biorthogonal

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1: 37.14 Orthogonal Polynomials on the Simplex

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§37.14(iii) Biorthogonal Bases

►The monic basis ${\{V_{𝝂}^{𝜶}\}}_{|𝝂|=n}$ of ${𝒱}_n^{𝜶}({\mathrm{△}}^d)$ and the co-monic basis ${\{U_{𝝂}^{𝜶}\}}_{|𝝂|=n}$, biorthogonal to the monic basis, can be explicitly given by … ►

37.14.9 $𝝂\in {\mathbb{N}}_0^d$, $|𝝂|=n$.

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Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $\in$: element of, $\mathbb{N}$: set of all positive integers, $n$: nonnegative integer, $d$: positive integer, usually $\ge 2$ , $\mathrm{\ell }$: positive integer, $𝐱$: vector in $d$-dimensional Euclidean space, ${\mathbb{N}}_0$: set of all nonnegative integers and ${\alpha }_{\mathrm{\ell }}$: real parameter ($>-1$)

Source:

Dunkl and Xu (2014, §5.3); See under the heading “Rodrigues formula and biorthogonal basis”. In the formula there replace ${\alpha }_{\mathrm{\ell }}$ by ${\alpha }_{\mathrm{\ell }}+\frac{1}{2}$ ($\mathrm{\ell }=1,\mathrm{\dots },d+1$).

Referenced by:

§37.14(iii), §37.15(vii)

Permalink:

http://dlmf.nist.gov/37.14.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §37.14(iii), §37.14, Part 37.III and Ch.37

►Formula (37.14.9) is an analogue of the Rodrigues formulas in §18.5(ii). ►

Biorthogonality Relation

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2: 37.15 Orthogonal Polynomials on the Ball

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§37.15(iv) Biorthogonal Bases

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Biorthogonality Relation

… ►See Dunkl and Xu (2014, §5.2.2) for expressions of of these biorthogonal polynomials in terms of Lauricella’s hypergeometric function $F_B$. … ►In particular, the various explicit bases of orthogonal or biorthogonal polynomials on ${\mathrm{△}}^d$ are related to similar explicit bases on ${𝔹}^d$ by quadratic transformations. … ►Second, the biorthogonal bases (37.14.9) and (37.14.8) on ${\mathrm{△}}^d$ are related to the biorthogonal bases (37.15.10) and (37.15.11) on ${𝔹}^d$ by the quadratic transformations …

3: 31.9 Orthogonality

… ►and the integration paths ${ℒ}_1$, ${ℒ}_2$ are Pochhammer double-loop contours encircling distinct pairs of singularities $\{0,1\}$, $\{0,a\}$, $\{1,a\}$. …

4: Bibliography I

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$q$-Hermite polynomials, biorthogonal rational functions, and $q$-beta integrals. Trans. Amer. Math. Soc. 346 (1), pp. 63–116.

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External Links:

ISSN 0002-9947, FullText, MathReview (R. A. Askey), ZentralBlatt

Referenced by:

§18.28(vii).

Encodings:

BibTeX

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5: 18.33 Polynomials Orthogonal on the Unit Circle

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§18.33(v) Biorthogonal Polynomials on the Unit Circle

… ►See Al-Salam and Ismail (1994) for special biorthogonal rational functions on the unit circle. …

6: 37.3 Triangular Region with Weight Function

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§37.3(iii) Biorthogonal Bases

►The monic basis of ${𝒱}_n^{\alpha ,\beta ,\gamma }$ and the co-monic basis , biorthogonal to the monic basis, can be explicitly given as follows. … ►

37.3.12

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Defines:

: co-monic basis element of two-variable Jacobi polynomial on the triangular region (locally)

Symbols:

$F_2(\alpha ;\beta ,{\beta }^{\prime };\gamma ,{\gamma }^{\prime };x,y)$: second Appell function, $F_3(\alpha ,{\alpha }^{\prime };\beta ,{\beta }^{\prime };\gamma ;x,y)$: third Appell function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $(a,b)$: open interval, $\partial x$: partial differential of $x$ , $k$: nonnegative integer, $n$: nonnegative integer, $d$: positive integer, usually $\ge 2$ , $x$: real variable, $y$: real variable, : real parameter ($>-1$), : real parameter ($>-1$) and : real parameter ($>-1$)

Proveds:

Appell and Kampé de Fériet (1926, Chap. VI); Dunkl and Xu (2014, Proposition 5.3.3); for $d=2$(proved)

Source:

Erdélyi et al. (1953b, 12.4(4))

Referenced by:

(37.3.24), (37.3.26), §37.3(iii), §37.3(v), §37.4(vii), §37.5(iii)

Permalink:

http://dlmf.nist.gov/37.3.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §37.3(iii), §37.3, Part 37.II and Ch.37

… ►The biorthogonality of the two bases is given by … ►For the orthogonal basis (37.3.3) and for the biorthogonal bases (37.3.11) and (37.3.12) these equations can be given more explicitly. …

7: 37.4 Disk with Weight Function ${\left(1-x^2-y^2\right)}^{\alpha }$

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§37.4(v) Biorthogonal Bases

… ►Clearly, …The biorthogonality of the two bases is given by … ►In particular, the various explicit bases of orthogonal or biorthogonal polynomials on $\mathrm{△}$ are related to similar explicit bases on $𝔻$ by quadratic transformations. … ►Third the biorthogonal bases (37.3.11) and (37.3.12) on $\mathrm{△}$ are related to the biorthogonal bases (37.4.23) and (37.4.24) on $𝔻$ by …

8: null

error generating summary

9: 37.17 Hermite Polynomials on ${ℝ}^d$

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Biorthogonal Bases

… ►Then the polynomials $H_{𝝂}(𝐱;𝐀)$ ($|𝝂|=n$) and $G_{𝝂}(𝐱;𝐀)$ ($|𝝂|=n$) both give a basis of ${𝒱}_n({ℝ}^d)$ and the two bases are biorthogonal: …

10: 37.2 General Orthogonal Polynomials of Two Variables

… ►Two bases ${\{P_{k,n}\}}_{k=0}^n$ and ${\{Q_{k,n}\}}_{k=0}^n$ of ${𝒱}_n$ are biorthogonal if ${⟨P_{k,n},Q_{j,n}⟩}_W=0$ whenever $k\ne j$. …Every basis ${\{P_{k,n}\}}_{k=0}^n$ belongs to a pair of biorthogonal bases, where the partner basis ${\{Q_{k,n}\}}_{k=0}^n$ is unique up to constant factors. …