forward elimination

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11: 4.43 Cubic Equations

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12: 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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13: 10.74 Methods of Computation

… ►In the interval , $J_{\nu }\left(x\right)$ needs to be integrated in the forward direction and $Y_{\nu }\left(x\right)$ in the backward direction, with initial values for the former obtained from the power-series expansion (10.2.2) and for the latter from asymptotic expansions (§§10.17(i) and 10.20(i)). … ►Similarly, to maintain stability in the interval the integration direction has to be forwards in the case of $I_{\nu }\left(x\right)$ and backwards in the case of $K_{\nu }\left(x\right)$, with initial values obtained in an analogous manner to those for $J_{\nu }\left(x\right)$ and $Y_{\nu }\left(x\right)$. … ►Then $J_n\left(x\right)$ and $Y_n\left(x\right)$ can be generated by either forward or backward recurrence on $n$ when , but if $n>x$ then to maintain stability $J_n\left(x\right)$ has to be generated by backward recurrence on $n$, and $Y_n\left(x\right)$ has to be generated by forward recurrence on $n$. …

14: 1.13 Differential Equations

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Elimination of First Derivative by Change of Dependent Variable

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Elimination of First Derivative by Change of Independent Variable

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1.13.17 $$\frac{d^{2}w}{{d\eta }^2}+g(z)\exp \left(2\int f(z)dz\right)w=0.$$

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Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $dx$: differential of $x$ , $\exp z$: exponential function, $\int$: integral, $z$: variable, $w(z)$: solution, $f(z)$: analytic coefficient, $\eta$: change of variable and $g(z)$: analytic coefficient

Permalink:

http://dlmf.nist.gov/1.13.E17

Encodings:

pMML, png, TeX

See also:

Annotations for §1.13(iv), §1.13(iv), §1.13 and Ch.1

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15: 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)). …

16: 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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17: 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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18: 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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19: 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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20: 10.63 Recurrence Relations and Derivatives

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