s $p_n(x)$ where the role of $\frac{d}{dx}$ is played by ${\mathrm{\Delta }}_x$ or ${\nabla }_x$ or ${\delta }_x$ (see §18.1(i) for the definition of these operators). The Hahn class consists of four discrete and two continuous families.

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Wilson class (or quadratic lattice class). These are OP’s $p_n(x)=p_n(\lambda (y))$ ($p_n(x)$ of degree $n$ in $x$, $\lambda (y)$ quadratic in $y$) where the role of the differentiation operator is played by $\frac{{\mathrm{\Delta }}_y}{{\mathrm{\Delta }}_y(\lambda (y))}$ or $\frac{{\nabla }_y}{{\nabla }_y(\lambda (y))}$ or $\frac{{\delta }_y}{{\delta }_y(\lambda (y))}$. The Wilson class consists of two discrete and two continuous families.

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13: 10.21 Zeros

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$$\mathrm{\Delta }{\rho }_{\nu }(t)={\rho }_{\nu }(t+1)-{\rho }_{\nu }(t),$$

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$${\mathrm{\Delta }}^2{\rho }_{\nu }(t)=\mathrm{\Delta }{\rho }_{\nu }(t+1)-\mathrm{\Delta }{\rho }_{\nu }(t),\mathrm{\dots },$$

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14: 1.17 Integral and Series Representations of the Dirac Delta

§1.17 Integral and Series Representations of the Dirac Delta

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§1.17(i) Delta Sequences

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Sine and Cosine Functions

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Coulomb Functions (§33.14(iv))

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Airy Functions (§9.2)

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15: 32.2 Differential Equations

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32.2.13 $$z(1-z)𝐼\left({\int }_{\mathrm{\infty }}^w\frac{dt}{\sqrt{t(t-1)(t-z)}}\right)=\sqrt{w(w-1)(w-z)}\left(\alpha +\frac{\beta z}{w^2}+\frac{\gamma (z-1)}{{(w-1)}^2}+(\delta -\frac{1}{2})\frac{z(z-1)}{{(w-z)}^2}\right),$$

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Symbols:

$dx$: differential of $x$ , $\int$: integral, $z$: real, $\alpha$: arbitrary constant, $𝐼$: operator, $\beta$: arbitrary constant, $\gamma$: arbitrary constant and $\delta$: arbitrary constant

Permalink:

http://dlmf.nist.gov/32.2.E13

Encodings:

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See also:

Annotations for §32.2(iv), §32.2 and Ch.32

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16: 13.27 Mathematical Applications

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13.27.1

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Permalink:

http://dlmf.nist.gov/13.27.E1

Encodings:

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See also:

Annotations for §13.27 and Ch.13

►where $\alpha$, $\beta$, $\gamma$, $\delta$ are real numbers, and $\gamma >0$. Vilenkin (1968, Chapter 8) constructs irreducible representations of this group, in which the diagonal matrices correspond to operators of multiplication by an exponential function. The other group elements correspond to integral operators whose kernels can be expressed in terms of Whittaker functions. …

17: 18.39 Applications in the Physical Sciences

… ►The bound state $L^2$ eigenfunctions of the radial Coulomb Schrödinger operator are discussed in §§18.39(i) and 18.39(ii), and the $\delta$-function normalized (non-$L^2$) in Chapter 33, where the solutions appear as Whittaker functions. …

18: 2.8 Differential Equations with a Parameter

… ►In addition, ${𝒱}_{{𝒬}_j}\left(A_1\right)$ and ${𝒱}_{{𝒬}_j}\left(A_n\right)$ must be bounded on ${𝚫}_j({\alpha }_j)$. …

19: 31.10 Integral Equations and Representations

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31.10.4 $${𝒟}_z=z(z-1)(z-a)({\partial }^2/{\partial z}^2)+\left(\gamma (z-1)(z-a)+\delta z(z-a)+ϵz(z-1)\right)(\partial /\partial z)+\alpha \beta z.$$

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Defines:

${𝒟}_z$: Heun’s operator (locally)

Symbols:

$\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $z$: complex variable, $\gamma$: real or complex parameter, $\delta$: real or complex parameter, $ϵ$: real or complex parameter, $a$: complex parameter, $\alpha$: real or complex parameter and $\beta$: real or complex parameter

Referenced by:

§31.10(ii)

Permalink:

http://dlmf.nist.gov/31.10.E4

Encodings:

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See also:

Annotations for §31.10(i), §31.10 and Ch.31

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20: 1.3 Determinants, Linear Operators, and Spectral Expansions

§1.3 Determinants, Linear Operators, and Spectral Expansions

… ►Here ${\delta }_{j,k}$ is the Kronecker delta. … ►

§1.3(iv) Matrices as Linear Operators

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Linear Operators in Finite Dimensional Vector Spaces

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Self-Adjoint Operators on ${𝐄}_n$

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