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=== Section 9.1 [[Wilson|Wilson]] ===
=== Section 9.1 [[Wilson|Wilson]] ===


==== [[Definition:normWilsonWtilde|normWilsonWtilde]]<ref>[[Formula:KLS:09.01:06]]</ref> ====
==== [[Definition:normWilsonWtilde|normWilsonWtilde]] ====
<math display=block>
<math display=block>
\normWilsonWtilde{n}@{x^2}{a}{b}{c}{d}:=\frac{\Wilson{n}@{x^2}{a}{b}{c}{d}}{\pochhammer{a+b}{n}\pochhammer{a+c}{n}\pochhammer{a+d}{n}}
\normWilsonWtilde{n}@{x^2}{a}{b}{c}{d}:=\frac{\Wilson{n}@{x^2}{a}{b}{c}{d}}{\pochhammer{a+b}{n}\pochhammer{a+c}{n}\pochhammer{a+d}{n}}
</math>
</math><ref>[[Formula:KLS:09.01:06]]</ref>  
==== [[Definition:monicWilson|monicWilson]]<ref>[[Formula:KLS:09.01:10]]</ref> ====
==== [[Definition:monicWilson|monicWilson]] ====
<math display=block>
<math display=block>
\Wilson{n}@{x^2}{a}{b}{c}{d}=:(-1)^n\pochhammer{n+a+b+c+d-1}{n}\monicWilson{n}@@{x^2}{a}{b}{c}{d}.
\Wilson{n}@{x^2}{a}{b}{c}{d}=:(-1)^n\pochhammer{n+a+b+c+d-1}{n}\monicWilson{n}@@{x^2}{a}{b}{c}{d}.
</math>
</math><ref>[[Formula:KLS:09.01:10]]</ref>


=== Section 9.2 [[Racah|Racah]] ===
=== Section 9.2 [[Racah|Racah]] ===
==== [[Definition:monicRacah|monicRacah]]<ref>[[Formula:KLS:09.02:14]]</ref> ====
==== [[Definition:monicRacah|monicRacah]] ====


<math display=block>
\Racah{n}@{\lambda(x)}{\alpha}{\beta}{\gamma}{\delta}=:
\frac{\pochhammer{n+\alpha+\beta+1}{n}}{\pochhammer{\alpha+1}{n}\pochhammer{\beta+\delta+1}{n}\pochhammer{\gamma+1}{n}}\monicRacah{n}@@{\lambda(x)}{\alpha}{\beta}{\gamma}{\delta}
</math><ref>[[Formula:KLS:09.02:14]]</ref>
=== Section 9.3 [[Continuous dual Hahn|Continuous dual Hahn]] ===
=== Section 9.3 [[Continuous dual Hahn|Continuous dual Hahn]] ===


=== Section 9.4 [[Continuous Hahn|Continuous Hahn]] ===
=== Section 9.4 [[Continuous Hahn|Continuous Hahn]] ===
==== [[Definition:normctsHahnptilde|normctsHahnptilde]] ====
<math display=block>
\normctsHahnptilde{n}@@{x}{a}{b}{c}{d}:=\normctsHahnptilde{n}@{x}{a}{b}{c}{d}=\frac{n!}{i^n\pochhammer{a+c}{n}\pochhammer{a+d}{n}}\ctsHahn{n}@{x}{a}{b}{c}{d}.
</math><ref>[[Formula:KLS:09.04:04]]</ref>


=== Section 9.5 [[Hahn|Hahn]] ===
=== Section 9.5 [[Hahn|Hahn]] ===

Latest revision as of 19:43, 13 July 2017

Symbols in KLS Chapter 9

Section 9.1 Wilson

normWilsonWtilde

W ~ n ⁡ ( x 2 ; a , b , c , d ) := W n ⁡ ( x 2 ; a , b , c , d ) ( a + b ) n ⁢ ( a + c ) n ⁢ ( a + d ) n assign Wilson-polynomial-normalized-W-tilde 𝑛 superscript 𝑥 2 𝑎 𝑏 𝑐 𝑑 Wilson-polynomial-W 𝑛 superscript 𝑥 2 𝑎 𝑏 𝑐 𝑑 Pochhammer-symbol 𝑎 𝑏 𝑛 Pochhammer-symbol 𝑎 𝑐 𝑛 Pochhammer-symbol 𝑎 𝑑 𝑛 {\displaystyle{\tilde{W}}_{n}\!\left(x^{2};a,b,c,d\right):=\frac{W_{n}\!\left(% x^{2};a,b,c,d\right)}{{\left(a+b\right)_{n}}{\left(a+c\right)_{n}}{\left(a+d% \right)_{n}}}} [1]

