DLMF:18.12.E2

MaRDI QIDQ6341

• Label: DLMF:18.12.E2

Digital Library of Mathematical Functions ID 18.12.E2

${\displaystyle{\displaystyle\left(\tfrac{1}{2}(1-x)z\right)^{-\frac{1}{2}\alpha}J_{\alpha}\left(\sqrt{2(1-x)z}\right)\*\left(\tfrac{1}{2}(1+x)z\right)^{-\frac{1}{2}\beta}I_{\beta}\left(\sqrt{2(1+x)z}\right)=\sum_{n=0}^{\infty}\frac{P^{(\alpha,\beta)}_{n}\left(x\right)}{\Gamma\left(n+\alpha+1\right)\Gamma\left(n+\beta+1\right)}z^{n}.}}$

Symbols List

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

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