with two variables

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1: 37.2 General Orthogonal Polynomials of Two Variables

§37.2 General Orthogonal Polynomials of Two Variables

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§37.2(iv) Zeros

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2: 37.10 Other Orthogonal Polynomials of Two Variables

§37.10 Other Orthogonal Polynomials of Two Variables

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§37.10(ii) Orthogonal Polynomials on an Annulus

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§37.10(iii) Bernstein–Szegő Polynomials of Two Variables

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§37.10(iv) Hahn polynomials of Two Variables

… ►As an example we give the Hahn polynomials of two variables: …

3: 37.20 Mathematical Applications

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Numerical Integration and Interpolation

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$x$ , $y$	real variables.
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$P, Q$	polynomials of two variables.

5: 37.4 Disk with Weight Function $\left(1-x^{2}-y^{2}\right)^{\alpha}$

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

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37.4.2

W_{\alpha}(x,y)=(1-x^{2}-y^{2})^{\alpha},

\alpha>-1

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Defines:: $W_{\alpha}$ : two-variable weight function on $\mathbb{D}$ (locally)
Symbols:: $x$ : real variable, $y$ : real variable, $\mathbb{D}$ : unit disk and $\alpha$ : real parameter
Referenced by:: §37.4(iv), §37.6(iv)
Permalink:: http://dlmf.nist.gov/37.4.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §37.4(i), §37.4, Part 37.II and Ch.37

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37.4.3

\langle{f},{g}\rangle_{\alpha}=\frac{\alpha+1}{\pi}\int_{\mathbb{D}}f(x,y)g(x,% y)W_{\alpha}(x,y)\,\mathrm{d}x\,\mathrm{d}y,

\alpha>-1

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Defines:: $\langle{f},{g}\rangle_{\alpha}$ : inner product on disk (locally)
Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $(\NVar{a},\NVar{b})$ : open interval, $x$ : real variable, $y$ : real variable, $\mathbb{D}$ : unit disk, $W_{\alpha}$ : two-variable weight function on $\mathbb{D}$ , $\alpha$ : real parameter, $f$ : two-variable function on $\mathbb{D}$ and $g$ : two-variable function on $\mathbb{D}$
Referenced by:: §37.4(i), §37.4(iii)
Permalink:: http://dlmf.nist.gov/37.4.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §37.4(i), §37.4, Part 37.II and Ch.37

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37.4.28

\left[(1-x^{2}){D}_{xx}-2xy{D}_{xy}+(1-y^{2}){D}_{yy}-(2\alpha+3)\left(x{D}_{x% }+y{D}_{y}\right)\right]{}u(x,y)=-n(n+2\alpha+2)u(x,y),

u\in\mathcal{V}_{n}^{\alpha}

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Defines:: $u$ : polynomial in two variables (locally)
Symbols:: $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $\in$ : element of, $n$ : nonnegative integer, $d$ : positive integer, usually $\geq 2$ , $x$ : real variable, $y$ : real variable, $\alpha$ : real parameter and $\mathcal{V}_{n}^{\alpha}$ : vector space of two-variable polynomials of degree $n$ on the disk $\mathbb{D}$
Proved:: Dunkl and Xu (2014, (5.2.3) and Theorem 8.1.3); for $d=2$ , $\mu=\alpha+\frac{1}{2}$ (proved)
Referenced by:: item 2, §37.4(i), §37.4(vi)
Permalink:: http://dlmf.nist.gov/37.4.E28
Encodings:: pMML, png, TeX
See also:: Annotations for §37.4(vi), §37.4, Part 37.II and Ch.37

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6: 37.7 Parabolic Biangular Region with Weight Function $\left(1-x\right)^{\alpha}\left(x-y^{2}\right)^{\beta}$

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

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37.7.4

P_{k,n}^{\alpha,\beta}(x,y)=\sum_{m=0}^{n-k}\sum_{j=0}^{\left\lfloor\frac{1}{2% }k\right\rfloor}d_{m,j}x^{n-k-m+j}y^{k-2j}.

