[tgtdtucz] Fourier series for ellipse in polar coordinates

below is the formula for an ellipse in polar coordinates, with eccentricity e and one focus at the origin (multiply by scaling factor to embiggen):

r = 1 / (1 - e*cos(t))

the formula has cosine in denominator.  we compute its Fourier transform to get a formula with the sines and cosines only in the numerators.  (only cos is needed.)

Mathematica:

FourierCosSeries[1/(1-e*Cos[t]), t, 7, Assumptions-> e < 1 && 0 <= e] //Timing

{245.35853, 1/Sqrt[1 - e^2] + (2*(-1 + 1/Sqrt[1 - e^2])*Cos[t])/e - (2*(-2 + e^2 + 2*Sqrt[1 - e^2])*Cos[2*t])/(e^2*Sqrt[1 - e^2]) + (2*(-4 + 4/Sqrt[1 - e^2] + e^2*(1 - 3/Sqrt[1 - e^2]))*Cos[3*t])/e^3 + (2*(8 + e^4 - 8*Sqrt[1 - e^2] + 4*e^2*(-2 + Sqrt[1 - e^2]))*Cos[4*t])/(e^4*Sqrt[1 - e^2]) - (2*(16 - 16/Sqrt[1 - e^2] + e^4*(1 - 5/Sqrt[1 - e^2]) + e^2*(-12 + 20/Sqrt[1 - e^2]))*Cos[5*t])/e^5 - (2*(e^6 + e^2*(48 - 32*Sqrt[1 - e^2]) + 6*e^4*(-3 + Sqrt[1 - e^2]) + 32*(-1 + Sqrt[1 - e^2]))*Cos[6*t])/(e^6*Sqrt[1 - e^2]) + (2*(e^6*(-7 + Sqrt[1 - e^2]) - 64*(-1 + Sqrt[1 - e^2]) - 8*e^4*(-7 + 3*Sqrt[1 - e^2]) + 16*e^2*(-7 + 5*Sqrt[1 - e^2]))*Cos[7*t])/(e^7*Sqrt[1 - e^2])}

what is a general formula for the coefficients (to get an infinite series)?  this has almost certainly already been solved, but I could not find such a formula published anywhere.  it seems like the kind of thing Joseph Fourier might have done himself.

investigate Fourier transforms of generalizations of the original formula with cosine in the denominator, for example, r(theta) = 1/(sum(n, A[n] * cos(n*theta + B[n]))).  what is the Fourier transform of the reciprocal of a Fourier transform?  previously, polar plots of regular (not reciprocal) Fourier series.  under what conditions is the shape convex?