$$\ \int_0^{\pi} \sin^{2n-2k+1} e^{a\cos x} dx =\sqrt{\pi }\, 2^{-k+n+\frac{1}{2}} a^{\frac{1}{2} (2 k-2 n-1)} \Gamma (-k+n+1) I_{\frac{1}{2} (-2 k+2 n+1)}(a),\;\;n-k+1>0.$$
Carlo Beenakker
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