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Symbols:: $\left\lfloor\NVar{x}\right\rfloor$ : floor of $x$ , $j$ : nonnegative integer, $k$ : nonnegative integer, $m$ : nonnegative integer, $n$ : nonnegative integer, $P$ : polynomial of two variables, $d$ : positive integer, usually $\geq 2$ , $x$ : real variable and $y$ : real variable
Permalink:: http://dlmf.nist.gov/37.7.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §37.7(i), §37.7, Part 37.II and Ch.37

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37.7.12

\left(L+n(n+\alpha+\beta+\tfrac{3}{2})\right)P_{2k,n+k}^{\alpha,\beta}(x,y)=0,

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Symbols:: $n$ : nonnegative integer, $x$ : real variable, $y$ : real variable and $P_{2}$ : polynomial of two variables with real coefficients
Permalink:: http://dlmf.nist.gov/37.7.E12
Encodings:: pMML, png, TeX
See also:: Annotations for §37.7(iii), §37.7, Part 37.II and Ch.37

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37.7.13

\left(L+(n+\tfrac{1}{2})(n+\alpha+\beta+2)\right)P_{2k+1,n+k+1}^{\alpha,\beta}% (x,y)=0,

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Symbols:: $n$ : nonnegative integer, $x$ : real variable, $y$ : real variable and $P_{2}$ : polynomial of two variables with real coefficients
Permalink:: http://dlmf.nist.gov/37.7.E13
Encodings:: pMML, png, TeX
See also:: Annotations for §37.7(iii), §37.7, Part 37.II and Ch.37

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37.7.23

x{D}_{xx}+\tfrac{1}{2}{D}_{yy}+(\alpha+1-x){D}_{x}-y{D}_{y}.

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Symbols:: $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $x$ : real variable and $y$ : real variable
Referenced by:: item 5
Permalink:: http://dlmf.nist.gov/37.7.E23
Encodings:: pMML, png, TeX
See also:: Annotations for §37.7(v), §37.7, Part 37.II and Ch.37

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Z. M. Yan (1992) Generalized Hypergeometric Functions and Laguerre Polynomials in Two Variables. In Hypergeometric Functions on Domains of Positivity, Jack Polynomials, and Applications (Tampa, FL, 1991), Contemporary Mathematics, Vol. 138, pp. 239–259.

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External Links:: MathReview (B. M. Agrawal), ZentralBlatt
Referenced by:: §35.10.
Encodings:: BibTeX

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8: 37.3 Triangular Region with Weight Function $x^{\alpha}y^{\beta}(1-x-y)^{\gamma}$

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§37.3(i) Orthogonal Decomposition

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37.3.1

W_{\alpha,\beta,\gamma}(x,y)=x^{\alpha}y^{\beta}(1-x-y)^{\gamma}

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Defines:: $W_{\alpha,\beta,\gamma}$ : weight function in two-variables on $\triangle$ (locally)
Symbols:: $\triangle$ : triangular region, $x$ : real variable, $y$ : real variable, $\alpha$ : real parameter ( $>-1$ ), $\beta$ : real parameter ( $>-1$ ) and $\gamma$ : real parameter ( $>-1$ )
Referenced by:: §37.5(iii)
Permalink:: http://dlmf.nist.gov/37.3.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §37.3(i), §37.3, Part 37.II and Ch.37

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37.3.2

\langle{f},{g}\rangle_{\alpha,\beta,\gamma}=\frac{\Gamma\left(\alpha+\beta+% \gamma+3\right)}{\Gamma\left(\alpha+1\right)\Gamma\left(\beta+1\right)\Gamma% \left(\gamma+1\right)}\*\int_{\triangle}f(x,y)g(x,y)W_{\alpha,\beta,\gamma}(x,% y)\,\mathrm{d}x\,\mathrm{d}y,