monicWilson

W n ⁡ ( x 2 ; a , b , c , d ) = : ( - 1 ) n ( n + a + b + c + d - 1 ) n W ^ n ⁡ ( x 2 ) . fragments Wilson-polynomial-W 𝑛 superscript 𝑥 2 𝑎 𝑏 𝑐 𝑑 : superscript fragments ( 1 ) 𝑛 Pochhammer-symbol 𝑛 𝑎 𝑏 𝑐 𝑑 1 𝑛 Wilson-polynomial-monic 𝑛 superscript 𝑥 2 𝑎 𝑏 𝑐 𝑑 . {\displaystyle W_{n}\!\left(x^{2};a,b,c,d\right)=:(-1)^{n}{\left(n+a+b+c+d-1% \right)_{n}}{\widehat{W}}_{n}\!\left(x^{2}\right).} [2]

Section 9.2 Racah

monicRacah

R n ⁡ ( λ ⁢ ( x ) ; α , β , γ , δ ) = : ( n + α + β + 1 ) n ( α + 1 ) n ⁢ ( β + δ + 1 ) n ⁢ ( γ + 1 ) n R ^ n ⁡ ( λ ⁢ ( x ) ) fragments Racah-polynomial-R 𝑛 𝜆 𝑥 𝛼 𝛽 𝛾 𝛿 : Pochhammer-symbol 𝑛 𝛼 𝛽 1 𝑛 Pochhammer-symbol 𝛼 1 𝑛 Pochhammer-symbol 𝛽 𝛿 1 𝑛 Pochhammer-symbol 𝛾 1 𝑛 Racah-polynomial-monic-p 𝑛 𝜆 𝑥 𝛼 𝛽 𝛾 𝛿 {\displaystyle R_{n}\!\left(\lambda(x);\alpha,\beta,\gamma,\delta\right)=:% \frac{{\left(n+\alpha+\beta+1\right)_{n}}}{{\left(\alpha+1\right)_{n}}{\left(% \beta+\delta+1\right)_{n}}{\left(\gamma+1\right)_{n}}}{\widehat{R}}_{n}\!\left% (\lambda(x)\right)} [3]

Section 9.3 Continuous dual Hahn

Section 9.4 Continuous Hahn

normctsHahnptilde

p ~ n ⁡ ( x ) := p ~ n ⁡ ( x ; a , b , c , d ) = n ! i n ⁢ ( a + c ) n ⁢ ( a + d ) n ⁢ p n ⁡ ( x ; a , b , c , d ) . assign continuous-Hahn-polynomial-normalized-p-tilde 𝑛 𝑥 𝑎 𝑏 𝑐 𝑑 continuous-Hahn-polynomial-normalized-p-tilde 𝑛 𝑥 𝑎 𝑏 𝑐 𝑑 𝑛 superscript 𝑖 𝑛 Pochhammer-symbol 𝑎 𝑐 𝑛 Pochhammer-symbol 𝑎 𝑑 𝑛 continuous-Hahn-polynomial 𝑛 𝑥 𝑎 𝑏 𝑐 𝑑 {\displaystyle{\tilde{p}}_{n}\!\left(x\right):={\tilde{p}}_{n}\!\left(x;a,b,c,% d\right)=\frac{n!}{i^{n}{\left(a+c\right)_{n}}{\left(a+d\right)_{n}}}p_{n}\!% \left(x;a,b,c,d\right).} [4]

Section 9.5 Hahn

Section 9.6 Dual Hahn

Section 9.7 Meixner-Pollaczek

Section 9.8 Jacobi

Section 9.9Jacobi: Special cases

Section 9.10 Pseudo Jacobi

Section 9.11 Meixner

Section 9.12 Krawtchouk

Section 9.13 Laguerre

Section 9.14 Bessel

Section 9.15 Charlier

Section 9.16 Hermite

  1. ↑ Formula:KLS:09.01:06
  2. ↑ Formula:KLS:09.01:10
  3. ↑ Formula:KLS:09.02:14
  4. ↑ Formula:KLS:09.04:04
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