DLMF:18.27.E23 (Q5974): Difference between revisions

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Property / Symbols used
 
Property / Symbols used: Q11727 / rank
 
Normal rank
Property / Symbols used: Q11727 / qualifier
 
Defining formula:

x 𝑥 {\displaystyle{\displaystyle x}}

x
Property / Symbols used: Q11727 / qualifier
 
xml-id: C18.S2.XMD3.m1pdec

Latest revision as of 01:52, 2 January 2020

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DLMF:18.27.E23
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    Statements

    Defining formula
    h ~ n ⁡ ( x ; q ) = ( q ; q ) n ⁢ ∑ ℓ = 0 ⌊ n / 2 ⌋ ( - 1 ) ℓ ⁢ q - 2 ⁢ n ⁢ ℓ ⁢ q ℓ ⁢ ( 2 ⁢ ℓ + 1 ) ⁢ x n - 2 ⁢ ℓ ( q 2 ; q 2 ) ℓ ⁢ ( q ; q ) n - 2 ⁢ ℓ = x n ⁢ ϕ 1 2 ⁡ ( q - n , q - n + 1 0 ; q 2 , - x - 2 ⁢ q 2 ) . discrete-q-Hermite-polynomial-h-II 𝑛 𝑥 𝑞 q-Pochhammer-symbol 𝑞 𝑞 𝑛 superscript subscript ℓ 0 𝑛 2 superscript 1 ℓ superscript 𝑞 2 𝑛 ℓ superscript 𝑞 ℓ 2 ℓ 1 superscript 𝑥 𝑛 2 ℓ q-Pochhammer-symbol superscript 𝑞 2 superscript 𝑞 2 ℓ q-Pochhammer-symbol 𝑞 𝑞 𝑛 2 ℓ superscript 𝑥 𝑛 q-hypergeometric-rphis 2 1 superscript 𝑞 𝑛 superscript 𝑞 𝑛 1 0 superscript 𝑞 2 superscript 𝑥 2 superscript 𝑞 2 {\displaystyle{\displaystyle\tilde{h}_{n}\left(x;q\right)=\left(q;q\right)_{n}% \sum_{\ell=0}^{\left\lfloor n/2\right\rfloor}\frac{(-1)^{\ell}q^{-2n\ell}q^{% \ell(2\ell+1)}x^{n-2\ell}}{\left(q^{2};q^{2}\right)_{\ell}\left(q;q\right)_{n-% 2\ell}}=x^{n}{{}_{2}\phi_{1}}\left({q^{-n},q^{-n+1}\atop 0};q^{2},-x^{-2}q^{2}% \right).}}
    0 references
    DLMF id
    DLMF:18.27.E23
    0 references
    Symbols used
    Q11169
    Defining formula
    ⌊ x ⌋ 𝑥 {\displaystyle{\displaystyle\left\lfloor\NVar{x}\right\rfloor}}
    xml-id
    introduction.Sx4.p1.t1.r17.m4aadec
    0 references
    Q11298
    Defining formula
    ( a ; q ) n q-Pochhammer-symbol 𝑎 𝑞 𝑛 {\displaystyle{\displaystyle\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}}}
    xml-id
    C17.S2.SS1.p1.m2aidec
    0 references
    basic hypergeometric (or $$q$$ -hypergeometric) function
    Defining formula
    ϕ s r + 1 ⁡ ( a 0 , … , a r ; b 1 , … , b s ; q , z ) q-hypergeometric-rphis 𝑟 1 𝑠 subscript 𝑎 0 … subscript 𝑎 𝑟 subscript 𝑏 1 … subscript 𝑏 𝑠 𝑞 𝑧 {\displaystyle{\displaystyle{{}_{\NVar{r+1}}\phi_{\NVar{s}}}\left(\NVar{a_{0},% \dots,a_{r}};\NVar{b_{1},\dots,b_{s}};\NVar{q},\NVar{z}\right)}}
    xml-id
    C17.S4.E1.m2afdec
    0 references
    Q11728
    Defining formula
    q 𝑞 {\displaystyle{\displaystyle q}}
    xml-id
    C18.S1.XMD3.m1udec
    0 references
    Q11744
    Defining formula
    ℓ ℓ {\displaystyle{\displaystyle\ell}}
    xml-id
    C18.S1.XMD4.m1edec
    0 references
    Q11726
    Defining formula
    n 𝑛 {\displaystyle{\displaystyle n}}
    xml-id
    C18.S1.XMD6.m1rdec
    0 references
    Q11727
    Defining formula
    x 𝑥 {\displaystyle{\displaystyle x}}
    xml-id
    C18.S2.XMD3.m1pdec
    0 references
     
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