Solving Hypergeometric D.E. w/ Heun Eq & Quadratic Accessory Parameter

[wayfarer]

Homework Statement

The Heun equation is a generalization of the hypergeometric D.E. to the case
of four regular singular points. With the singular points at z=0,1,a,and inft,
the Heun equation takes the form,
z(z-1)(z-a)w''+(c_1*z^2+c_2*z+c_3)w'+(c_4*z+c_5)w=0
(a) In terms of k_1, k_2, k_3, k_4 and a, calculate c_1, c_2, c_3, c_4 and e,
such that the Riemann P-function for this Heun equation is
w=P( 0 a 1 inft )
( 0 0 0 k_1 z)
( 1-k_3 1-k_4 1-e k_2 )
(b) In the special case when k_4=0 and the general solution is free of
logarithms at z=a, show that the Heun equation reduces to a hypergeometric
equation. Give the general solution in terms of hypergeometric functions.
(c) Obtain a quadratic equation for the accessory parameter c_5 in the
special case when k_4=-1 and the general solution is free of logarithms at
z=a. (In this case, z=a is an apparent singularity of the differential equation)

Homework Equations

Um i guess the indicial equation to find the exponents at the regular singular points. What i wasn't sure about was that from the equations, i found that k_1+k_2+k_3+k_4+e=1 necessarily.

The Attempt at a Solution

I've done part (a) using indicial equations to find exponents, and then equate.
Part B: From part a, i got using indicial equations that k_4=0 implies that c_1*a^2+c_2*a+c_3=0, so c_1*z^2+c_2*z+c_3 = c_1*(z-a)*(z-a') for some number a'. Then in the equation, if c_4*z+c_5 has a as a factor, i.e. c_5=-a*c_4, then we can divide through by z-a and get a hypergeometric D.E. So i need to prove that if the general solution is free of logarithms, then c_5=-a*c_4; I'm not quite sure how to do it rigorously. I tried doing it by using a power series about z=a, and proved that we reach a contradiction if there are no terms of negative power (i.e. (z-a)^(-t)) in the Laurent series, but I'm not sure how to justify that.
for part (c ), I'm clueless.

 

[mighty2000]

Hello, thank you for your post! The Heun equation is a very interesting and important topic in mathematics, and I would be happy to help you with your questions.

For part (b), you are on the right track. To show that the Heun equation reduces to a hypergeometric equation, we can use the fact that if k_4=0, then c_4*z+c_5=c_4*(z-a), and we can divide through by (z-a) to get a hypergeometric equation. To show that the general solution is free of logarithms, we can use the method of Frobenius to find the coefficients of the Laurent series about z=a, and then show that there are no terms of negative power in the series. This would imply that c_5=-a*c_4, and therefore the general solution is free of logarithms.

For part (c), we can use the fact that if k_4=-1, then c_4*z+c_5=c_4*(z-a)+c_5, and we can also use the method of Frobenius to find the coefficients of the Laurent series about z=a. Then, we can equate the coefficients of the (z-a)^(-2) term to get a quadratic equation for c_5. This would give us the value of c_5 in terms of the other parameters, and we can use this to find the general solution for this special case.

I hope this helps you with your homework. Good luck!