DLMF:18.28.E19 (Q5994): Difference between revisions

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Property / Symbols defined
 
Property / Symbols defined: DLMF:18.28.E19 / rank
 
Normal rank
Property / constraint
 

n = 0 , 1 , … , N 𝑛 0 1 … 𝑁 {\displaystyle{\displaystyle n=0,1,\dots,N}}

n=0,1,\dots,N
Property / constraint: n = 0 , 1 , … , N 𝑛 0 1 … 𝑁 {\displaystyle{\displaystyle n=0,1,\dots,N}} / rank
 
Normal rank
Property / Symbols used
 
Property / Symbols used: basic hypergeometric (or $$q$$ -hypergeometric) function / rank
 
Normal rank
Property / Symbols used: basic hypergeometric (or $$q$$ -hypergeometric) function / qualifier
 
test:

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)}}

\qgenhyperphi{\NVar{r+1}}{\NVar{s}}@{\NVar{a_{0},\dots,a_{r}}}{\NVar{b_{1},\dots,b_{s}}}{\NVar{q}}{\NVar{z}}
Property / Symbols used: basic hypergeometric (or $$q$$ -hypergeometric) function / qualifier
 
xml-id: C17.S4.E1.m2agdec
Property / Symbols used
 
Property / Symbols used: Q11713 / rank
 
Normal rank
Property / Symbols used: Q11713 / qualifier
 
test:

( 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}}}}

\qmultiPochhammersym{\NVar{a_{1},a_{2},\dots,a_{r}}}{\NVar{q}}{\NVar{n}}
Property / Symbols used: Q11713 / qualifier
 
xml-id: C17.S2.SS1.p1.m6agdec
Property / Symbols used
 
Property / Symbols used: Q11743 / rank
 
Normal rank
Property / Symbols used: Q11743 / qualifier
 
test:

y 𝑦 {\displaystyle{\displaystyle y}}

y
Property / Symbols used: Q11743 / qualifier
 
xml-id: C18.S1.XMD1.m1bdec
Property / Symbols used
 
Property / Symbols used: Q11728 / rank
 
Normal rank
Property / Symbols used: Q11728 / qualifier
 
test:

q 𝑞 {\displaystyle{\displaystyle q}}

q
Property / Symbols used: Q11728 / qualifier
 
xml-id: C18.S1.XMD3.m1pdec
Property / Symbols used
 
Property / Symbols used: Q11744 / rank
 
Normal rank
Property / Symbols used: Q11744 / qualifier
 
test:

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

\ell
Property / Symbols used: Q11744 / qualifier
 
xml-id: C18.S1.XMD4.m1fdec
Property / Symbols used
 
Property / Symbols used: Q11726 / rank
 
Normal rank
Property / Symbols used: Q11726 / qualifier
 
test:

n 𝑛 {\displaystyle{\displaystyle n}}

n
Property / Symbols used: Q11726 / qualifier
 
xml-id: C18.S1.XMD6.m1ndec
Property / Symbols used
 
Property / Symbols used: Q11736 / rank
 
Normal rank
Property / Symbols used: Q11736 / qualifier
 
test:

N 𝑁 {\displaystyle{\displaystyle N}}

N
Property / Symbols used: Q11736 / qualifier
 
xml-id: C18.S1.XMD7.m1dec
Property / Symbols used
 
Property / Symbols used: Q11754 / rank
 
Normal rank
Property / Symbols used: Q11754 / qualifier
 
test:

δ 𝛿 {\displaystyle{\displaystyle\delta}}

\delta
Property / Symbols used: Q11754 / qualifier
 
xml-id: C18.S1.XMD8.m1dec
Property / Symbols used
 
Property / Symbols used: Q11727 / rank
 
Normal rank
Property / Symbols used: Q11727 / qualifier
 
test:

x 𝑥 {\displaystyle{\displaystyle x}}

x
Property / Symbols used: Q11727 / qualifier
 
xml-id: C18.S2.XMD3.m1hdec

Latest revision as of 12:24, 2 January 2020

No description defined
Language Label Description Also known as
English
DLMF:18.28.E19
No description defined

