Chapter 11.7, Problem 14AP

Consider the hypergeometric equation

$x\left(1-x\right)y^{″}+\left[c-\left(a-b+1\right)x\right]y^{\prime }-aby=0$, … $\left(\mathrm{11.7.8}\right)$

where $a$, $b$, $c$ are constants.

(a) Verify that $x=0$ is a regular singular point of $\left(\mathrm{11.7.8}\right)$.

(b) Verify that the roots of the indicial equation are $r_1=0$, $r_2=1-c$.

(c) Show that the coefficients in a series solution to Equation $\left(\mathrm{11.7.8}\right)$ centered at $x=0$ must satisfy the recurrence relation:

$a_{n+1}=\frac{\left(n+r+a\right)\left(n+r+b\right)}{\left(n+r+1\right)\left(n+r+c\right)}{a}_n$,

$n=0,1,\dots$.

(d) Assuming that $c$ is not an integer, derive the following solutions to $\left(\mathrm{11.7.8}\right)$:

$\begin{array}{l}{y}_1\left(x\right)=F\left(a,b,c;x\right)=1+\frac{ab}{c}x+\frac{a\left(a+1\right)b\left(b+1\right)}{c\left(c+1\right)}\frac{x^2}{2!}\\ +\frac{a\left(a+1\right)\dots \left(a+n-1\right)b\left(b+1\right)\dots \left(b+n-1\right)}{\left(c+1\right)\dots \left(c+n-1\right)}\frac{x^n}{n!}+\dots \end{array}$,

$y_2\left(x\right)=x^{1-c}F\left(a+1-c,b+1-c,2-c;x\right)$.