Notes on Whittaker & Watson, Chapter XVI, part 1

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

16.1 The confluence of two singularities of Riemann’s equation.

Riemann’s $P$​​-function $$ P\begin{Bmatrix}0 & \infty & c\\ \frac12+m & -c & c-k & z\\ \frac12-m & 0 & k\end{Bmatrix} $$ in the limit $c\to\infty$ is the solution to the differential equation $$ \frac{d^2u}{dz^2}+\frac{du}{dz}+\left(\frac{k}{z}+\frac{\frac14-m^2}{z^2}\right)u=0. $$ Substituting $u=e^{-\frac12z}W_{k,m}$​​​, we have the equation for $W$ $$ \frac{d^2W}{dz^2}+\left(-\frac14+\frac{k}{z}+\frac{\frac14-m^2}{z^2}\right)W=0 $$ which has a regular singularity at $z=0$ and an irregular singularity at $z=\infty$. When $2m\notin\mathbb{Z}$, we two fundamental solutions $$ \begin{gather*} M_{k,m}(z)=z^{\frac12+m}e^{-\frac12z}\left(1+\frac{\frac12+m-k}{(2m+1)}z+\frac{(\frac12+m-k)(\frac32+m-k)}{2!(2m+1)(2m+2)}z^2+\cdots\right),\\ M_{k,-m}(z)=z^{\frac12-m}e^{-\frac12z}\left(1+\frac{\frac12-m-k}{(1-2m)}z+\frac{(\frac12-m-k)(\frac32-m-k)}{2!(1-2m)(2-2m)}z^2+\cdots\right). \end{gather*} $$ 16.11 Kummer’s formulae.

When $2m$​​​​​ is not a negative integer, we have Kummer’s first formula $$ \begin{gather*}z^{-\frac12-m}M_{k,m}(z)=(-z)^{-\frac12-m}M_{-k,m}(-z), \text{or} d\\ e^{-z}{}_1F_1\left(\frac12+m-k;2m+1;z\right) = {}_1F_1\left(\frac12+m+k;2m+1;-z\right) \end{gather*} $$ which can be obtained by expanding $e^{-z}$​ in power series $$ \begin{align*}&{}e^{-z}{}_1F_1\left(\frac12+m-k;2m+1;z\right) \\ = & \left(1+\sum_{n=1}^\infty\frac{(-z)^n}{n!}\right)\left(1+\sum_{n=1}^\infty\frac{(\frac12+m-k)\cdots(m-k+n-\frac12)}{n!(2m+1)\cdots(2m+n)}z^n\right) \\ =& 1+ \sum_{n=1}^\infty\frac{(-z)^n}{n!}\sum_{l=0}^n{n\choose l}\frac{(-1)^l(\frac12+m-k)\cdots(m-k+l-\frac12)}{(2m+1)\cdots(2m+l)} \\ = & 1+\sum_{n=1}^\infty\frac{(-z)^n}{n!}F\left(-n,\frac12+m-k;2m+1;1\right) \\ = & 1+\sum_{n=1}^\infty\frac{(-z)^n}{n!}\frac{\Gamma(2m+1)\Gamma(\frac12+m+k+n)}{\Gamma(2m+n+1)\Gamma(\frac12+m+k)} \\ = & {}_1F_1\left(\frac12+m+k;2m+1;-z\right). \end{align*} $$ Kummer’s second formula $$ \begin{gather*} M_{0,m}=z^{\frac12+m}\left(1+\sum_{p=1}^\infty\frac{z^{2p}}{2^{4p}p!(m+1)(m+2)\cdots(m+p)}\right), \text{or} \\ e^{-\frac12z}{}_1F_1\left(\frac12+m;2m+1;z\right) = {}_0F_1\left(m+1;\frac{z^2}{2^4}\right) \end{gather*} $$ which can be obtained by considering the coefficients of $z^{n+m+\frac12}$​ in the power series expansion $$ \begin{align*} &{}\frac{(\frac12+m)(\frac32+m)\cdots(n+m-\frac12)}{n!(2m+1)\cdots(2m+n)}F\left(-n,-2m-n;-n+\frac12-m,\frac12\right)\\ =&\frac{(\frac12+m)(\frac32+m)\cdots(n+m-\frac12)}{n!(2m+1)\cdots(2m+n)}F\left(-\frac n2,-m-\frac n2;-n+\frac12-m,1\right) & \href{https://mathworld.wolfram.com/KummersRelation.html}{\text{Kummer’s relation}}\\ =& \frac{(\frac12+m)(\frac32+m)\cdots(n+m-\frac12)}{n!(2m+1)\cdots(2m+n)}\frac{\Gamma(-n+\frac12-m)\Gamma(\frac12)}{\Gamma(\frac12-m-\frac n2)\Gamma(\frac12-\frac n2)} \\ =&\frac{(-1)^n\Gamma(\frac12-m)\Gamma(\frac12)}{n!(2m+1)\cdots(2m+n)\Gamma(\frac12-m-\frac n2)\Gamma(\frac12-\frac n2)} \end{align*} $$ which vanishes when $n$ is odd, and evaluates to the summand on the RHS when $n=2p$.

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16.12 Definition of the function $W_{k,m}(z)$​.

With $W_{k,m}=e^{\frac12z}u$​, and $u$​​ obtained by the contour solution to Riemann’s differential equation (§14.6), we have $$ W_{k,m}=-\frac{1}{2\pi i}\Gamma\left(k+\frac12-m\right)e^{-\frac12z}z^k\int_\infty^{(0+)}(-t)^{-k-\frac12+m}\left(1+\frac tz\right)^{k-\frac12+m}e^{-t}dt, $$ defined on a Hanekel’s contour (around the cut between $t=0$​​ and $\infty$​​) with $t=-z$​​​ is outside the contour, and $\arg(-t$​​​) chosen in $[-\pi,\pi]$​​​, and $\arg(1+t/z)\to0$​​ continuous as $t\to0$​​​. This defines the function except for when $k-\frac12+m$​​​ is negative integer. For $\Re(k-\frac12+m)\le0$​​​, we define $$ W_{k,m}=\frac{e^{-\frac12z}z^k}{\Gamma\left(\frac12-k+m\right)}\int_0^\infty t^{-k-\frac12+m}\left(1+\frac tz\right)^{k-\frac12+m}e^{-t}dt. $$