drmf-kls9.ocd

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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

[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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