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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:

pMML, png, TeX

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: 18.26 Wilson Class: Continued

… ►For comments on the use of the forward-difference operator ${\mathrm{\Delta }}_x$, the backward-difference operator ${\nabla }_x$, and the central-difference operator ${\delta }_x$, see §18.2(ii). … ►

18.26.16 $$\frac{{\mathrm{\Delta }}_y\left(R_n(y(y+\gamma +\delta +1);\alpha ,\beta ,\gamma ,\delta )\right)}{{\mathrm{\Delta }}_y\left(y(y+\gamma +\delta +1)\right)}=\frac{n(n+\alpha +\beta +1)}{(\alpha +1)(\beta +\delta +1)(\gamma +1)}{R}_{n-1}(y(y+\gamma +\delta +2);\alpha +1,\beta +1,\gamma +1,\delta ).$$

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Symbols:

$R_n(x;\alpha ,\beta ,\gamma ,\delta )$: Racah polynomial, $\mathrm{\Delta }$: forward difference operator, $y$: real variable, $\delta$: arbitrary small positive constant and $n$: nonnegative integer

Referenced by:

§16.4(iii)

Permalink:

http://dlmf.nist.gov/18.26.E16

Encodings:

pMML, png, TeX

See also:

Annotations for §18.26(iii), §18.26 and Ch.18

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18.26.17 $$\frac{{\mathrm{\Delta }}_y\left(R_n(y(y+\gamma +\delta +1);\gamma ,\delta ,N)\right)}{{\mathrm{\Delta }}_y\left(y(y+\gamma +\delta +1)\right)}=-\frac{n}{(\gamma +1)N}{R}_{n-1}(y(y+\gamma +\delta +2);\gamma +1,\delta ,N-1).$$

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Symbols:

$R_n(x;\gamma ,\delta ,N)$: dual Hahn polynomial, $\mathrm{\Delta }$: forward difference operator, $y$: real variable, $N$: positive integer, $\delta$: arbitrary small positive constant and $n$: nonnegative integer

Referenced by:

§18.26(iii)

Permalink:

http://dlmf.nist.gov/18.26.E17

Encodings:

pMML, png, TeX

See also:

Annotations for §18.26(iii), §18.26 and Ch.18

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13: 26.8 Set Partitions: Stirling Numbers

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26.8.31 $$\frac{1}{k!}\frac{d^k}{{dx}^k}f(x)=\sum _{n=k}^{\mathrm{\infty }}\frac{s(n,k)}{n!}{\mathrm{\Delta }}^{n}f(x),$$

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Symbols:

$s(n,k)$: Stirling number of the first kind, $\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $!$: factorial (as in $n!$), $\mathrm{\Delta }$: forward difference operator, $x$: real variable, $k$: nonnegative integer and $n$: nonnegative integer

A&S Ref:

24.1.3

Referenced by:

§26.8(v)

Permalink:

http://dlmf.nist.gov/26.8.E31

Encodings:

pMML, png, TeX

See also:

Annotations for §26.8(v), §26.8 and Ch.26

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26.8.32 $$\mathrm{\Delta }f(x)=f(x+1)-f(x);$$

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Symbols:

$\mathrm{\Delta }$: forward difference operator and $x$: real variable

Permalink:

http://dlmf.nist.gov/26.8.E32

Encodings:

pMML, png, TeX

See also:

Annotations for §26.8(v), §26.8 and Ch.26

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26.8.37 $$\frac{1}{k!}{\mathrm{\Delta }}^{k}f(x)=\sum _{n=k}^{\mathrm{\infty }}\frac{S(n,k)}{n!}\frac{d^n}{{dx}^n}f(x),$$

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Symbols:

$S(n,k)$: Stirling number of the second kind, $\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $!$: factorial (as in $n!$), $\mathrm{\Delta }$: forward difference operator, $x$: real variable, $k$: nonnegative integer and $n$: nonnegative integer

A&S Ref:

24.1.4

Referenced by:

§26.8(v)

Permalink:

http://dlmf.nist.gov/26.8.E37

Encodings:

pMML, png, TeX

See also:

Annotations for §26.8(v), §26.8 and Ch.26

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14: Mathematical Introduction

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$ℂ$	complex plane (excluding infinity).
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$\mathrm{\Delta }$ (or ${\mathrm{\Delta }}_x$)	forward difference operator: $\mathrm{\Delta }f(x)=f(x+1)-f(x)$.
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15: 3.2 Linear Algebra

… ►With $𝐲={[y_1,y_2,\mathrm{\dots },y_n]}^{\mathrm{T}}$ the process of solution can then be regarded as first solving the equation $𝐋𝐲=𝐛$ for $𝐲$ (forward elimination), followed by the solution of $𝐔𝐱=𝐲$ for $𝐱$ (back substitution). … ►In solving $𝐀𝐱={[1,1,1]}^{\mathrm{T}}$, we obtain by forward elimination $𝐲={[1,-1,3]}^{\mathrm{T}}$, and by back substitution $𝐱={[\frac{1}{6},\frac{1}{6},\frac{1}{6}]}^{\mathrm{T}}$. … ►Forward elimination for solving $𝐀𝐱=𝐟$ then becomes $y_1=f_1$, …

16: 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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17: 18.20 Hahn Class: Explicit Representations

… ►For comments on the use of the forward-difference operator ${\mathrm{\Delta }}_x$, the backward-difference operator ${\nabla }_x$, and the central-difference operator ${\delta }_x$, see §18.2(ii). …

18: 30.8 Expansions in Series of Ferrers Functions

… ►For $k=-N,-N+1,\mathrm{\dots },-R-1$ they are determined from (30.8.4) by forward recursion using $a_{n,-N-1}^m({\gamma }^2)=0$. …It should be noted that if the forward recursion (30.8.4) beginning with $f_{-N-1}=0$, $f_{-N}=1$ leads to $f_{-R}=0$, then $a_{n,k}^m({\gamma }^2)$ is undefined for and ${\mathrm{𝖰𝗌}}_n^m(x,{\gamma }^2)$ does not exist. …

19: 18.25 Wilson Class: Definitions

… ►For the Wilson class OP’s $p_n(x)$ with $x=\lambda (y)$: if the $y$-orthogonality set is $\{0,1,\mathrm{\dots },N\}$, then the role of the differentiation operator $d/dx$ in the Jacobi, Laguerre, and Hermite cases is played by the operator ${\mathrm{\Delta }}_y$ followed by division by ${\mathrm{\Delta }}_y(\lambda (y))$, or by the operator ${\nabla }_y$ followed by division by ${\nabla }_y(\lambda (y))$. …

20: 10.21 Zeros

… ►For sign properties of the forward differences that are defined by ►

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

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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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