Formula:KLS:14.17:22

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R n ⁡ ( μ ⁢ ( x ) ; 0 , 0 , q - N - 1 , c | q ) = K n ⁡ ( λ ⁢ ( x ) ; c , N | q ) q-Racah-polynomial-R 𝑛 𝜇 𝑥 0 0 superscript 𝑞 𝑁 1 𝑐 𝑞 dual-q-Krawtchouk-polynomial-K 𝑛 𝜆 𝑥 𝑐 𝑁 𝑞 {\displaystyle{\displaystyle{\displaystyle R_{n}\!\left(\mu(x);0,0,q^{-N-1},c% \,|\,q\right)=K_{n}\!\left(\lambda(x);c,N|q\right)}}}

Substitution(s)

λ ⁢ ( n ) = q - n - p ⁢ q n 𝜆 𝑛 superscript 𝑞 𝑛 𝑝 superscript 𝑞 𝑛 {\displaystyle{\displaystyle{\displaystyle\lambda(n)=q^{-n}-pq^{n}}}} &

λ ⁢ ( x ) = q - x + c ⁢ q x - N 𝜆 𝑥 superscript 𝑞 𝑥 𝑐 superscript 𝑞 𝑥 𝑁 {\displaystyle{\displaystyle{\displaystyle\lambda(x)=q^{-x}+cq^{x-N}}}} &
μ ⁢ ( x ) = λ ⁢ ( x ) = q - x + c ⁢ q x - N 𝜇 𝑥 𝜆 𝑥 superscript 𝑞 𝑥 𝑐 superscript 𝑞 𝑥 𝑁 {\displaystyle{\displaystyle{\displaystyle\mu(x)=\lambda(x)=q^{-x}+cq^{x-N}}}} &
λ ⁢ ( n ) = q - n - p ⁢ q n 𝜆 𝑛 superscript 𝑞 𝑛 𝑝 superscript 𝑞 𝑛 {\displaystyle{\displaystyle{\displaystyle\lambda(n)=q^{-n}-pq^{n}}}} &
λ ⁢ ( x ) = q - x + c ⁢ q x - N 𝜆 𝑥 superscript 𝑞 𝑥 𝑐 superscript 𝑞 𝑥 𝑁 {\displaystyle{\displaystyle{\displaystyle\lambda(x)=q^{-x}+cq^{x-N}}}} &
λ ⁢ ( x ) := q - x + c ⁢ q x - N assign 𝜆 𝑥 superscript 𝑞 𝑥 𝑐 superscript 𝑞 𝑥 𝑁 {\displaystyle{\displaystyle{\displaystyle\lambda(x):=q^{-x}+cq^{x-N}}}} &
λ ⁢ ( n ) = q - n - p ⁢ q n 𝜆 𝑛 superscript 𝑞 𝑛 𝑝 superscript 𝑞 𝑛 {\displaystyle{\displaystyle{\displaystyle\lambda(n)=q^{-n}-pq^{n}}}} &

λ ⁢ ( x ) = q - x + c ⁢ q x - N 𝜆 𝑥 superscript 𝑞 𝑥 𝑐 superscript 𝑞 𝑥 𝑁 {\displaystyle{\displaystyle{\displaystyle\lambda(x)=q^{-x}+cq^{x-N}}}}


Proof

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Symbols List

& : logical and
R n subscript 𝑅 𝑛 {\displaystyle{\displaystyle{\displaystyle R_{n}}}}  : q 𝑞 {\displaystyle{\displaystyle{\displaystyle q}}} -Racah polynomial : http://dlmf.nist.gov/18.28#E19
K n subscript 𝐾 𝑛 {\displaystyle{\displaystyle{\displaystyle K_{n}}}}  : dual q 𝑞 {\displaystyle{\displaystyle{\displaystyle q}}} -Krawtchouk polynomial : http://drmf.wmflabs.org/wiki/Definition:dualqKrawtchouk

Bibliography

Equation in Section 14.17 of KLS.

URL links

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