Group Representations, Differential Equations, and Special Functions

\[\Gamma(x) = \int_0^{\infty} = t^{x - 1}\exp -t \mathrm{d}t \Rightarrow\Gamma(x + 1) = x\Gamma(x).\]

\[x^2 \frac{\mathrm{d}^2 y}{\mathrm{d}x^2} + x \frac{\mathrm{d}y}{\mathrm{d}x} + (x^2 - \nu^2)y = 0,\; x> 0.\]

\[J_\nu(x) = \sum_{m = 0}^{\infty} \frac{(-1)^m}{m!\Gamma(m + \nu + 1)} \left( \frac{x}{2} \right)^{2m + \nu}.\] When ν is a natural # ν = n = 0, 1, 2, …: \[J_n(x) = \sum_{m = 0}^{\infty} \frac{(-1)^m}{m!(m + n)!} \left( \frac{x}{2} \right)^{2m + n}.\]

• \(\nu\ne n\): \[Y_\nu(x) = \frac{J_\nu(x) \cos\nu\pi - J_{\nu}(x)}{\sin\nu\pi}.\]
• \(\nu = n\): \[Y_{\nu}(x)=\lim_{\alpha\rightarrow\nu} \frac{J_{\alpha}\cos\alpha\pi - J_{-\alpha}(x)}{\sin\alpha\pi}.\]

• For integer order \(\nu = n\), \(J_n)\) is often defined via a Laurent series for a generating function: \[\exp \left[ \frac{x}{2}\left( r - \frac{1}{r} \right) \right] = \sum_{n = -\infty}^{\infty}J_n(x)r^n.\]
• ∀ ν \[\frac{\mathrm{d}}{\mathrm{d}x}\left[x^{\nu}J_{\nu}(x)\right] = x^{\nu}J_{\nu-1}(x), \;\frac{\mathrm{d}}{\mathrm{d}x}\left[x^{-\nu}J_{\nu}(x)\right] = -x^{-\nu}J_{\nu+1}(x).\] Hence we have \[J^{\prime}_{\nu}(x) = \frac{1}{2}\left[ J_{\nu - 1}(x)-J_{\nu+1}(x) \right], \;J_{\nu-1}(x) + J_{\nu+1}(x) = \frac{2\nu}{x}J_{\nu}(x).\] \[\int x^{\nu+1}J_{\nu}(x)\mathrm{d}x = x^{\nu + 1}J_{\nu + 1}(x).\]