\alpha,\beta,\gamma>-1

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Defines:: $\langle{f},{g}\rangle_{\alpha,\beta,\gamma}$ : inner product on triangular region (locally), $\alpha$ : real parameter ( $>-1$ ) (locally), $\beta$ : real parameter ( $>-1$ ) (locally) and $\gamma$ : real parameter ( $>-1$ ) (locally)
Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $\triangle$ : triangular region, $x$ : real variable, $y$ : real variable, $W_{\alpha,\beta,\gamma}$ : weight function in two-variables on $\triangle$ , $f$ : function of two variables and $g$ : function of two variables
Referenced by:: (37.3.18), (37.3.19), (37.3.20), §37.3(i)
Permalink:: http://dlmf.nist.gov/37.3.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §37.3(i), §37.3, Part 37.II and Ch.37

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37.3.15

W_{\alpha,\beta,\gamma}(x,y)^{-1}\Big{(}{D}_{x}[W_{\alpha+1,\beta,\gamma+1}(x,% y){D}_{x}u(x,y)]+{D}_{y}[W_{\alpha,\beta+1,\gamma+1}(x,y){D}_{y}u(x,y)]+{D}_{z% }[W_{\alpha+1,\beta+1,\gamma}(x,y){D}_{z}u(x,y)]\Big{)}=-n(n+\alpha+\beta+% \gamma+2)u(x,y),

u\in\mathcal{V}_{n}

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Symbols:: $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $\in$ : element of, $n$ : nonnegative integer, $z$ : complex number, $x$ : real variable, $y$ : real variable, $W_{\alpha,\beta,\gamma}$ : weight function in two-variables on $\triangle$ , $u$ : polynomial in two variables, $\alpha$ : real parameter ( $>-1$ ), $\beta$ : real parameter ( $>-1$ ), $\gamma$ : real parameter ( $>-1$ ) and $\mathcal{V}_{n}$ : vector space of two-variable polynomials of degree $n$ on triangular region
Proof sketch:: By direct computation the left-hand side of the present equality equals the left-hand side of (37.3.14).
Permalink:: http://dlmf.nist.gov/37.3.E15
Encodings:: pMML, png, TeX
See also:: Annotations for §37.3(iv), §37.3, Part 37.II and Ch.37

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9: null

error generating summary

10: 19.21 Connection Formulas

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19.21.1

R_{F}\left(0,z+1,z\right)R_{D}\left(0,z+1,1\right)+R_{D}\left(0,z+1,z\right)R_% {F}\left(0,z+1,1\right)=3\pi/(2z),

z\in\mathbb{C}\setminus(-\infty,0]

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Symbols:: $R_{D}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : elliptic integral symmetric in only two variables, $R_{F}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : symmetric elliptic integral of first kind, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathbb{C}$ : complex plane, $\in$ : element of, $(\NVar{a},\NVar{b}]$ : half-closed interval and $\setminus$ : set subtraction
Referenced by:: §19.21(i), §19.21(i), §19.7(i)
Permalink:: http://dlmf.nist.gov/19.21.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §19.21(i), §19.21 and Ch.19

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19.21.2

3R_{F}\left(0,y,z\right)=zR_{D}\left(0,y,z\right)+yR_{D}\left(0,z,y\right).

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Symbols:: $R_{D}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : elliptic integral symmetric in only two variables and $R_{F}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : symmetric elliptic integral of first kind
Referenced by:: §19.21(i)
Permalink:: http://dlmf.nist.gov/19.21.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §19.21(i), §19.21 and Ch.19

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19.21.3

6R_{G}\left(0,y,z\right)=yz(R_{D}\left(0,y,z\right)+R_{D}\left(0,z,y\right))=3% zR_{F}\left(0,y,z\right)+z(y-z)R_{D}\left(0,y,z\right).

