DLMF:5.18.E12 (Q2199): Difference between revisions

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Property / constraint
 

ℜ ⁡ a > 0 𝑎 0 {\displaystyle{\displaystyle\Re a>0}}

\Re a>0
Property / constraint: ℜ ⁡ a > 0 𝑎 0 {\displaystyle{\displaystyle\Re a>0}} / rank
 
Normal rank
Property / constraint
 

ℜ ⁡ b > 0 𝑏 0 {\displaystyle{\displaystyle\Re b>0}}

\Re b>0
Property / constraint: ℜ ⁡ b > 0 𝑏 0 {\displaystyle{\displaystyle\Re b>0}} / rank
 
Normal rank
Property / Symbols used
 
Property / Symbols used: Q10771 / rank
 
Normal rank
Property / Symbols used: Q10771 / qualifier
 
test:

∫ {\displaystyle{\displaystyle\int}}

\int
Property / Symbols used: Q10771 / qualifier
 
xml-id: C1.S4.SS4.m3adec
Property / Symbols used
 
Property / Symbols used: $$q$$ -beta function / rank
 
Normal rank
Property / Symbols used: $$q$$ -beta function / qualifier
 
test:

B q ⁡ ( a , b ) q-Beta 𝑞 𝑎 𝑏 {\displaystyle{\displaystyle\mathrm{B}_{\NVar{q}}\left(\NVar{a},\NVar{b}\right% )}}

\qBeta{\NVar{q}}@{\NVar{a}}{\NVar{b}}
Property / Symbols used: $$q$$ -beta function / qualifier
 
xml-id: C5.S18.E11.m2aadec
Property / Symbols used
 
Property / Symbols used: Q11298 / rank
 
Normal rank
Property / Symbols used: Q11298 / qualifier
 
test:

( a ; q ) n q-Pochhammer-symbol 𝑎 𝑞 𝑛 {\displaystyle{\displaystyle\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}}}

\qPochhammer{\NVar{a}}{\NVar{q}}{\NVar{n}}
Property / Symbols used: Q11298 / qualifier
 
xml-id: C17.S2.SS1.p1.m2addec
Property / Symbols used
 
Property / Symbols used: Q11299 / rank
 
Normal rank
Property / Symbols used: Q11299 / qualifier
 
test:

d q x q-differential 𝑞 𝑥 {\displaystyle{\displaystyle{\mathrm{d}}_{\NVar{q}}\NVar{x}}}

\qdiff{\NVar{q}}{\NVar{x}}
Property / Symbols used: Q11299 / qualifier
 
xml-id: C17.S2.SS5.p1.m3adec
Property / Symbols used
 
Property / Symbols used: Q10811 / rank
 
Normal rank
Property / Symbols used: Q10811 / qualifier
 
test:

ℜ ⁡ absent {\displaystyle{\displaystyle\Re}}

\realpart@@
Property / Symbols used: Q10811 / qualifier
 
xml-id: C1.S9.E2.m1adec
Property / Symbols used
 
Property / Symbols used: Q11280 / rank
 
Normal rank
Property / Symbols used: Q11280 / qualifier
 
test:

q 𝑞 {\displaystyle{\displaystyle q}}

q
Property / Symbols used: Q11280 / qualifier
 
xml-id: C5.S1.XMD10.m1kdec
Property / Symbols used
 
Property / Symbols used: Q11275 / rank
 
Normal rank
Property / Symbols used: Q11275 / qualifier
 
test:

a 𝑎 {\displaystyle{\displaystyle a}}

a
Property / Symbols used: Q11275 / qualifier
 
xml-id: C5.S1.XMD8.m1cdec
Property / Symbols used
 
Property / Symbols used: Q11283 / rank
 
Normal rank
Property / Symbols used: Q11283 / qualifier
 
test:

b 𝑏 {\displaystyle{\displaystyle b}}

b
Property / Symbols used: Q11283 / qualifier
 
xml-id: C5.S1.XMD9.m1adec

Latest revision as of 00:11, 2 January 2020

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DLMF:5.18.E12
No description defined

    Statements

    test
    B q ⁡ ( a , b ) = ∫ 0 1 t a - 1 ⁢ ( t ⁢ q ; q ) ∞ ( t ⁢ q b ; q ) ∞ ⁢ d q t , q-Beta 𝑞 𝑎 𝑏 superscript subscript 0 1 superscript 𝑡 𝑎 1 q-Pochhammer-symbol 𝑡 𝑞 𝑞 q-Pochhammer-symbol 𝑡 superscript 𝑞 𝑏 𝑞 q-differential 𝑞 𝑡 {\displaystyle{\displaystyle\mathrm{B}_{q}\left(a,b\right)=\int_{0}^{1}\frac{t% ^{a-1}\left(tq;q\right)_{\infty}}{\left(tq^{b};q\right)_{\infty}}{\mathrm{d}}_% {q}t,}}
    0 references
    DLMF id
    DLMF:5.18.E12
    0 references
    constraint
    ℜ ⁡ a > 0 𝑎 0 {\displaystyle{\displaystyle\Re a>0}}
    0 references
    ℜ ⁡ b > 0 𝑏 0 {\displaystyle{\displaystyle\Re b>0}}
    0 references
    0 < q < 1 0 𝑞 1 {\displaystyle{\displaystyle 0<q<1}}
    0 references
    ℜ ⁡ a > 0 𝑎 0 {\displaystyle{\displaystyle\Re a>0}}
    0 references
    ℜ ⁡ b > 0 𝑏 0 {\displaystyle{\displaystyle\Re b>0}}
    0 references
    Symbols used
    Q10771
    test
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3adec
    0 references
    $$q$$ -beta function
    test
    B q ⁡ ( a , b ) q-Beta 𝑞 𝑎 𝑏 {\displaystyle{\displaystyle\mathrm{B}_{\NVar{q}}\left(\NVar{a},\NVar{b}\right% )}}
    xml-id
    C5.S18.E11.m2aadec
    0 references
    Q11298
    test
    ( 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.m2addec
    0 references
    Q11299
    test
    d q x q-differential 𝑞 𝑥 {\displaystyle{\displaystyle{\mathrm{d}}_{\NVar{q}}\NVar{x}}}
    xml-id
    C17.S2.SS5.p1.m3adec
    0 references
    Q10811
    test
    ℜ ⁡ absent {\displaystyle{\displaystyle\Re}}
    xml-id
    C1.S9.E2.m1adec
    0 references
    Q11280
    test
    q 𝑞 {\displaystyle{\displaystyle q}}
    xml-id
    C5.S1.XMD10.m1kdec
    0 references
    Q11275
    test
    a 𝑎 {\displaystyle{\displaystyle a}}
    xml-id
    C5.S1.XMD8.m1cdec
    0 references
    Q11283
    test
    b 𝑏 {\displaystyle{\displaystyle b}}
    xml-id
    C5.S1.XMD9.m1adec
    0 references
     
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