disk polynomials

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1: 18.37 Classical OP’s in Two or More Variables

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§18.37(i) Disk Polynomials

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18.37.2 $m\ne j$ and/or $n\ne \mathrm{\ell }$.

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Symbols:

$dx$: differential of $x$ , $R_{m,n}^{(\alpha )}\left(z\right)$: disk polynomial, $\mathrm{i}$: imaginary unit, $y$: real variable, $\mathrm{\ell }$: nonnegative integer, $m$: nonnegative integer, $n$: nonnegative integer and $x$: real variable

Permalink:

http://dlmf.nist.gov/18.37.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §18.37(i), §18.37(i), §18.37 and Ch.18

… ►The following three conditions, taken together, determine $R_{m,n}^{(\alpha )}\left(z\right)$ uniquely: ►

18.37.3 $$R_{m,n}^{(\alpha )}\left(z\right)=\sum _{j=0}^{\min (m,n)}{c}_j{z}^{m-j}{\overline{z}}^{n-j},$$

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Symbols:

$\overline{z}$: complex conjugate, $R_{m,n}^{(\alpha )}\left(z\right)$: disk polynomial, $z$: complex variable, $m$: nonnegative integer and $n$: nonnegative integer

Permalink:

http://dlmf.nist.gov/18.37.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §18.37(i), §18.37(i), §18.37 and Ch.18

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18.37.5 $$R_{m,n}^{(\alpha )}\left(1\right)=1.$$

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Symbols:

$R_{m,n}^{(\alpha )}\left(z\right)$: disk polynomial, $m$: nonnegative integer and $n$: nonnegative integer

Permalink:

http://dlmf.nist.gov/18.37.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §18.37(i), §18.37(i), §18.37 and Ch.18

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

… ►There is also an orthogonal basis of ${𝒱}_n^{\alpha }$ consisting of polynomials $C_{k,n}^{(\alpha +\frac{1}{2})}(y,x)$ ($k=0,1,\mathrm{\dots },n$). … ►

Real Disk Polynomials

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$$R_{m,n}^{\alpha ,(\mathrm{c})}(r\cos \theta ,r\sin \theta )=R_n^{(\alpha ,m-n)}\left(2r^2-1\right)r^{m-n}\cos \left((m-n)\theta \right),$$ $m\ge n\ge 0$,

… ►The polynomials $R_{m,n-m}^{\alpha ,(\mathrm{c})}$ ($\frac{1}{2}n\le m\le n$) and $R_{m,n-m}^{\alpha ,(\mathrm{s})}$ () form together an orthogonal basis of ${𝒱}_n^{\alpha }$. … ►At ASML and Zeiss companies, where Zernike polynomials are applied in lithography, $R_{m,n}^{0,(\mathrm{c})}$ and $R_{m,n}^{0,(\mathrm{s})}$ in notation (37.4.15) are written as $Z_m^{m-n}$ ($m\ge n\ge 0$). …

3: 37.7 Parabolic Biangular Region with Weight Function ${\left(1-x\right)}^{\alpha }{\left(x-y^2\right)}^{\beta }$

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37.7.8 $$P_{k,n}^{-\frac{1}{2},\gamma }(1-x^2,y)=\frac{{(-1)}^{n-k}{\left(\gamma +1\right)}_k{\left(\frac{1}{2}\right)}_{n-k}}{{\left(2\gamma +1\right)}_k{\left(\gamma +k+1\right)}_{n-k}}{C}_{k,2n-k}^{(\gamma +\frac{1}{2})}(x,y),$$

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Symbols:

${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $C_{k,n}^{(\alpha +\frac{1}{2})}(x,y)$: two-variable ultraspherical polynomial on the disk, $n$: nonnegative integer, $k$: nonnegative integer, $x$: real variable, $y$: real variable and $P_{k,n}^{\alpha ,\beta }$: two-variable Jacobi polynomial on the parabolic biangular region $𝔸$

Permalink:

http://dlmf.nist.gov/37.7.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §37.7(ii), §37.7 and Ch.37

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37.7.9 $$x{P}_{k,n}^{\frac{1}{2},\gamma }(1-x^2,y)=\frac{{(-1)}^{n-k}{\left(\gamma +1\right)}_k{\left(\frac{1}{2}\right)}_{n-k+1}}{{\left(2\gamma +1\right)}_k{\left(\gamma +k+1\right)}_{n-k+1}}{C}_{k,2n-k+1}^{(\gamma +\frac{1}{2})}(x,y).$$

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Symbols:

