Are you actually referring to a coaxial waveguide with a center conductor (a very specialized waveguide item or also referring to some specialized fiber optics) -or- is this question in reference to air-dielectric coax in which the center conductor is offcenter?

The waveguide is a coaxial structure with outer and inner conductor and the dielectric happens to be teflon.  If the inner conductor is centered the impedance of the line is (138/sqrt(Er)) * log(D/d), where Er is dielectric constant, D is diameter of outer conductor and d is diameter is inner conductor.  If the inner conductor is moved away from the center, toward one side of the outer conductor, how is the impedance calculated?

In "Transmission Line Design Handbook" by Brian Wadell (Artech House), the formula for coax is given by Z= No/(2*pi*sqrt(er)) * ln(D/d) where No is free space (377 Ohms) and er is dielectric. Your formula is another approximation for this without using the natural log.

In this book, the Formula for eccentric round coaxial line is also given - Hummmm - well - ??? It would take a little time to put the explaniations here without a diagram.

A quick Google search for "Eccentric Round Coaxial Line" got a few hits which looked somewhat like the formula.

I am interested in your application for such a product.  

What you describe isn't a waveguide in the truest sense.  Waveguide is a balanced feed device and coaxial cable is unbalanced.  (remember, waveguide acts like twin-lead with an infinite number of ?/4 stubs connecting the two conductors...Oh, to be able to draw a picture.)

I remain,

The Old Soldering Gunslinger