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1: 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

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2: 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

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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:

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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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3: 37.21 Physical Applications

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Computer Tomography

►OPs on the unit disk are closely related to Radon transforms, and they are used to construct reconstruction algorithms for x-ray computer tomography. … ►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). …

4: 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 $𝕡$ and the ultraspherical polynomials (37.4.4) on $𝔻$ are related by the quadratic transformations ►

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, $k$: nonnegative integer, $n$: nonnegative integer, $P$: polynomial of two variables, $x$: real variable and $y$: real variable

Permalink:

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

Encodings:

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

… ►The Jacobi polynomials (37.7.3) on $𝕡$ 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, $k$: nonnegative integer, $n$: nonnegative integer, $P$: polynomial of two variables, $x$: real variable and $y$: real variable

Permalink:

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

Encodings:

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Annotations for §37.7(ii), §37.7, Part 37.II 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, $k$: nonnegative integer, $n$: nonnegative integer, $P$: polynomial of two variables, $x$: real variable and $y$: real variable

Permalink:

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

Encodings:

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

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5: 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, $k$: nonnegative integer and $n$: nonnegative integer

Permalink:

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

Encodings:

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See also:

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

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6: 11.12 Physical Applications

… ►Applications of Struve functions occur in water-wave and surface-wave problems (Hirata (1975) and Ahmadi and Widnall (1985)), unsteady aerodynamics (Shaw (1985) and Wehausen and Laitone (1960)), distribution of fluid pressure over a vibrating disk (McLachlan (1934)), resistive MHD instability theory (Paris and Sy (1983)), and optical diffraction (Levine and Schwinger (1948)). …

7: 37.10 Other Orthogonal Polynomials of Two Variables

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§37.10(i) Orthogonal polynomials on the Disk for Generalized Weight Function

… ►on the unit disk, the OPs are fully developed. …

8: 37.6 Plane with Weight Function ${\mathrm{e}}^{-x^2-y^2}$

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§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 ${\mathrm{e}}^{-x^2-y^2}$: … ►

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

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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, $k$: nonnegative integer, $n$: nonnegative integer, $x$: real variable and $y$: real variable

Permalink:

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

Encodings:

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See also:

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

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9: 18.1 Notation

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

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10: 16.23 Mathematical Applications

… ►The Bieberbach conjecture states that if ${\sum }_{n=0}^{\mathrm{\infty }}{a}_n{z}^n$ is a conformal map of the unit disk to any complex domain, then $|a_n|\le n|a_1|$. …