I have previously found truncated power series representations for solutions to differential equations with verified solutions for some differential equations like the classical Bessel functions, Airy-Bairy et.c.
I know that also for example complex exponential bases are very popular for this. However what I have not yet managed to figure out is if reciprocal bases like $$\{x^{-1},x^{-2},x^{-3},\cdots,x^{-N}\}$$ or maybe fractional reciprocal bases $$\{x^{\alpha},x^{\alpha-1},x^{\alpha-2},x^{\alpha-3},\cdots,x^{\alpha-N}\}, \\\alpha \in ]0,1[$$
could contain solutions to such differential equations.
We can see that similarly as in positive integer exponent power series case differentiation will connect this basis in a chain $$D(x^\beta) = \beta x^{\beta-1}$$
Systematically and uniquely connecting every basis function to another within the basis.
Are there any families of interesting functions satisfying differential equations which would admit a representation like
$$f(x) = \sum_k c_k x^{\alpha-k}$$
?