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1: 37.8 Jacobi Polynomials Associated with Root System $B{C}_2$

… ►the Jacobi polynomials associated with root system $B{C}_2$ are symmetric polynomials $p_{k,n}^{\alpha ,\beta ,\gamma }(x,y)$ ($0\le k\le n$) of the form …for all $m,j$ with $0\le j\le m$ such that $m\le n$, $m+j\le n+k$ and $(j,m)\ne (k,n)$. … ►The polynomials $P_{k,n}^{\alpha ,\beta ,\gamma }(u,v)$, defined in terms of the polynomials $p_{k,n}^{\alpha ,\beta ,\gamma }(x,y)$ by …on the region bounded by a parabolic arc and two line segments, see Fig. … ►More generally, the definition of the symmetric OPs $p_{k,n}^{\alpha ,\beta ,\gamma }(x,y)$ can be extended to symmetric OPs $p_{k,n}(x,y)$ for weight function $W(x,y)=w(x)w(y){(x-y)}^{2\gamma +1}$ () for any weight function $w$ on $ℝ$. …

2: 7.23 Tables

… ►

Abramowitz and Stegun (1964, Chapter 7) includes $\mathrm{erf}x$, $(2/\sqrt{\pi }){\mathrm{e}}^{-x^2}$, $x\in [0,2]$, 10D; $(2/\sqrt{\pi }){\mathrm{e}}^{-x^2}$, $x\in [2,10]$, 8S; $x{\mathrm{e}}^{x^2}\mathrm{erfc}x$, $x^{-2}\in [0,0.25]$, 7D; $2^n\mathrm{\Gamma }\left(\frac{1}{2}n+1\right){\mathrm{i}}^n\mathrm{erfc}\left(x\right)$, $n=1(1)6,10,11$, $x\in [0,5]$, 6S; $F\left(x\right)$, $x\in [0,2]$, 10D; $xF\left(x\right)$, $x^{-2}\in [0,0.25]$, 9D; $C\left(x\right)$, $S\left(x\right)$, $x\in [0,5]$, 7D; $\mathrm{f}\left(x\right)$, $\mathrm{g}\left(x\right)$, $x\in [0,1]$, $x^{-1}\in [0,1]$, 15D.

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Finn and Mugglestone (1965) includes the Voigt function $H(a,u)$, $u\in [0,22]$, $a\in [0,1]$, 6S.

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Zhang and Jin (1996, pp. 638, 640–641) includes the real and imaginary parts of $\mathrm{erf}z$, $x\in [0,5]$, $y=0.5(.5)3$, 7D and 8D, respectively; the real and imaginary parts of ${\int }_x^{\mathrm{\infty }}{\mathrm{e}}^{\pm \mathrm{i}{t}^2}dt$, $(1/\sqrt{\pi }){\mathrm{e}}^{\mp \mathrm{i}(x^2+(\pi /4))}{\int }_x^{\mathrm{\infty }}{\mathrm{e}}^{\pm \mathrm{i}{t}^2}dt$, $x=0(.5)20(1)25$, 8D, together with the corresponding modulus and phase to 8D and 6D (degrees), respectively.

…

3: 37.12 Orthogonal Polynomials on Quadratic Surfaces

… ►where $a,b\in ℝ\cup \{\pm \mathrm{\infty }\}$, $\varphi$ is either a linear polynomial that is nonnegative on the interval $(a,b)$, or the square root of a nonnegative polynomial on $(a,b)$ of degree at most $2$. … ►

Unit sphere: $\varphi (t)=\sqrt{1-t^2}$, $t\in (-1,1)$.

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Cylinder: $\varphi (t)=1$, $t\in (0,1)$.

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Conic surface: $\varphi (t)=t$, $t\in (0,1)$.

… ►Let $w$ be a weight function on $(a,b)$. …

4: 14.27 Zeros

… ► $P_{\nu }^{\mu }\left(x\pm \mathrm{i}0\right)$ (either side of the cut) has exactly one zero in the interval $(-\mathrm{\infty },-1)$ if either of the following sets of conditions holds: …For all other values of the parameters $P_{\nu }^{\mu }\left(x\pm \mathrm{i}0\right)$ has no zeros in the interval $(-\mathrm{\infty },-1)$. …

5: 26.15 Permutations: Matrix Notation

… ►If $(j,k)\in B$, then $\sigma (j)\ne k$. The number of derangements of $n$ is the number of permutations with forbidden positions $B=\{(1,1),(2,2),\mathrm{\dots },(n,n)\}$. … ►For $(j,k)\in B$, $B\setminus [j,k]$ denotes $B$ after removal of all elements of the form $(j,t)$ or $(t,k)$, $t=1,2,\mathrm{\dots },n$. $B\setminus (j,k)$ denotes $B$ with the element $(j,k)$ removed. … ►Let . …

6: 14.16 Zeros

… ►

§14.16(ii) Interval

… ►The zeros of ${𝖰}_{\nu }^{\mu }\left(x\right)$ in the interval $(-1,1)$ interlace those of ${𝖯}_{\nu }^{\mu }\left(x\right)$. … ►

§14.16(iii) Interval

► $P_{\nu }^{\mu }\left(x\right)$ has exactly one zero in the interval $(1,\mathrm{\infty })$ if either of the following sets of conditions holds: … ► ${𝑸}_{\nu }^{\mu }\left(x\right)$ has no zeros in the interval $(1,\mathrm{\infty })$ when $\nu >-1$, and at most one zero in the interval $(1,\mathrm{\infty })$ when .

