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11: Qiming Wang

… ►She has applied VRML and X3D techniques to several different fields including interactive mathematical function visualization, 3D human body modeling, and manufacturing-related modeling. …

12: Daniel W. Lozier

… ►Army Engineer Research and Development Laboratory in Virginia on finite-difference solutions of differential equations associated with nuclear weapons effects. Then he transferred to NIST (then known as the National Bureau of Standards), where he collaborated for several years with the Building and Fire Research Laboratory developing and applying finite-difference and spectral methods to differential equation models of fire growth. …

13: 3.10 Continued Fractions

… ►

Quotient-Difference Algorithm

… ► … ►We continue by means of the rhombus rule … ► … ►($\nabla$ is the backward difference operator.) …

14: Browsers

… ►MathML has long been considered to be important for representing mathematics on the web; it provides for much better accessibility, reusability and scalability to different displays. …

15: Annie A. M. Cuyt

… ►As a consequence her expertise spans a wide range of activities from pure abstract mathematics to computer arithmetic and different engineering applications. …

16: Wadim Zudilin

… ►His research interests are primarily focused on applications of special functions in different parts of number theory. …

17: 18.26 Wilson Class: Continued

… ►

§18.26(iii) Difference Relations

►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). ►For each family only the $y$-difference that lowers $n$ is given. … ►

18.26.14 $${\delta }_y\left(W_n(y^2;a,b,c,d)\right)/{\delta }_y(y^2)=-n(n+a+b+c+d-1)W_{n-1}(y^2;a+\frac{1}{2},b+\frac{1}{2},c+\frac{1}{2},d+\frac{1}{2}).$$

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Symbols:

$W_n(x;a,b,c,d)$: Wilson polynomial, ${\delta }_x$: central difference, $y$: real variable and $n$: nonnegative integer

Referenced by:

§16.4(iii), §18.26(iii)

Permalink:

http://dlmf.nist.gov/18.26.E14

Encodings:

pMML, png, TeX

See also:

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

►

18.26.15 $${\delta }_y\left(S_n(y^2;a,b,c)\right)/{\delta }_y(y^2)=-n{S}_{n-1}(y^2;a+\frac{1}{2},b+\frac{1}{2},c+\frac{1}{2}).$$

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Symbols:

${\delta }_x$: central difference, $S_n(x;a,b,c)$: continuous dual Hahn polynomial, $y$: real variable and $n$: nonnegative integer

Permalink:

http://dlmf.nist.gov/18.26.E15

Encodings:

pMML, png, TeX

See also:

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

…

18: 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). … ►

18.20.1 $$p_n(x)=\frac{1}{{\kappa }_n{w}_x}{\nabla }_x^n\left(w_x\prod _{\mathrm{\ell }=0}^{n-1}F(x+\mathrm{\ell })\right),$$ $x\in X$.

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Symbols:

${\nabla }_x$: backward difference, $\in$: element of, $p_n(x)$: polynomial of degree $n$ , $w_x$: weights, $X$: subset of $ℝ$ , $\mathrm{\ell }$: nonnegative integer, $n$: nonnegative integer, $x$: real variable, $F(x)$: coefficient and ${\kappa }_n$: coefficient

A&S Ref:

22.17.2

Referenced by:

§18.20(i), Table 18.20.1

Permalink:

http://dlmf.nist.gov/18.20.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §18.20(i), §18.20(i), §18.20 and Ch.18

… ►

18.20.3 $$w(x;a,b,\overline{a},\overline{b})p_n(x;a,b,\overline{a},\overline{b})=\frac{1}{n!}{\delta }_x^n\left(w(x;a+\frac{1}{2}n,b+\frac{1}{2}n,\overline{a}+\frac{1}{2}n,\overline{b}+\frac{1}{2}n)\right).$$

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Symbols:

${\delta }_x$: central difference, $\overline{z}$: complex conjugate, $p_n(x;a,b,\overline{a},\overline{b})$: continuous Hahn polynomial, $!$: factorial (as in $n!$), $n$: nonnegative integer and $x$: real variable

Referenced by:

§18.20(i)

Permalink:

http://dlmf.nist.gov/18.20.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §18.20(i), §18.20(i), §18.20 and Ch.18

… ►

18.20.4 $$w^{(\lambda )}(x;\varphi )P_n^{(\lambda )}(x;\varphi )=\frac{1}{n!}{\delta }_x^n\left(w^{(\lambda +\frac{1}{2}n)}(x;\varphi )\right).$$

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Symbols:

$P_n^{(\lambda )}(x;\varphi )$: Meixner–Pollaczek polynomial, ${\delta }_x$: central difference, $!$: factorial (as in $n!$), $n$: nonnegative integer and $x$: real variable

Referenced by:

§18.20(i)

Permalink:

http://dlmf.nist.gov/18.20.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §18.20(i), §18.20(i), §18.20 and Ch.18

…

19: 3.9 Acceleration of Convergence

… ►

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

… ►

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

…

20: 7.9 Continued Fractions

… ►

7.9.1 $$\sqrt{\pi }{\mathrm{e}}^{z^2}\mathrm{erfc}z=\frac{z}{z^2+}\frac{\frac{1}{2}}{1+}\frac{1}{z^2+}\frac{\frac{3}{2}}{1+}\frac{2}{z^2+}\mathrm{\cdots },$$ $\mathrm{\Re }z>0$,

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{erfc}z$: complementary error function, $\mathrm{e}$: base of natural logarithm, $\mathrm{\Re }$: real part and $z$: complex variable

A&S Ref:

7.1.14 (in different form)

Referenced by:

§7.9

Permalink:

http://dlmf.nist.gov/7.9.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §7.9 and Ch.7

… ►

7.9.3 $$w\left(z\right)=\frac{\mathrm{i}}{\sqrt{\pi }}\frac{1}{z-}\frac{\frac{1}{2}}{z-}\frac{1}{z-}\frac{\frac{3}{2}}{z-}\frac{2}{z-}\mathrm{\cdots },$$ $\mathrm{\Im }z>0$.

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $w\left(z\right)$: Faddeeva (or Faddeyeva) function, $\mathrm{\Im }$: imaginary part, $\mathrm{i}$: imaginary unit and $z$: complex variable

A&S Ref:

7.1.15 (in different form)

Permalink:

http://dlmf.nist.gov/7.9.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §7.9 and Ch.7

…