open disks around infinity

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1: 37.7 Parabolic Biangular Region with Weight Function ${\left(1-x\right)}^{\alpha }{\left(x-y^2\right)}^{\beta }$

… ►The Jacobi polynomials (37.7.3) on $𝕡$ are related to the real disk polynomials (37.4.15) by the quadratic transformations … ►If we denote the polynomials on the right-hand sides of (37.7.20) and (37.7.21) by $p_{k,n}(x,y)$ then $p_{2k,n+k}(x,y)$ and $p_{2k+1,n+k+1}(x,y)$ are eigenfunctions with respective eigenvalues $-n$ and $-n-\frac{1}{2}$ of the PDO …Note that the polynomials $p_{k,n}(x,y)=L_{n-k}^{(\alpha )}\left(x\right)H_k\left(y\right)$ on the right-hand side of (37.7.20) are eigenfunctions with eigenvalue $-n$ of the slightly changed PDO … ►The right-hand sides of both (37.7.24) and (37.7.25) are orthogonal bases of polynomials on the parabolic region for the weight function ${(x-y^2)}^{\beta }{\mathrm{e}}^{-x}$ ($\beta >-1$). … ►If we denote the polynomials on the right-hand sides of (37.7.24) and (37.7.25) by $p_{k,n}(x,y)$ then $p_{2k,n+k}(x,y)$ and $p_{2k+1,n+k+1}(x,y)$ are eigenfunctions with respective eigenvalues $-n$ and $-n-\frac{1}{2}$ of the PDO …

2: 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 ${\mathrm{e}}^{-x^2-y^2}$. … ►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. … ►

§37.6(iv) Limits

►The explicit basis functions in §37.4 of (bi)orthogonal polynomials on the unit disk for the weight function (37.4.2) all tend after rescaling, as $\alpha \to \mathrm{\infty }$, to basis functions given above of OPs on ${ℝ}^2$ for the weight function ${\mathrm{e}}^{-x^2-y^2}$: ►

37.6.15 $$\underset{\alpha \to \mathrm{\infty }}{lim}{\alpha }^{\frac{1}{2}(m+n)}{R}_{m,n}^{\alpha }({\alpha }^{-\frac{1}{2}}z,{\alpha }^{-\frac{1}{2}}\overline{z})=S_{m,n}(z,\overline{z}),$$

ⓘ

Symbols:

$\overline{z}$: complex conjugate, $(a,b)$: open interval, $m$: nonnegative integer, $n$: nonnegative integer and $z$: complex number

Proof sketch:

Use (18.7.22).

Referenced by:

(37.6.14)

Permalink:

http://dlmf.nist.gov/37.6.E15

Encodings:

pMML, png, TeX

See also:

Annotations for §37.6(iv), §37.6, Part 37.II and Ch.37

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3: 37.4 Disk with Weight Function ${\left(1-x^2-y^2\right)}^{\alpha }$

… ►On the unit disk … ►

Complex Disk Polynomials

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Real Disk Polynomials

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Formulas for Complex Disk Polynomials

… ► …

4: 37.10 Other Orthogonal Polynomials of Two Variables

… ►on the unit disk, the OPs are fully developed. … ►Thus the $p_n^{(k)}(x)$ in (37.2.27) are orthogonal on $(\rho ,1)$ with weight function $x^k$. Hence the polynomials $P_n(x)=p_n^{(k)}(\rho +(1-\rho )x)$ are OPs on $(0,1)$ with weight function ${(x+\frac{\rho }{1-\rho })}^k$. This is a special case of the OPs on $(0,1)$ with generalized Jacobi weight $x^{\alpha }{(1-x)}^{\beta }{(x+\tau )}^{\gamma }$ ($\tau >0$), see Magnus (1995, §5), Dai and Zhang (2010). … ►These polynomials are orthogonal on the triangular finite discrete set $\{(x,y)\in {\mathbb{Z}}^2\mid x,y\ge 0,x+y\le N\}$ with respect to the weights …

5: 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 $(a,b)$. …

6: 1.9 Calculus of a Complex Variable

… ►It is single-valued on $ℂ\setminus \{0\}$, except on the interval $(-\mathrm{\infty },0)$ where it is discontinuous and two-valued. … ►

Continuity

… ►at $(x,y)$. … ►A system of open disks around infinity is given by … ►Suppose ${\sum }_{n=0}^{\mathrm{\infty }}{f}_n(t)$ converges uniformly in any compact interval in $(a,b)$, and at least one of the following two conditions is satisfied: …

7: 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 . …

8: null

error generating summary

9: 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 $(-\mathrm{\infty },-1)$. …

10: 1.10 Functions of a Complex Variable

… ►We write $(f_1,D_1)$, $(f_2,D_2)$ to signify this continuation. … ►Suppose $f(z)$ is analytic in the annulus , , and $r\in (r_1,r_2)$. … ►If $D=ℂ\setminus (-\mathrm{\infty },0]$ and $z=r{\mathrm{e}}^{\mathrm{i}\theta }$, then one branch is $\sqrt{r}{\mathrm{e}}^{\mathrm{i}\theta /2}$, the other branch is $-\sqrt{r}{\mathrm{e}}^{\mathrm{i}\theta /2}$, with in both cases. … ►(Thus if $z_0$ is in the interval $(-1,1)$, then the logarithms are real.) … ►Let $F(x,z)$ have a converging power series expansion of the form …