7: 22.17 Moduli Outside the Interval [0,1]

§22.17 Moduli Outside the Interval [0,1]

… ►Jacobian elliptic functions with real moduli in the intervals $(-\mathrm{\infty },0)$ and $(1,\mathrm{\infty })$, or with purely imaginary moduli are related to functions with moduli in the interval $[0,1]$ by the following formulas. … ►For proofs of these results and further information see Walker (2003).

8: 1.4 Calculus of One Variable

… ► … ►Suppose $f(x)$ is defined on $[a,b]$. … ►Then for $f(x)$ continuous on $(a,b)$, … ►for any $c,d\in (a,b)$, and $t\in [0,1]$. …A similar definition applies to closed intervals $[a,b]$. …

9: 37.6 Plane with Weight Function ${\mathrm{e}}^{-x^2-y^2}$

… ►Then the polynomials $S_{m,n-m}(x+\mathrm{i}y,x-\mathrm{i}y)$ ($m=0,1,\mathrm{\dots },n$) form an orthogonal basis of the space ${𝒱}_n\mathrm{\otimes }ℂ$ of complex-valued orthogonal polynomials of degree $n$ on ${ℝ}^2$ with weight function ${\mathrm{e}}^{-x^2-y^2}$. … ►

$$z{S}_{m,n}(z,\overline{z})=S_{m+1,n}(z,\overline{z})+n{S}_{m,n-1}(z,\overline{z}),$$

►

$$\overline{z}{S}_{m,n}(z,\overline{z})=S_{m,n+1}(z,\overline{z})+m{S}_{m-1,n}(z,\overline{z}).$$

… ►

$$D_z{S}_{m,n}(z,\overline{z})=m{S}_{m-1,n}(z,\overline{z}),$$

… ►The definition of $S_{m,n}(z,\overline{z})$ can be extended to $S_{m,n}(z_1,z_2)$, where $z_1$ and $z_2$ are two independent complex variables. …

10: 37.5 Quarter Plane with Weight Function $x^{\alpha }{y}^{\beta }{\mathrm{e}}^{-x-y}$

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37.5.1 $${ℝ}_{+}^2=\{(x,y)\in {ℝ}^2\mid x,y>0\}$$

ⓘ

Defines:

${ℝ}_{+}^2$: quarter plane (locally)

Symbols:

$\in$: element of, $(a,b)$: open interval, $ℝ$: real line, $x$: real variable, ${ℝ}^2$: real plane and $y$: real variable

Permalink:

http://dlmf.nist.gov/37.5.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §37.5, Part 37.II and Ch.37

… ►

37.5.2 $$W_{\alpha ,\beta }(x,y)=x^{\alpha }{y}^{\beta }{\mathrm{e}}^{-x-y},$$ $\alpha ,\beta >-1$.

ⓘ

Symbols:

$\mathrm{e}$: base of natural logarithm, $(a,b)$: open interval, $x$: real variable and $y$: real variable

Referenced by:

§37.5(iii)

Permalink:

http://dlmf.nist.gov/37.5.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §37.5, Part 37.II and Ch.37

… ►

37.5.3 $${⟨f,g⟩}_{\alpha ,\beta }=\frac{1}{\mathrm{\Gamma }\left(\alpha +1\right)\mathrm{\Gamma }\left(\beta +1\right)}{\int }_{{ℝ}_{+}^2}f(x,y)g(x,y)W_{\alpha ,\beta }(x,y)dxdy,$$ $\alpha ,\beta >-1$,

ⓘ

Defines:

${⟨f,g⟩}_{\alpha ,\beta }$: inner product on quarter plane (locally)

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $dx$: differential of $x$ , $\int$: integral, $(a,b)$: open interval, $ℝ$: real line, $x$: real variable, $y$: real variable and ${ℝ}_{+}^2$: quarter plane

Referenced by:

§37.5

Permalink:

http://dlmf.nist.gov/37.5.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §37.5, Part 37.II and Ch.37

… ►

37.5.6 $${⟨Q_{k,n}^{(\alpha ,\beta )},Q_{j,m}^{(\alpha ,\beta )}⟩}_{\alpha ,\beta }=h_{k,n}^{(\alpha ,\beta )}{\delta }_{n,m}{\delta }_{k,j},$$

ⓘ

Symbols:

${\delta }_{j,k}$: Kronecker delta, $(a,b)$: open interval, $j$: nonnegative integer, $k$: nonnegative integer, $m$: nonnegative integer, $n$: nonnegative integer, $Q$: polynomial of two variables and ${⟨f,g⟩}_{\alpha ,\beta }$: inner product on quarter plane

Permalink:

http://dlmf.nist.gov/37.5.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §37.5(i), §37.5, Part 37.II and Ch.37

… ►and, for the orthogonal basis (37.5.5) of polynomials $Q_{k,n}^{(\alpha ,\beta )}$, …