Overlapping power-of-cosine windows with constant weight
your result is interesting but you should take a look at :

F. J. Harris, "On the use of windows for harmonic analysis with the discrete Fourier transform," in Proceedings of the IEEE, vol. 66, no. 1, pp. 51-83, Jan. 1978.

doi: 10.1109/PROC.1978.10837

Abstract: This paper makes available a concise review of data windows and their affect on the detection of harmonic signals in the presence of broad-band noise, and in the presence of nearby strong harmonic interference. We also call attention to a number of common errors in the application of windows when used with the fast Fourier transform. This paper includes a comprehensive catalog of data windows along with their significant performance parameters from which the different windows can be compared. Finally, an example demonstrates the use and value of windows to resolve closely spaced harmonic signals characterized by large differences in amplitude.
keywords: {Discrete Fourier transforms;Fourier transforms;Frequency;Harmonic analysis;Oceans;Parameter estimation;Signal processing;Signal resolution;Signal sampling;Smoothing methods},
URL: http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&amp;arnumber=1455106&amp;isnumber=31261

There a number of Figures of Merits (FOMs) that are discussed for windows including $cos^k(),\; 
 k=\{1,2,3,4\}$ but not at the generalized overlaps you show.  Looking at the main table in Harris's paper, as $k$ increases, many of the FOMs like the ENBW get worse.  Also while the constant energy is preserved for your result, the coherent gain of the DFT decreases.  The only FOM that touches on overlap is Overlap Correlation. 

If you may permit me being blunt, but $cos^{20}$ as windows go, doesn't seem to offer much advantage in general terms. I think that your result is interesting and someone might publish it, but you might hold off going down to the Porsche dealership.