DLMF:14.3.E16 (Q4705)

From DRMF
Jump to navigation Jump to search
No description defined
Language Label Description Also known as
English
DLMF:14.3.E16
No description defined

    Statements

    Defining formula
    cos ⁡ ( ν ⁢ π ) ⁢ P ν - μ ⁡ ( x ) = 2 ν ⁢ π 1 / 2 ⁢ x ν - μ ⁢ ( x 2 - 1 ) μ / 2 Γ ⁡ ( ν + μ + 1 ) ⁢ 𝐅 ⁡ ( 1 2 ⁢ μ - 1 2 ⁢ ν , 1 2 ⁢ μ - 1 2 ⁢ ν + 1 2 ; 1 2 - ν ; 1 x 2 ) - π 1 / 2 ⁢ ( x 2 - 1 ) μ / 2 2 ν + 1 ⁢ Γ ⁡ ( μ - ν ) ⁢ x ν + μ + 1 ⁢ 𝐅 ⁡ ( 1 2 ⁢ ν + 1 2 ⁢ μ + 1 , 1 2 ⁢ ν + 1 2 ⁢ μ + 1 2 ; ν + 3 2 ; 1 x 2 ) , 𝜈 𝜋 Legendre-P-first-kind 𝜇 𝜈 𝑥 superscript 2 𝜈 superscript 𝜋 1 2 superscript 𝑥 𝜈 𝜇 superscript superscript 𝑥 2 1 𝜇 2 Euler-Gamma 𝜈 𝜇 1 scaled-hypergeometric-bold-F 1 2 𝜇 1 2 𝜈 1 2 𝜇 1 2 𝜈 1 2 1 2 𝜈 1 superscript 𝑥 2 superscript 𝜋 1 2 superscript superscript 𝑥 2 1 𝜇 2 superscript 2 𝜈 1 Euler-Gamma 𝜇 𝜈 superscript 𝑥 𝜈 𝜇 1 scaled-hypergeometric-bold-F 1 2 𝜈 1 2 𝜇 1 1 2 𝜈 1 2 𝜇 1 2 𝜈 3 2 1 superscript 𝑥 2 {\displaystyle{\displaystyle\cos\left(\nu\pi\right)P^{-\mu}_{\nu}\left(x\right% )=\frac{2^{\nu}\pi^{1/2}x^{\nu-\mu}\left(x^{2}-1\right)^{\mu/2}}{\Gamma\left(% \nu+\mu+1\right)}\mathbf{F}\left(\tfrac{1}{2}\mu-\tfrac{1}{2}\nu,\tfrac{1}{2}% \mu-\tfrac{1}{2}\nu+\tfrac{1}{2};\tfrac{1}{2}-\nu;\frac{1}{x^{2}}\right)-\frac% {\pi^{1/2}\left(x^{2}-1\right)^{\mu/2}}{2^{\nu+1}\Gamma\left(\mu-\nu\right)x^{% \nu+\mu+1}}\mathbf{F}\left(\tfrac{1}{2}\nu+\tfrac{1}{2}\mu+1,\tfrac{1}{2}\nu+% \tfrac{1}{2}\mu+\tfrac{1}{2};\nu+\tfrac{3}{2};\frac{1}{x^{2}}\right),}}
    0 references
    DLMF id
    DLMF:14.3.E16
    0 references
    Symbols used
    gamma function
    Defining formula
    Γ ⁡ ( z ) Euler-Gamma 𝑧 {\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}
    xml-id
    C5.S2.E1.m2aidec
    0 references
    Q11481
    Defining formula
    P ν μ ⁡ ( z ) Legendre-P-first-kind 𝜇 𝜈 𝑧 {\displaystyle{\displaystyle P^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{z}\right)}}
    xml-id
    C14.S21.SS1.p1.m1addec
    0 references
    the ratio of the circumference of a circle to its diameter
    Defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2aedec
    0 references
    cosine function
    Defining formula
    cos ⁡ z 𝑧 {\displaystyle{\displaystyle\cos\NVar{z}}}
    xml-id
    C4.S14.E2.m2acdec
    0 references
    $$={{}_{2}{\mathbf{F}}_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$$ Olver’s hypergeometric function
    Defining formula
    𝐅 ⁡ ( a , b ; c ; z ) scaled-hypergeometric-bold-F 𝑎 𝑏 𝑐 𝑧 {\displaystyle{\displaystyle\mathbf{F}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z% }\right)}}
    xml-id
    C15.S2.E2.m2aldec
    0 references
    Q11589
    Defining formula
    x 𝑥 {\displaystyle{\displaystyle x}}
    xml-id
    C14.S1.XMD1.m1odec
    0 references
    Q11591
    Defining formula
    μ 𝜇 {\displaystyle{\displaystyle\mu}}
    xml-id
    C14.S1.XMD7.m1kdec
    0 references
    Q11590
    Defining formula
    ν 𝜈 {\displaystyle{\displaystyle\nu}}
    xml-id
    C14.S1.XMD8.m1ndec
    0 references
     
        Retrieved from "https://drmf-beta.wmflabs.org/index.php?title=Item:Q4705&oldid=30410"