Formula:6410

Digital Library of Mathematical Functions ID 18.15.E19

${\displaystyle{\displaystyle L^{(\alpha)}_{n}\left(\nu x\right)=\frac{e^{\frac{1}{2}\nu x}}{2^{\alpha}x^{\frac{1}{2}\alpha+\frac{1}{4}}(1-x)^{\frac{1}{4}}}\left(\xi^{\frac{1}{2}}J_{\alpha}\left(\nu\xi\right)\sum_{m=0}^{M-1}\frac{A_{m}(\xi)}{\nu^{2m}}+\xi^{-\frac{1}{2}}J_{\alpha+1}\left(\nu\xi\right)\sum_{m=0}^{M-1}\frac{B_{m}(\xi)}{\nu^{2m+1}}+\xi^{\frac{1}{2}}\mathrm{env}\mskip-2.0mu J_{\alpha}\left(\nu\xi\right)O\left(\frac{1}{\nu^{2M-1}}\right)\right),}}$

Symbols List

• ${\displaystyle{\displaystyle J_{\NVar{\nu}}\left(\NVar{z}\right)}}$: Bessel function of the first kind
• ${\displaystyle{\displaystyle O\left(\NVar{x}\right)}}$: order not exceeding
• ${\displaystyle{\displaystyle L^{(\NVar{\alpha})}_{\NVar{n}}\left(\NVar{x}\right)}}$: Laguerre (or generalized Laguerre) polynomial
• ${\displaystyle{\displaystyle\mathrm{env}\mskip-2.0mu J_{\NVar{\nu}}\left(\NVar{x}\right)}}$:
• ${\displaystyle{\displaystyle\mathrm{e}}}$: base of natural logarithm
• ${\displaystyle{\displaystyle m}}$: nonnegative integer
• ${\displaystyle{\displaystyle n}}$: nonnegative integer
• ${\displaystyle{\displaystyle\nu}}$: 18.15.E17
• ${\displaystyle{\displaystyle\xi}}$: 18.15.E18
• ${\displaystyle{\displaystyle A_{m}(\xi)}}$: coefficient
• ${\displaystyle{\displaystyle B_{m}(\xi)}}$: coefficient (locally)
• ${\displaystyle{\displaystyle x}}$: real variable
 

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