DLMF:18.15.E6 (Q5696)

From DRMF
Jump to navigation Jump to search
No description defined
Language Label Description Also known as
English
DLMF:18.15.E6
No description defined

    Statements

    Defining formula
    ( sin ⁡ 1 2 ⁢ θ ) α + 1 2 ⁢ ( cos ⁡ 1 2 ⁢ θ ) β + 1 2 ⁢ P n ( α , β ) ⁡ ( cos ⁡ θ ) = Γ ⁡ ( n + α + 1 ) 2 1 2 ⁢ ρ α ⁢ n ! ⁢ ( θ 1 2 ⁢ J α ⁡ ( ρ ⁢ θ ) ⁢ ∑ m = 0 M A m ⁢ ( θ ) ρ 2 ⁢ m + θ 3 2 ⁢ J α + 1 ⁡ ( ρ ⁢ θ ) ⁢ ∑ m = 0 M - 1 B m ⁢ ( θ ) ρ 2 ⁢ m + 1 + ε M ⁢ ( ρ , θ ) ) , superscript 1 2 𝜃 𝛼 1 2 superscript 1 2 𝜃 𝛽 1 2 Jacobi-polynomial-P 𝛼 𝛽 𝑛 𝜃 Euler-Gamma 𝑛 𝛼 1 superscript 2 1 2 superscript 𝜌 𝛼 𝑛 superscript 𝜃 1 2 Bessel-J 𝛼 𝜌 𝜃 superscript subscript 𝑚 0 𝑀 subscript 𝐴 𝑚 𝜃 superscript 𝜌 2 𝑚 superscript 𝜃 3 2 Bessel-J 𝛼 1 𝜌 𝜃 superscript subscript 𝑚 0 𝑀 1 subscript 𝐵 𝑚 𝜃 superscript 𝜌 2 𝑚 1 subscript 𝜀 𝑀 𝜌 𝜃 {\displaystyle{\displaystyle(\sin\tfrac{1}{2}\theta)^{\alpha+\frac{1}{2}}(\cos% \tfrac{1}{2}\theta)^{\beta+\frac{1}{2}}P^{(\alpha,\beta)}_{n}\left(\cos\theta% \right)=\frac{\Gamma\left(n+\alpha+1\right)}{2^{\frac{1}{2}}\rho^{\alpha}n!}\*% \left(\theta^{\frac{1}{2}}J_{\alpha}\left(\rho\theta\right)\sum_{m=0}^{M}% \dfrac{A_{m}(\theta)}{\rho^{2m}}+\theta^{\frac{3}{2}}J_{\alpha+1}\left(\rho% \theta\right)\sum_{m=0}^{M-1}\dfrac{B_{m}(\theta)}{\rho^{2m+1}}+\varepsilon_{M% }(\rho,\theta)\right),}}
    0 references
    DLMF id
    DLMF:18.15.E6
    0 references
    Symbols used
    Bessel function of the first kind
    Defining formula
    J ν ⁡ ( z ) Bessel-J 𝜈 𝑧 {\displaystyle{\displaystyle J_{\NVar{\nu}}\left(\NVar{z}\right)}}
    xml-id
    C10.S2.E2.m2adec
    0 references
    gamma function
    Defining formula
    Γ ⁡ ( z ) Euler-Gamma 𝑧 {\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}
    xml-id
    C5.S2.E1.m2adec
    0 references
    Q11660
    Defining formula
    P n ( α , β ) ⁡ ( x ) Jacobi-polynomial-P 𝛼 𝛽 𝑛 𝑥 {\displaystyle{\displaystyle P^{(\NVar{\alpha},\NVar{\beta})}_{\NVar{n}}\left(% \NVar{x}\right)}}
    xml-id
    C18.S3.T1.t1.r2.m2aadec
    0 references
    cosine function
    Defining formula
    cos ⁡ z 𝑧 {\displaystyle{\displaystyle\cos\NVar{z}}}
    xml-id
    C4.S14.E2.m2abdec
    0 references
    Q10755
    Defining formula
    ! {\displaystyle{\displaystyle!}}
    xml-id
    introduction.Sx4.p1.t1.r15.m5aadec
    0 references
    sine function
    Defining formula
    sin ⁡ z 𝑧 {\displaystyle{\displaystyle\sin\NVar{z}}}
    xml-id
    C4.S14.E1.m2abdec
    0 references
    Q11730
    Defining formula
    m 𝑚 {\displaystyle{\displaystyle m}}
    xml-id
    C18.S1.XMD5.m1ddec
    0 references
    Q11726
    Defining formula
    n 𝑛 {\displaystyle{\displaystyle n}}
    xml-id
    C18.S1.XMD6.m1ddec
    0 references
    DLMF:18.15.E5
    Defining formula
    ρ 𝜌 {\displaystyle{\displaystyle\rho}}
    xml-id
    C18.S15.XMD4.m1adec
    0 references
    DLMF:18.15.E7
    Defining formula
    ε M ⁢ ( ρ , θ ) subscript 𝜀 𝑀 𝜌 𝜃 {\displaystyle{\displaystyle\varepsilon_{M}(\rho,\theta)}}
    xml-id
    C18.S15.XMD5.m1dec
    0 references
    Q11752
    Defining formula
    A m ⁢ ( θ ) subscript 𝐴 𝑚 𝜃 {\displaystyle{\displaystyle A_{m}(\theta)}}
    xml-id
    C18.S15.XMD6.m1dec
    0 references
    Q11753
    Defining formula
    B m ⁢ ( θ ) subscript 𝐵 𝑚 𝜃 {\displaystyle{\displaystyle B_{m}(\theta)}}
    xml-id
    C18.S15.XMD7.m1dec
    0 references
     
        Retrieved from "https://drmf-beta.wmflabs.org/index.php?title=Item:Q5696&oldid=59416"