DLMF:18.27.E19 (Q5970): Difference between revisions

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Property / Symbols used
 
Property / Symbols used: Q10768 / rank
 
Normal rank
Property / Symbols used: Q10768 / qualifier
 
Defining formula:

δ j , k Kronecker 𝑗 𝑘 {\displaystyle{\displaystyle\delta_{\NVar{j},\NVar{k}}}}

\Kroneckerdelta{\NVar{j}}{\NVar{k}}
Property / Symbols used: Q10768 / qualifier
 
xml-id: introduction.Sx4.p1.t1.r4.m5aedec
Property / Symbols used
 
Property / Symbols used: Stieltjes–Wigert polynomial / rank
 
Normal rank
Property / Symbols used: Stieltjes–Wigert polynomial / qualifier
 
Defining formula:

S n ⁡ ( x ; q ) Stieltjes-Wigert-polynomial-S 𝑛 𝑥 𝑞 {\displaystyle{\displaystyle S_{\NVar{n}}\left(\NVar{x};\NVar{q}\right)}}

\StieltjesWigertpolyS{\NVar{n}}@{\NVar{x}}{\NVar{q}}
Property / Symbols used: Stieltjes–Wigert polynomial / qualifier
 
xml-id: C18.S27.E18.m2aadec
Property / Symbols used
 
Property / Symbols used: Q10770 / rank
 
Normal rank
Property / Symbols used: Q10770 / qualifier
 
Defining formula:

d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}

\diff{\NVar{x}}
Property / Symbols used: Q10770 / qualifier
 
xml-id: C1.S4.SS4.m1aadec
Property / Symbols used
 
Property / Symbols used: Q10771 / rank
 
Normal rank
Property / Symbols used: Q10771 / qualifier
 
Defining formula:

∫ {\displaystyle{\displaystyle\int}}

\int
Property / Symbols used: Q10771 / qualifier
 
xml-id: C1.S4.SS4.m3aadec
Property / Symbols used
 
Property / Symbols used: principal branch of logarithm function / rank
 
Normal rank
Property / Symbols used: principal branch of logarithm function / qualifier
 
Defining formula:

ln ⁡ z 𝑧 {\displaystyle{\displaystyle\ln\NVar{z}}}

\ln@@{\NVar{z}}
Property / Symbols used: principal branch of logarithm function / qualifier
 
xml-id: C4.S2.E2.m2adec
Property / Symbols used
 
Property / Symbols used: Q11298 / rank
 
Normal rank
Property / Symbols used: Q11298 / qualifier
 
Defining formula:

( 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.m2aedec
Property / Symbols used
 
Property / Symbols used: Q11713 / rank
 
Normal rank
Property / Symbols used: Q11713 / qualifier
 
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}}}}

\qmultiPochhammersym{\NVar{a_{1},a_{2},\dots,a_{r}}}{\NVar{q}}{\NVar{n}}
Property / Symbols used: Q11713 / qualifier
 
xml-id: C17.S2.SS1.p1.m6aadec
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.m1qdec
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.m1edec
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.m1ndec
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.m1mdec

Latest revision as of 01:51, 2 January 2020

No description defined
Language Label Description Also known as
English
DLMF:18.27.E19
No description defined

    Statements

    Defining formula
    ∫ 0 ∞ S n ⁡ ( x ; q ) ⁢ S m ⁡ ( x ; q ) ( - x , - q ⁢ x - 1 ; q ) ∞ ⁢ d x = ln ⁡ ( q - 1 ) q n ⁢ ( q ; q ) ∞ ( q ; q ) n ⁢ δ n , m , superscript subscript 0 Stieltjes-Wigert-polynomial-S 𝑛 𝑥 𝑞 Stieltjes-Wigert-polynomial-S 𝑚 𝑥 𝑞 q-multiple-Pochhammer 𝑥 𝑞 superscript 𝑥 1 𝑞 𝑥 superscript 𝑞 1 superscript 𝑞 𝑛 q-Pochhammer-symbol 𝑞 𝑞 q-Pochhammer-symbol 𝑞 𝑞 𝑛 Kronecker 𝑛 𝑚 {\displaystyle{\displaystyle\int_{0}^{\infty}\frac{S_{n}\left(x;q\right)S_{m}% \left(x;q\right)}{\left(-x,-qx^{-1};q\right)_{\infty}}\mathrm{d}x=\frac{\ln% \left(q^{-1}\right)}{q^{n}}\frac{\left(q;q\right)_{\infty}}{\left(q;q\right)_{% n}}\delta_{n,m},}}
    0 references
    DLMF id
    DLMF:18.27.E19
    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.m5aedec
    0 references
    Stieltjes–Wigert polynomial
    Defining formula
    S n ⁡ ( x ; q ) Stieltjes-Wigert-polynomial-S 𝑛 𝑥 𝑞 {\displaystyle{\displaystyle S_{\NVar{n}}\left(\NVar{x};\NVar{q}\right)}}
    xml-id
    C18.S27.E18.m2aadec
    0 references
    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1aadec
    0 references
    Q10771
    Defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3aadec
    0 references
    principal branch of logarithm function
    Defining formula
    ln ⁡ z 𝑧 {\displaystyle{\displaystyle\ln\NVar{z}}}
    xml-id
    C4.S2.E2.m2adec
    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.m2aedec
    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.m6aadec
    0 references
    Q11728
    Defining formula
    q 𝑞 {\displaystyle{\displaystyle q}}
    xml-id
    C18.S1.XMD3.m1qdec
    0 references
    Q11730
    Defining formula
    m 𝑚 {\displaystyle{\displaystyle m}}
    xml-id
    C18.S1.XMD5.m1edec
    0 references
    Q11726
    Defining formula
    n 𝑛 {\displaystyle{\displaystyle n}}
    xml-id
    C18.S1.XMD6.m1ndec
    0 references
    Q11727
    Defining formula
    x 𝑥 {\displaystyle{\displaystyle x}}
    xml-id
    C18.S2.XMD3.m1mdec
    0 references
     
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