DLMF:18.11.E5

MaRDI QIDQ6336

• Label: DLMF:18.11.E5

Digital Library of Mathematical Functions ID 18.11.E5

${\displaystyle{\displaystyle\lim_{n\to\infty}\frac{1}{n^{\alpha}}P^{(\alpha,\beta)}_{n}\left(1-\frac{z^{2}}{2n^{2}}\right)=\lim_{n\to\infty}\frac{1}{n^{\alpha}}P^{(\alpha,\beta)}_{n}\left(\cos\frac{z}{n}\right)=\frac{2^{\alpha}}{z^{\alpha}}J_{\alpha}\left(z\right).}}$

Symbols List

• ${\displaystyle{\displaystyle J_{\NVar{\nu}}\left(\NVar{z}\right)}}$: Bessel function of the first kind
• ${\displaystyle{\displaystyle P^{(\NVar{\alpha},\NVar{\beta})}_{\NVar{n}}\left(\NVar{x}\right)}}$: Jacobi polynomial
• ${\displaystyle{\displaystyle\cos\NVar{z}}}$: cosine function
• ${\displaystyle{\displaystyle z}}$: complex variable
• ${\displaystyle{\displaystyle n}}$: nonnegative integer
 

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