DLMF:18.27.E24 (Q5975): Difference between revisions

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Property / Symbols used
 
Property / Symbols used: Q11728 / rank
 
Normal rank
Property / Symbols used: Q11728 / qualifier
 
Defining formula:

q 𝑞 {\displaystyle{\displaystyle q}}

q
Property / Symbols used: Q11728 / qualifier
 
xml-id: C18.S1.XMD3.m1vdec
Property / Symbols used
 
Property / Symbols used: Q11744 / rank
 
Normal rank
Property / Symbols used: Q11744 / qualifier
 
Defining formula:

ℓ ℓ {\displaystyle{\displaystyle\ell}}

\ell
Property / Symbols used: Q11744 / qualifier
 
xml-id: C18.S1.XMD4.m1fdec
Property / Symbols used
 
Property / Symbols used: Q11730 / rank
 
Normal rank
Property / Symbols used: Q11730 / qualifier
 
Defining formula:

m 𝑚 {\displaystyle{\displaystyle m}}

m
Property / Symbols used: Q11730 / qualifier
 
xml-id: C18.S1.XMD5.m1hdec
Property / Symbols used
 
Property / Symbols used: Q11726 / rank
 
Normal rank
Property / Symbols used: Q11726 / qualifier
 
Defining formula:

n 𝑛 {\displaystyle{\displaystyle n}}

n
Property / Symbols used: Q11726 / qualifier
 
xml-id: C18.S1.XMD6.m1sdec

Latest revision as of 01:52, 2 January 2020

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DLMF:18.27.E24
No description defined

    Statements

    Defining formula
    ∑ ℓ = - ∞ ∞ ( h ~ n ⁡ ( c ⁢ q ℓ ; q ) ⁢ h ~ m ⁡ ( c ⁢ q ℓ ; q ) + h ~ n ⁡ ( - c ⁢ q ℓ ; q ) ⁢ h ~ m ⁡ ( - c ⁢ q ℓ ; q ) ) ⁢ q ℓ ( - c 2 ⁢ q 2 ⁢ ℓ ; q 2 ) ∞ = 2 ⁢ ( q 2 , - c 2 ⁢ q , - c - 2 ⁢ q ; q 2 ) ∞ ( q , - c 2 , - c - 2 ⁢ q 2 ; q 2 ) ∞ ⁢ ( q ; q ) n q n 2 ⁢ δ n , m , superscript subscript ℓ discrete-q-Hermite-polynomial-h-II 𝑛 𝑐 superscript 𝑞 ℓ 𝑞 discrete-q-Hermite-polynomial-h-II 𝑚 𝑐 superscript 𝑞 ℓ 𝑞 discrete-q-Hermite-polynomial-h-II 𝑛 𝑐 superscript 𝑞 ℓ 𝑞 discrete-q-Hermite-polynomial-h-II 𝑚 𝑐 superscript 𝑞 ℓ 𝑞 superscript 𝑞 ℓ q-Pochhammer-symbol superscript 𝑐 2 superscript 𝑞 2 ℓ superscript 𝑞 2 2 q-multiple-Pochhammer superscript 𝑞 2 superscript 𝑐 2 𝑞 superscript 𝑐 2 𝑞 superscript 𝑞 2 q-multiple-Pochhammer 𝑞 superscript 𝑐 2 superscript 𝑐 2 superscript 𝑞 2 superscript 𝑞 2 q-Pochhammer-symbol 𝑞 𝑞 𝑛 superscript 𝑞 superscript 𝑛 2 Kronecker 𝑛 𝑚 {\displaystyle{\displaystyle\sum_{\ell=-\infty}^{\infty}\left(\tilde{h}_{n}% \left(cq^{\ell};q\right)\tilde{h}_{m}\left(cq^{\ell};q\right)+\tilde{h}_{n}% \left(-cq^{\ell};q\right)\tilde{h}_{m}\left(-cq^{\ell};q\right)\right)\frac{q^% {\ell}}{\left(-c^{2}q^{2\ell};q^{2}\right)_{\infty}}=2\frac{\left(q^{2},-c^{2}% q,-c^{-2}q;q^{2}\right)_{\infty}}{\left(q,-c^{2},-c^{-2}q^{2};q^{2}\right)_{% \infty}}\frac{\left(q;q\right)_{n}}{q^{n^{2}}}\delta_{n,m},}}
    0 references
    DLMF id
    DLMF:18.27.E24
    0 references
    constraint
    c > 0 𝑐 0 {\displaystyle{\displaystyle c>0}}
    0 references
    Symbols used
    Q10768
    Defining formula
    δ j , k Kronecker 𝑗 𝑘 {\displaystyle{\displaystyle\delta_{\NVar{j},\NVar{k}}}}
    xml-id
    introduction.Sx4.p1.t1.r4.m5ahdec
    0 references
    discrete $$q$$ -Hermite II polynomial
    Defining formula
    h ~ n ⁡ ( x ; q ) discrete-q-Hermite-polynomial-h-II 𝑛 𝑥 𝑞 {\displaystyle{\displaystyle\tilde{h}_{\NVar{n}}\left(\NVar{x};\NVar{q}\right)}}
    xml-id
    C18.S27.E23.m2aadec
    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.m2ajdec
    0 references
    Q11713
    Defining formula
    ( a 1 , a 2 , … , a r ; q ) n q-multiple-Pochhammer subscript 𝑎 1 subscript 𝑎 2 … subscript 𝑎 𝑟 𝑞 𝑛 {\displaystyle{\displaystyle\left(\NVar{a_{1},a_{2},\dots,a_{r}};\NVar{q}% \right)_{\NVar{n}}}}
    xml-id
    C17.S2.SS1.p1.m6acdec
    0 references
    Q11728
    Defining formula
    q 𝑞 {\displaystyle{\displaystyle q}}
    xml-id
    C18.S1.XMD3.m1vdec
    0 references
    Q11744
    Defining formula
    ℓ ℓ {\displaystyle{\displaystyle\ell}}
    xml-id
    C18.S1.XMD4.m1fdec
    0 references
    Q11730
    Defining formula
    m 𝑚 {\displaystyle{\displaystyle m}}
    xml-id
    C18.S1.XMD5.m1hdec
    0 references
    Q11726
    Defining formula
    n 𝑛 {\displaystyle{\displaystyle n}}
    xml-id
    C18.S1.XMD6.m1sdec
    0 references
     
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