Formula:KLS:14.17:19: Difference between revisions

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Latest revision as of 08:38, 22 December 2019


w ⁢ ( x ; c , N | q ) ⁢ K n ⁡ ( λ ⁢ ( x ) ; c , N | q ) = ( 1 - q ) n ⁢ ( c ⁢ q - N ; q ) n ⁢ ( ∇ λ ) n ⁢ [ w ⁢ ( x ; c , N - n | q ) ] 𝑤 𝑥 𝑐 conditional 𝑁 𝑞 dual-q-Krawtchouk-polynomial-K 𝑛 𝜆 𝑥 𝑐 𝑁 𝑞 superscript 1 𝑞 𝑛 q-Pochhammer-symbol 𝑐 superscript 𝑞 𝑁 𝑞 𝑛 superscript subscript ∇ 𝜆 𝑛 delimited-[] 𝑤 𝑥 𝑐 𝑁 conditional 𝑛 𝑞 {\displaystyle{\displaystyle{\displaystyle w(x;c,N|q)K_{n}\!\left(\lambda(x);c% ,N|q\right){}=(1-q)^{n}\left(cq^{-N};q\right)_{n}\left(\nabla_{\lambda}\right)% ^{n}\left[w(x;c,N-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}}}} &
w ⁢ ( x ; c , N | q ) = ( q - N , c ⁢ q - N ; q ) x ( q , c ⁢ q ; q ) x ⁢ c - x ⁢ q 2 ⁢ N ⁢ x - x ⁢ ( x - 1 ) 𝑤 𝑥 𝑐 conditional 𝑁 𝑞 q-Pochhammer-symbol superscript 𝑞 𝑁 𝑐 superscript 𝑞 𝑁 𝑞 𝑥 q-Pochhammer-symbol 𝑞 𝑐 𝑞 𝑞 𝑥 superscript 𝑐 𝑥 superscript 𝑞 2 𝑁 𝑥 𝑥 𝑥 1 {\displaystyle{\displaystyle{\displaystyle w(x;c,N|q)=\frac{\left(q^{-N},cq^{-% N};q\right)_{x}}{\left(q,cq;q\right)_{x}}c^{-x}q^{2Nx-x(x-1)}}}} &
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
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
( a ; q ) n subscript 𝑎 𝑞 𝑛 {\displaystyle{\displaystyle{\displaystyle(a;q)_{n}}}}  : q 𝑞 {\displaystyle{\displaystyle{\displaystyle q}}} -Pochhammer symbol : http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/17.2#SS1.p1

Bibliography

Equation in Section 14.17 of KLS.

URL links

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