The Heun's differential equation is a second order linear differential equation of Fuchs type
$\frac{d^2 w}{dz^2} + (\frac{\gamma}{z}+\frac{\delta}{z-1}+\frac{\epsilon}{z-a})\frac{dw}{dz} + \frac{\alpha\beta z – q}{z(z-1)(z-a)} w = 0$. It has four regular singularities at $0,1,a,\infty$. I am searching for references of this equation, mainly for two things
(1) any equation with four regular singularities can be transformed into the form of Heun's
(2) solutions that are analytic at two singularities, say $0$ and $1$
It seems the canonical reference is the book "Heun's differential equations" edited by Ronveaux. Unfortunately this book is not available in local libraries or bookstores in my country, no electronic edition to buy online. I highly appreciate any suggestion of references that cover the two topics I mentioned above.
Best Answer
I would say "Second order differential equations - special functions and their classification" of Gerhard Kristensson is a good book covering the most important on Heun functions and its relation with Lame and Hypergeomtric equations.