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To prove the identity \( J_{n-1}(x) + J_{n+1}(x) = \frac{2n}{x} J_n(x) \) using the generating function \( J_n(x) \), we will proceed step-by-step. ### Step 1: Understanding the Generating Function of \( J_n(x) \) The generating function for the Bessel functions \( J_n(x) \) is given by: \[ \sum_{n=-\infty}^{\infty} J_n(x) t^n = \exp\left(\frac{x}{2} \left(t - \frac{1}{t}\right)\right) \] This function allows us to _____ _____ ______ __________ _________ __________ ___ ______.
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