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Symbols:: $R_{D}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : elliptic integral symmetric in only two variables, $R_{F}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : symmetric elliptic integral of first kind and $R_{G}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : symmetric elliptic integral of second kind
Referenced by:: §19.21(i), §19.22(i), §19.34
Permalink:: http://dlmf.nist.gov/19.21.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §19.21(i), §19.21 and Ch.19

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19.21.8

R_{D}\left(y,z,x\right)+R_{D}\left(z,x,y\right)+R_{D}\left(x,y,z\right)=3x^{-1% /2}y^{-1/2}z^{-1/2},

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Symbols:: $R_{D}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : elliptic integral symmetric in only two variables
Referenced by:: §19.21(ii), §19.33(iii), Erratum (V1.1.3) for Chapter 19
Permalink:: http://dlmf.nist.gov/19.21.E8
Encodings:: pMML, png, TeX
Correction (effective with 1.1.3):: The factors inside the square root on the right-hand side were written as products to ensure the correct multivalued behavior.
Suggested 2021-06-07 by Luc Maisonobe
See also:: Annotations for §19.21(ii), §19.21 and Ch.19

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19.21.9

xR_{D}\left(y,z,x\right)+yR_{D}\left(z,x,y\right)+zR_{D}\left(x,y,z\right)=3R_% {F}\left(x,y,z\right).

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Symbols:: $R_{D}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : elliptic integral symmetric in only two variables and $R_{F}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : symmetric elliptic integral of first kind
Referenced by:: §19.21(i), §19.21(ii)
Permalink:: http://dlmf.nist.gov/19.21.E9
Encodings:: pMML, png, TeX
See also:: Annotations for §19.21(ii), §19.21 and Ch.19

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with two variables

1—10 of 181 matching pages

1: 37.2 General Orthogonal Polynomials of Two Variables

§37.2 General Orthogonal Polynomials of Two Variables

§37.2(iv) Zeros

2: 37.10 Other Orthogonal Polynomials of Two Variables

§37.10 Other Orthogonal Polynomials of Two Variables

§37.10(ii) Orthogonal Polynomials on an Annulus

§37.10(iii) Bernstein–Szegő Polynomials of Two Variables

§37.10(iv) Hahn polynomials of Two Variables

3: 37.20 Mathematical Applications

Numerical Integration and Interpolation

4: 37.1 Notation

5: 37.4 Disk with Weight Function $\left(1-x^{2}-y^{2}\right)^{\alpha}$

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

6: 37.7 Parabolic Biangular Region with Weight Function $\left(1-x\right)^{\alpha}\left(x-y^{2}\right)^{\beta}$

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

7: Bibliography Y

8: 37.3 Triangular Region with Weight Function $x^{\alpha}y^{\beta}(1-x-y)^{\gamma}$

§37.3(i) Orthogonal Decomposition

9: null

10: 19.21 Connection Formulas

with two variables

1—10 of 181 matching pages

1: 37.2 General Orthogonal Polynomials of Two Variables

§37.2 General Orthogonal Polynomials of Two Variables

§37.2(iv) Zeros

2: 37.10 Other Orthogonal Polynomials of Two Variables

§37.10 Other Orthogonal Polynomials of Two Variables

§37.10(ii) Orthogonal Polynomials on an Annulus

§37.10(iii) Bernstein–Szegő Polynomials of Two Variables

§37.10(iv) Hahn polynomials of Two Variables

3: 37.20 Mathematical Applications

Numerical Integration and Interpolation

4: 37.1 Notation

5: 37.4 Disk with Weight Function ( 1 − x 2 − y 2 ) α

§37.4 Disk with Weight Function ( 1 − x 2 − y 2 ) α

6: 37.7 Parabolic Biangular Region with Weight Function ( 1 − x ) α ⁢ ( x − y 2 ) β

§37.7 Parabolic Biangular Region with Weight Function ( 1 − x ) α ⁢ ( x − y 2 ) β

7: Bibliography Y

8: 37.3 Triangular Region with Weight Function x α ⁢ y β ⁢ ( 1 − x − y ) γ

§37.3(i) Orthogonal Decomposition

9: null

10: 19.21 Connection Formulas

5: 37.4 Disk with Weight Function $\left(1-x^{2}-y^{2}\right)^{\alpha}$

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

6: 37.7 Parabolic Biangular Region with Weight Function $\left(1-x\right)^{\alpha}\left(x-y^{2}\right)^{\beta}$

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

8: 37.3 Triangular Region with Weight Function $x^{\alpha}y^{\beta}(1-x-y)^{\gamma}$