DLMF:18.23.E3 (Q5904)

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DLMF:18.23.E3
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    Statements

    Defining formula
    ( 1 - 1 - p p ⁢ z ) x ⁢ ( 1 + z ) N - x = ∑ n = 0 N ( N n ) ⁢ K n ⁡ ( x ; p , N ) ⁢ z n , superscript 1 1 𝑝 𝑝 𝑧 𝑥 superscript 1 𝑧 𝑁 𝑥 superscript subscript 𝑛 0 𝑁 binomial 𝑁 𝑛 Krawtchouk-polynomial-K 𝑛 𝑥 𝑝 𝑁 superscript 𝑧 𝑛 {\displaystyle{\displaystyle\left(1-\frac{1-p}{p}z\right)^{x}(1+z)^{N-x}=\sum_% {n=0}^{N}\genfrac{(}{)}{0.0pt}{}{N}{n}K_{n}\left(x;p,N\right)z^{n},}}
    0 references
    DLMF id
    DLMF:18.23.E3
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    constraint
    x = 0 , 1 , … , N 𝑥 0 1 … 𝑁 {\displaystyle{\displaystyle x=0,1,\dots,N}}
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    Symbols used
    Q11662
    Defining formula
    K n ⁡ ( x ; p , N ) Krawtchouk-polynomial-K 𝑛 𝑥 𝑝 𝑁 {\displaystyle{\displaystyle K_{\NVar{n}}\left(\NVar{x};\NVar{p},\NVar{N}% \right)}}
    xml-id
    C18.S19.T1.t1.r3.p1.m2adec
    0 references
    Q10754
    Defining formula
    ( m n ) binomial 𝑚 𝑛 {\displaystyle{\displaystyle\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}}}
    xml-id
    C1.S2.SS1.m1adec
    0 references
    Q11750
    Defining formula
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C18.S1.XMD2.m1bdec
    0 references
     
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