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DLMF:18.28.E14 - MaRDI portal

DLMF:18.28.E14 (Q6685)

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DLMF:18.28.E14
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    Statements

    DLMF 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
    Symbols used
    cosine function
    DLMF defining formula
    cos ⁡ z 𝑧 {\displaystyle{\displaystyle\cos\NVar{z}}}
    xml-id
    C4.S14.E2.m2aedec
    0 references
    base of natural logarithm
    DLMF defining formula
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2aedec
    0 references
    imaginary unit
    DLMF defining formula
    i imaginary-unit {\displaystyle{\displaystyle\mathrm{i}}}
    xml-id
    C1.S9.E1.m2afdec
    0 references
    Q11269
    DLMF 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
    DLMF 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
    DLMF 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
    real variable
    DLMF defining formula
    q 𝑞 {\displaystyle{\displaystyle q}}
    xml-id
    C18.S1.XMD3.m1kdec
    0 references
    nonnegative integer
    DLMF defining formula
    n 𝑛 {\displaystyle{\displaystyle n}}
    xml-id
    C18.S1.XMD6.m1idec
    0 references

    Identifiers

     
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