Notes on Whittaker & Watson, Chapter XIV, part 1

Whittaker & Watson, A Course of Modern Analysis. Chapter XIV, The Hypergeometric Function.

14.1 The hypergeometric series $$ 1+\frac{a\cdot b}{1\cdot c}z + \frac{a(a+1)b(b+1)}{1\cdot2\cdot c(c+1)}z^2 + \frac{a(a+1)(a+2)b(b+1)(b+2)}{1\cdot2\cdot3\cdot c(c+1)(c+2)}z^3+\cdots $$ defines an analytic function $F(a,b;c;z)$ for $|z|<1$.

Some examples: $$ \begin{gather*} (1+z)^n=F(-n,\beta;\beta;-z),\newline \log(1+z)=zF(1,1;2;-z),\newline e^z=\lim_{\beta\to\infty}F(1,\beta;1;z/\beta). \end{gather*} $$ 14.11 The values of $F(a,b;c;1)$ when $\Re(c-a-b)>0$.

Consider the series $$ \begin{align*} & c\big(c-1-(2c-a-b-1)x\big)F(a,b;c;x) + (c-a)(c-b)xF(a,b;c+1;x) \newline ={}& c(c-1)(1-x)F(a,b;c-1;x) \newline ={}& c(c-1)\left(1+\sum_{n=1}^\infty(u_n-u_{n-1})x^n\right) \qquad\qquad F(a,b;c-1;x)=\sum_{n=0}^{\infty}u_nx^n \end{align*} $$ in which, the RHS tends to 0 as $x\to1$. Thus $$ \begin{align*} F(a,b;c;1) & =\frac{(c-a)(c-b)}{c(c-a-b)}F(a,b;c+1;1)\newline & = \prod_{n=0}^{m-1}\frac{(c-a+n)(c-b+n)}{(c+n)(c-a-b+n)}F(a,b;c+m;1)\newline & = \left(\lim_{m\to\infty}\prod_{n=0}^{m-1}\frac{(c-a+n)(c-b+n)}{(c+n)(c-a-b+n)}\right)\lim_{m\to\infty}F(a,b;c+m;1). \end{align*} $$ The first limit tends to $\frac{\Gamma(c)\Gamma(c-a-b)}{\Gamma(c-a)\Gamma(c-b)}$. The second limit tends to 1 since for $m>|c|+|a|+|b|-1$ and $m>|c|$ $$ \begin{align*} |F(a,b;c+m;1)-1| & \le\sum_{n=1}^{\infty}u_n(|a|, |b|;m-|c|)\newline & <\frac{|ab|}{m-|c|}\sum_{n=1}^{\infty}u_n(|a|+1, |b|+1;m-|c|+1)\to0 & m\to\infty. \end{align*} $$ Hence $$ F(a,b;c;1) =\frac{\Gamma(c)\Gamma(c-a-b)}{\Gamma(c-a)\Gamma(c-b)}. $$ 14.2 The differential equation satisfied by $F(a,b;c;z)$.

The hypergeometric equation $$ z(1-z)\frac{d^2u}{dz^2}+\big(c-(a+b+1)z\big)\frac{du}{dz}-abu=0 $$ which has regular singularities at $0,1,\infty$.

14.3 Riemann’s equation (see §10.7) $$ \begin{split} \frac{d^2u}{dz^2}+\left(\frac{1-\alpha-\alpha'}{z-a}+\frac{1-\beta-\beta'}{z-b}+\frac{1-\gamma-\gamma'}{z-c}\right)\frac{du}{dz}\newline + \left(\frac{\alpha\alpha'(a-c)(a-b)}{z-a}+\frac{\beta\beta'(b-a)(b-c)}{z-b}+\frac{\gamma\gamma'(c-a)(c-b)}{z-c}\right)\newline\times\frac{u}{(z-a)(z-b)(z-c)}=0 \end{split} $$ can be reduced to a hypergeometric equation with solution $$ \left(\frac{z-a}{z-b}\right)^\alpha\left(\frac{z-c}{z-b}\right)^\gamma F\left(\alpha+\beta+\gamma, \alpha+\beta'+\gamma; 1+\alpha-\alpha';\frac{(z-a)(c-b)}{(z-b)(c-a)}\right). $$ Under the transpositions $(\ \alpha\ \alpha'\ )$ and $(\ \gamma\ \gamma'\ )$, and the 6 permutations ($S_3$) on $\{a,b,c\}$, the equation remains invariant, resulting in total 24 expressions of solutions, which are due to Kummer.

14.4 Relations between particular solutions of the hypergeometric equation.

Being a second-order linear ODE, Riemann’s equation admits 2 linearly independent solutions. Therefore Kummer’s 24 solutions are not linearly independent. It can be shown that these 24 expressions can be grouped into 6 sets of 4, with one set (complex) scalar multiples of $$ P^{(\alpha)}=(z-a)^{\alpha}\left(1+\sum_{n=1}^\infty e_n(z-a)^n\right), $$ and other sets scalar multiples of $P^{(\alpha')}, P^{(\beta)}, P^{(\beta')}, P^{(\gamma)}, P^{(\gamma')}$ respectively.