${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $C_{k,n}^{(\alpha +\frac{1}{2})}(x,y)$: two-variable ultraspherical polynomial on the disk, $n$: nonnegative integer, $k$: nonnegative integer, $x$: real variable, $y$: real variable and $P_{k,n}^{\alpha ,\beta }$: two-variable Jacobi polynomial on the parabolic biangular region $𝔸$

Permalink:

http://dlmf.nist.gov/37.7.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §37.7(ii), §37.7 and Ch.37

►The Jacobi polynomials (37.7.3) on $𝔸$ are related to the real disk polynomials (37.4.15) by the quadratic transformations ►

37.7.10 $$P_{k,n}^{\alpha ,-\frac{1}{2}}(x^2+y^2,x)=\frac{{\left(\frac{1}{2}\right)}_k{\left(\alpha +1\right)}_{n-k}}{k!(n-k)!}{R}_{n,n-k}^{\alpha ,(\mathrm{c})}(x,y),$$

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Symbols:

${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $!$: factorial (as in $n!$), $R_{m,n}^{\alpha ,(\mathrm{c})}(r\cos \left(\theta \right),r\sin \left(\theta \right))$: two-variable real cosine disk polynomial, $n$: nonnegative integer, $k$: nonnegative integer, $x$: real variable, $y$: real variable and $P_{k,n}^{\alpha ,\beta }$: two-variable Jacobi polynomial on the parabolic biangular region $𝔸$

Permalink:

http://dlmf.nist.gov/37.7.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §37.7(ii), §37.7 and Ch.37

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37.7.11 $$y{P}_{k-1,n-1}^{\alpha ,\frac{1}{2}}(x^2+y^2,x)=2\frac{{\left(\frac{1}{2}\right)}_k{\left(\alpha +1\right)}_{n-k}}{k!(n-k)!}{R}_{n,n-k}^{\alpha ,(\mathrm{s})}(x,y).$$

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Symbols:

${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $!$: factorial (as in $n!$), $R_{m,n}^{\alpha ,(\mathrm{s})}(r\cos \left(\theta \right),r\sin \left(\theta \right))$: two-variable real sine disk polynomial, $n$: nonnegative integer, $k$: nonnegative integer, $x$: real variable, $y$: real variable and $P_{k,n}^{\alpha ,\beta }$: two-variable Jacobi polynomial on the parabolic biangular region $𝔸$

Permalink:

http://dlmf.nist.gov/37.7.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §37.7(ii), §37.7 and Ch.37

…

4: 37.15 Orthogonal Polynomials on the Ball

… ►For $d=2$ the polynomial $C_{{\nu }_1,{\nu }_2}^{(\alpha +\frac{1}{2})}(x_1,x_2)$ as defined by (37.15.4) becomes the polynomial $C_{{\nu }_2,{\nu }_1+{\nu }_2}^{(\alpha +\frac{1}{2})}(x_1,x_2)$ as given by (37.4.5). … ►For $d=2$ ball polynomials yield complex and real disk polynomials (37.4.11), (37.4.15): ►

37.15.9

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Symbols:

$R_{m,n}^{\alpha }(z,\overline{z})$: two-variable complex disk polynomial, $\cos z$: cosine function, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $(a,b)$: open interval, $R_{m,n}^{\alpha ,(\mathrm{c})}(r\cos \left(\theta \right),r\sin \left(\theta \right))$: two-variable real cosine disk polynomial, $R_{m,n}^{\alpha ,(\mathrm{s})}(r\cos \left(\theta \right),r\sin \left(\theta \right))$: two-variable real sine disk polynomial, $\sin z$: sine function, $n$: nonnegative integer and $k$: nonnegative integer

Permalink:

http://dlmf.nist.gov/37.15.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §37.15(iii), §37.15 and Ch.37

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5: 37.6 Plane with Weight Function ${\mathrm{e}}^{-x^2-y^2}$

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

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Symbols:

$S_{m,n}(z,\overline{z})$: two-variable complex circular Hermite polynomial, $R_{m,n}^{\alpha }(z,\overline{z})$: two-variable complex disk polynomial, $\overline{z}$: complex conjugate, $n$: nonnegative integer, $z$: complex number and $m$: nonnegative integer

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 and Ch.37

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37.6.16 $$\underset{\alpha \to \mathrm{\infty }}{lim}{\alpha }^{-\frac{1}{2}n}{C}_{k,n}^{(\alpha +\frac{1}{2})}({\alpha }^{-\frac{1}{2}}x,{\alpha }^{-\frac{1}{2}}y)=\frac{1}{(n-k)!k!}{H}_{n-k}\left(x\right)H_k\left(y\right),$$

