The Fast Fourier Transform Recursive Equations for Arbitrary Length Records

where Sr(a), S,(b), Sr'(a) and S,'(b) stand for the sum of all possible combinations of r factors taken at a time from the sequence of numbers q ql qa2 gar; bl b2 br 1-bl l-b2 1-ba 1-a2 I-ar q ,q ,*... yq ;q,q yq ,** ..,q and qll, q )2... qrrespectively,we can sum up both the series on the right-hand side of (4.6) and obtain (ar)Z =-M [1 qN?l][q(a,) N + 1] rT4r -q 1+(b,)j -N [z 11[ql1+(br); Nl (8) [1 qM?l][q (br); M + 2] q(M+2)[r+Z(br)-Z(a,) [qr+Z(tbr)-(ar) z] x [ql (ar); M + l]zM+l