DLMF:18.28.E14 (Q5989): Difference between revisions

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Property / Symbols used
 
Property / Symbols used: base of natural logarithm / rank
 
Normal rank
Property / Symbols used: base of natural logarithm / qualifier
 
Defining formula:

e {\displaystyle{\displaystyle\mathrm{e}}}

\expe
Property / Symbols used: base of natural logarithm / qualifier
 
xml-id: C4.S2.E11.m2aedec
Property / Symbols used
 
Property / Symbols used: imaginary unit / rank
 
Normal rank
Property / Symbols used: imaginary unit / qualifier
 
Defining formula:

i imaginary-unit {\displaystyle{\displaystyle\mathrm{i}}}

\iunit
Property / Symbols used: imaginary unit / qualifier
 
xml-id: C1.S9.E1.m2afdec
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.m2aidec
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
 
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)}}

\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.m2addec
Property / Symbols used
 
Property / Symbols used: continuous $$q$$ -ultraspherical polynomial / rank
 
Normal rank
Property / Symbols used: continuous $$q$$ -ultraspherical polynomial / qualifier
 
Defining formula:

C n ⁡ ( x ; β | q ) q-ultraspherical-polynomial 𝑛 𝑥 𝛽 𝑞 {\displaystyle{\displaystyle C_{\NVar{n}}\left(\NVar{x};\NVar{\beta}\,|\,\NVar% {q}\right)}}

\contqultrasphpoly{\NVar{n}}@{\NVar{x}}{\NVar{\beta}}{\NVar{q}}
Property / Symbols used: continuous $$q$$ -ultraspherical polynomial / qualifier
 
xml-id: C18.S28.E13.m2aadec
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.m1kdec
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.m1idec

Latest revision as of 13:23, 2 January 2020

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DLMF:18.28.E14
No description defined

    Statements

    Defining formula
    C n ⁡ ( cos ⁡ θ ; β | q ) = ( β 2 ; q ) n ( q ; q ) n ⁢ β 1 2 ⁢ n ⁢ ϕ 3 4 ⁡ ( q - n , β 2 ⁢ q n , β 1 2 ⁢ e i ⁢ θ , β 1 2 ⁢ e - i ⁢ θ β ⁢ q 1 2 , - β , - β ⁢ q 1 2 ; q , q ) . q-ultraspherical-polynomial 𝑛 𝜃 𝛽 𝑞 q-Pochhammer-symbol superscript 𝛽 2 𝑞 𝑛 q-Pochhammer-symbol 𝑞 𝑞 𝑛 superscript 𝛽 1 2 𝑛 q-hypergeometric-rphis 4 3 superscript 𝑞 𝑛 superscript 𝛽 2 superscript 𝑞 𝑛 superscript 𝛽 1 2 superscript 𝑒 imaginary-unit 𝜃 superscript 𝛽 1 2 superscript 𝑒 imaginary-unit 𝜃 𝛽 superscript 𝑞 1 2 𝛽 𝛽 superscript 𝑞 1 2 𝑞 𝑞 {\displaystyle{\displaystyle C_{n}\left(\cos\theta;\beta\,|\,q\right)=\frac{% \left(\beta^{2};q\right)_{n}}{\left(q;q\right)_{n}\beta^{\frac{1}{2}n}}{{}_{4}% \phi_{3}}\left({q^{-n},\beta^{2}q^{n},\beta^{\frac{1}{2}}e^{\mathrm{i}\theta},% \beta^{\frac{1}{2}}e^{-\mathrm{i}\theta}\atop\beta q^{\frac{1}{2}},-\beta,-% \beta q^{\frac{1}{2}}};q,q\right).}}
    0 references
    DLMF id
    DLMF:18.28.E14
    0 references
    Symbols used
    cosine function
    Defining formula
    cos ⁡ z 𝑧 {\displaystyle{\displaystyle\cos\NVar{z}}}
    xml-id
    C4.S14.E2.m2aedec
    0 references
    base of natural logarithm
    Defining formula
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2aedec
    0 references
    imaginary unit
    Defining formula
    i imaginary-unit {\displaystyle{\displaystyle\mathrm{i}}}
    xml-id
    C1.S9.E1.m2afdec
    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.m2addec
    0 references
    continuous $$q$$ -ultraspherical polynomial
    Defining formula
    C n ⁡ ( x ; β | q ) q-ultraspherical-polynomial 𝑛 𝑥 𝛽 𝑞 {\displaystyle{\displaystyle C_{\NVar{n}}\left(\NVar{x};\NVar{\beta}\,|\,\NVar% {q}\right)}}
    xml-id
    C18.S28.E13.m2aadec
    0 references
    Q11728
    Defining formula
    q 𝑞 {\displaystyle{\displaystyle q}}
    xml-id
    C18.S1.XMD3.m1kdec
    0 references
    Q11726
    Defining formula
    n 𝑛 {\displaystyle{\displaystyle n}}
    xml-id
    C18.S1.XMD6.m1idec
    0 references
     
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