forward differences

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1: 3.9 Acceleration of Convergence

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3.9.2 $$S=\sum _{k=0}^{\mathrm{\infty }}{(-1)}^k{2}^{-k-1}{\mathrm{\Delta }}^k{a}_0,$$

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Symbols:

$\mathrm{\Delta }$: forward difference operator and $S$: a convergent series

Permalink:

http://dlmf.nist.gov/3.9.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §3.9(ii), §3.9 and Ch.3

… ►Here $\mathrm{\Delta }$ is the forward difference operator: ►

3.9.3 $${\mathrm{\Delta }}^k{a}_0={\mathrm{\Delta }}^{k-1}{a}_1-{\mathrm{\Delta }}^{k-1}{a}_0,$$ $k=1,2,\mathrm{\dots }$.

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Symbols:

$\mathrm{\Delta }$: forward difference operator

A&S Ref:

3.6.27

Permalink:

http://dlmf.nist.gov/3.9.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §3.9(ii), §3.9 and Ch.3

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3.9.4 $${\mathrm{\Delta }}^k{a}_0=\sum _{m=0}^k{(-1)}^m\left(\genfrac{}{}{0.0pt}{}{k}{m}\right)a_{k-m}.$$

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Symbols:

$\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient and $\mathrm{\Delta }$: forward difference operator

A&S Ref:

3.6.27

Permalink:

http://dlmf.nist.gov/3.9.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §3.9(ii), §3.9 and Ch.3

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2: 3.6 Linear Difference Equations

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3.6.2 $$a_n{\mathrm{\Delta }}^2{w}_{n-1}+(2a_n-b_n)\mathrm{\Delta }{w}_{n-1}+(a_n-b_n+c_n)w_{n-1}=d_n,$$

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Symbols:

$\mathrm{\Delta }$: forward difference operator, $w_n$: sequence, $a_n$: coefficient, $b_n$: coefficient, $c_n$: coefficient and $d_n$: coefficient

Permalink:

http://dlmf.nist.gov/3.6.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §3.6(i), §3.6 and Ch.3

►where $\mathrm{\Delta }{w}_{n-1}=w_n-w_{n-1}$, ${\mathrm{\Delta }}^2{w}_{n-1}=\mathrm{\Delta }{w}_n-\mathrm{\Delta }{w}_{n-1}$, and $n\in \mathbb{Z}$. … ►If, as $n\to \mathrm{\infty }$, the wanted solution $w_n$ grows (decays) in magnitude at least as fast as any solution of the corresponding homogeneous equation, then forward (backward) recursion is stable. …

3: 18.1 Notation

… ►Forward differences: ►

$${\mathrm{\Delta }}_x\left(f(x)\right)=f(x+1)-f(x),$$

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$${\mathrm{\Delta }}_x^{n+1}\left(f(x)\right)={\mathrm{\Delta }}_x\left({\mathrm{\Delta }}_x^n(f(x))\right).$$

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4: 18.22 Hahn Class: Recurrence Relations and Differences

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18.22.19 $${\mathrm{\Delta }}_x{Q}_n(x;\alpha ,\beta ,N)=-\frac{n(n+\alpha +\beta +1)}{(\alpha +1)N}{Q}_{n-1}(x;\alpha +1,\beta +1,N-1),$$

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Symbols:

$Q_n(x;\alpha ,\beta ,N)$: Hahn polynomial, $\mathrm{\Delta }$: forward difference operator, $N$: positive integer, $n$: nonnegative integer and $x$: real variable

Referenced by:

§16.4(iii), §18.22(iii)

Permalink:

http://dlmf.nist.gov/18.22.E19

Encodings:

pMML, png, TeX

See also:

Annotations for §18.22(iii), §18.22(iii), §18.22 and Ch.18

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18.22.21 $${\mathrm{\Delta }}_x{K}_n(x;p,N)=-\frac{n}{pN}{K}_{n-1}(x;p,N-1),$$

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Symbols:

$K_n(x;p,N)$: Krawtchouk polynomial, $\mathrm{\Delta }$: forward difference operator, $N$: positive integer, $n$: nonnegative integer and $x$: real variable

Permalink:

http://dlmf.nist.gov/18.22.E21

Encodings:

pMML, png, TeX

See also:

Annotations for §18.22(iii), §18.22(iii), §18.22 and Ch.18

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18.22.23 $${\mathrm{\Delta }}_x{M}_n(x;\beta ,c)=-\frac{n(1-c)}{\beta c}{M}_{n-1}(x;\beta +1,c),$$

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Symbols:

$M_n(x;\beta ,c)$: Meixner polynomial, $\mathrm{\Delta }$: forward difference operator, $n$: nonnegative integer and $x$: real variable

Permalink:

http://dlmf.nist.gov/18.22.E23

Encodings:

pMML, png, TeX

See also:

Annotations for §18.22(iii), §18.22(iii), §18.22 and Ch.18

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18.22.25 $${\mathrm{\Delta }}_x{C}_n(x;a)=-\frac{n}{a}{C}_{n-1}(x;a),$$

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Symbols:

$C_n(x;a)$: Charlier polynomial, $\mathrm{\Delta }$: forward difference operator, $n$: nonnegative integer and $x$: real variable

Referenced by:

§18.22(iii)

Permalink:

http://dlmf.nist.gov/18.22.E25

Encodings:

pMML, png, TeX

See also:

Annotations for §18.22(iii), §18.22(iii), §18.22 and Ch.18

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5: 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)). …

6: 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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7: 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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8: 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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9: 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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10: 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 ${\nabla }_y(\lambda (y))$. …