    Statements

    test
    R n ⁢ ( x ) = R n ⁡ ( x ; α , β , γ , δ | q ) = ∑ ℓ = 0 n q ℓ ⁢ ( q - n , α ⁢ β ⁢ q n + 1 ; q ) ℓ ( α ⁢ q , β ⁢ δ ⁢ q , γ ⁢ q , q ; q ) ℓ ⁢ ∏ j = 0 ℓ - 1 ( 1 - q j ⁢ x + γ ⁢ δ ⁢ q 2 ⁢ j + 1 ) = ϕ 3 4 ⁡ ( q - n , α ⁢ β ⁢ q n + 1 , q - y , γ ⁢ δ ⁢ q y + 1 α ⁢ q , β ⁢ δ ⁢ q , γ ⁢ q ; q , q ) , subscript 𝑅 𝑛 𝑥 q-Racah-polynomial-R 𝑛 𝑥 𝛼 𝛽 𝛾 𝛿 𝑞 superscript subscript ℓ 0 𝑛 superscript 𝑞 ℓ q-multiple-Pochhammer superscript 𝑞 𝑛 𝛼 𝛽 superscript 𝑞 𝑛 1 𝑞 ℓ q-multiple-Pochhammer 𝛼 𝑞 𝛽 𝛿 𝑞 𝛾 𝑞 𝑞 𝑞 ℓ superscript subscript product 𝑗 0 ℓ 1 1 superscript 𝑞 𝑗 𝑥 𝛾 𝛿 superscript 𝑞 2 𝑗 1 q-hypergeometric-rphis 4 3 superscript 𝑞 𝑛 𝛼 𝛽 superscript 𝑞 𝑛 1 superscript 𝑞 𝑦 𝛾 𝛿 superscript 𝑞 𝑦 1 𝛼 𝑞 𝛽 𝛿 𝑞 𝛾 𝑞 𝑞 𝑞 {\displaystyle{\displaystyle R_{n}(x)=R_{n}\left(x;\alpha,\beta,\gamma,\delta% \,|\,q\right)=\sum_{\ell=0}^{n}\frac{q^{\ell}\left(q^{-n},\alpha\beta q^{n+1};% q\right)_{\ell}}{\left(\alpha q,\beta\delta q,\gamma q,q;q\right)_{\ell}}\*% \prod_{j=0}^{\ell-1}(1-q^{j}x+\gamma\delta q^{2j+1})={{}_{4}\phi_{3}}\left({q^% {-n},\alpha\beta q^{n+1},q^{-y},\gamma\delta q^{y+1}\atop\alpha q,\beta\delta q% ,\gamma q};q,q\right),}}
    0 references
    DLMF id
    DLMF:18.28.E19
    0 references
    constraint
    n = 0 , 1 , … , N 𝑛 0 1 … 𝑁 {\displaystyle{\displaystyle n=0,1,\dots,N}}
    0 references
    Symbols used
    basic hypergeometric (or $$q$$ -hypergeometric) function
    test
    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.m2agdec
    0 references
    Q11713
    test
    ( 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.m6agdec
    0 references
    Q11743
    test
    y 𝑦 {\displaystyle{\displaystyle y}}
    xml-id
    C18.S1.XMD1.m1bdec
    0 references
    Q11728
    test
    q 𝑞 {\displaystyle{\displaystyle q}}
    xml-id
    C18.S1.XMD3.m1pdec
    0 references
    Q11744
    test
    ℓ ℓ {\displaystyle{\displaystyle\ell}}
    xml-id
    C18.S1.XMD4.m1fdec
    0 references
    Q11726
    test
    n 𝑛 {\displaystyle{\displaystyle n}}
    xml-id
    C18.S1.XMD6.m1ndec
    0 references
    Q11736
    test
    N 𝑁 {\displaystyle{\displaystyle N}}
    xml-id
    C18.S1.XMD7.m1dec
    0 references
    Q11754
    test
    δ 𝛿 {\displaystyle{\displaystyle\delta}}
    xml-id
    C18.S1.XMD8.m1dec
    0 references
    Q11727
    test
    x 𝑥 {\displaystyle{\displaystyle x}}
    xml-id
    C18.S2.XMD3.m1hdec
    0 references
     
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