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Symbols:

$H_n\left(x\right)$: Hermite polynomial, $!$: factorial (as in $n!$), $C_{k,n}^{(\alpha +\frac{1}{2})}(x,y)$: two-variable ultraspherical polynomial on the disk, $n$: nonnegative integer, $k$: nonnegative integer, $x$: real variable and $y$: real variable

Permalink:

http://dlmf.nist.gov/37.6.E16

Encodings:

pMML, png, TeX

See also:

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

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37.6.17 $$\underset{\alpha \to \mathrm{\infty }}{lim}{\alpha }^{\frac{1}{2}n}{V}_{k,n}^{(\alpha +\frac{1}{2})}({\alpha }^{-\frac{1}{2}}x,{\alpha }^{-\frac{1}{2}}y)=2^{-n}{H}_k\left(x\right)H_{n-k}\left(y\right),$$

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Symbols:

$H_n\left(x\right)$: Hermite polynomial, $V_{k,n}^{((\alpha +\frac{1}{2}))}(x,y)$: monic basis disk $V$ , $n$: nonnegative integer, $k$: nonnegative integer, $x$: real variable and $y$: real variable

Permalink:

http://dlmf.nist.gov/37.6.E17

Encodings:

pMML, png, TeX

See also:

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

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37.6.18 $$\underset{\alpha \to \mathrm{\infty }}{lim}{\alpha }^{-\frac{1}{2}n}{U}_{k,n}^{(\alpha +\frac{1}{2})}({\alpha }^{-\frac{1}{2}}x,{\alpha }^{-\frac{1}{2}}y)={(-1)}^n{H}_k\left(x\right)H_{n-k}\left(y\right).$$

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Symbols:

$H_n\left(x\right)$: Hermite polynomial, $U_{k,n}^{((\alpha +\frac{1}{2}))}(x,y)$: co-monic basis disk $U$ , $n$: nonnegative integer, $k$: nonnegative integer, $x$: real variable and $y$: real variable

Permalink:

http://dlmf.nist.gov/37.6.E18

Encodings:

pMML, png, TeX

See also:

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

6: 18.1 Notation

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Disk: $R_{m,n}^{(\alpha )}\left(z\right)$.

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7: 37.12 Orthogonal Polynomials on Quadratic Surfaces

… ►Then there are quadratic transformations for the polynomials (37.12.9) and (37.12.14) in terms of complex disk polynomials (37.4.11) and complex circular Hermite polynomials (37.6.3), respectively: ►

$S_{\mathrm{\ell },m}^n(z^2,{\left|z\right|}^2;-1,\gamma )=\left\{\begin{array}{c}{R}_{n+m,n-m}^{\gamma }(z,\overline{z})\\ R_{n-m,n+m}^{\gamma }(z,\overline{z})\end{array}\right\}$

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$S_{\mathrm{\ell },m}^n(z^2,{\left|z\right|}^2;0,\gamma )=\left\{\begin{array}{c}{z}^{-1}{R}_{n+m+1,n-m}^{\gamma }(z,\overline{z})\\ {\overline{z}}^{-1}{R}_{n-m,n+m+1}^{\gamma }(z,\overline{z})\end{array}\right\}$ , $0\le m\le n$, $z\in 𝔻$.

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$S_{\mathrm{\ell },m}^n(z^2,{\left|z\right|}^2;0)=\left\{\begin{array}{c}{z}^{-1}{S}_{n+m+1,n-m}(z,\overline{z})\\ {\overline{z}}^{-1}{S}_{n-m,n+m+1}(z,\overline{z})\end{array}\right\}$ , $0\le m\le n$, $z\in ℂ$.

8: 37.21 Physical Applications

… ►Applications in optics were already the motivation for Zernike (1934) to introduce the real disk polynomials (37.4.15) for $\alpha =0$. …This makes it necessary to work with OPs on an annulus instead of a disk: the Tatian polynomials given in §37.10(ii), see de Winter et al. (2020) and Bilski et al. (2022). …

9: 37.11 Spherical Harmonics

… ►In particular, the complex disk polynomial (37.4.11) in the form $Y(z)=R_{m,n}^{d-2}(z_1,\overline{z_1})$, $z\in {𝕊}^{2d-1}$, is such a complex spherical harmonic. …

10: Bibliography C

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Stability properties of disk polynomials. Numer. Algorithms.

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External Links:

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Referenced by:

(18.14.3_5).

Encodings:

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