DLMF:18.12.E2 (Q5644): Difference between revisions

From DRMF
Jump to navigation Jump to search
← Older edit
‎Changed an Item: Add constraint
‎Changed an Item: Add constraint
 
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.m1adec

Latest revision as of 15:30, 2 January 2020

No description defined
Language Label Description Also known as
English
DLMF:18.12.E2
No description defined

    Statements

    Defining formula
    ( 1 2 ⁢ ( 1 - x ) ⁢ z ) - 1 2 ⁢ α ⁢ J α ⁡ ( 2 ⁢ ( 1 - x ) ⁢ z ) ⁢ ( 1 2 ⁢ ( 1 + x ) ⁢ z ) - 1 2 ⁢ β ⁢ I β ⁡ ( 2 ⁢ ( 1 + x ) ⁢ z ) = ∑ n = 0 ∞ P n ( α , β ) ⁡ ( x ) Γ ⁡ ( n + α + 1 ) ⁢ Γ ⁡ ( n + β + 1 ) ⁢ z n . superscript 1 2 1 𝑥 𝑧 1 2 𝛼 Bessel-J 𝛼 2 1 𝑥 𝑧 superscript 1 2 1 𝑥 𝑧 1 2 𝛽 modified-Bessel-first-kind 𝛽 2 1 𝑥 𝑧 superscript subscript 𝑛 0 Jacobi-polynomial-P 𝛼 𝛽 𝑛 𝑥 Euler-Gamma 𝑛 𝛼 1 Euler-Gamma 𝑛 𝛽 1 superscript 𝑧 𝑛 {\displaystyle{\displaystyle\left(\tfrac{1}{2}(1-x)z\right)^{-\frac{1}{2}% \alpha}J_{\alpha}\left(\sqrt{2(1-x)z}\right)\*\left(\tfrac{1}{2}(1+x)z\right)^% {-\frac{1}{2}\beta}I_{\beta}\left(\sqrt{2(1+x)z}\right)=\sum_{n=0}^{\infty}% \frac{P^{(\alpha,\beta)}_{n}\left(x\right)}{\Gamma\left(n+\alpha+1\right)% \Gamma\left(n+\beta+1\right)}z^{n}.}}
    0 references
    DLMF id
    DLMF:18.12.E2
    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
    modified Bessel function of the first kind
    Defining formula
    I ν ⁡ ( z ) modified-Bessel-first-kind 𝜈 𝑧 {\displaystyle{\displaystyle I_{\NVar{\nu}}\left(\NVar{z}\right)}}
    xml-id
    C10.S25.E2.m2adec
    0 references
    Q11750
    Defining formula
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C18.S1.XMD2.m1adec
    0 references
    Q11726
    Defining formula
    n 𝑛 {\displaystyle{\displaystyle n}}
    xml-id
    C18.S1.XMD6.m1adec
    0 references
    Q11727
    Defining formula
    x 𝑥 {\displaystyle{\displaystyle x}}
    xml-id
    C18.S2.XMD3.m1adec
    0 references
     
        Retrieved from "https://drmf-beta.wmflabs.org/index.php?title=Item:Q5644&oldid=58413"