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Carlo Beenakker
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I have not found a proof of precisely this identity, but it follows quickly from the equations in Orthogonal polynomials on the unit circle associated with the Laguerre polynomials . The Fourier cosine and sine transforms of $\phi_m$ are equal to the real and imaginary part of $(1-z)z^m$, with $z=(2k-i)/(2k+i)$.

Carlo Beenakker
  • 188.2k
  • 18
  • 448
  • 651