# 合流超几何函数

• Kummer 函数第一类合流超几何函数M(a,b,z) 是 Kummer 方程的解。注意有另一个相异且无关的函数也被称为 Kummer 函数；
• Tricomi 函数第二类合流超几何函数U(a,b,z)是 Kummer 方程的另一个线性无关的解，有时会写成 Ψ(a,b,z)；
• 惠泰克函数 是惠泰克方程的解，惠泰克方程里的参数与 Kummer 方程的参数所对应的李代数参数相关[注 1]

## Kummer 方程

${\displaystyle z\left(z{\frac {\rm {d}}{{\rm {d}}z}}+a\right)w=z{\frac {\rm {d}}{{\rm {d}}z}}\left(z{\frac {\rm {d}}{{\rm {d}}z}}+b-1\right)w}$ .

${\displaystyle z{\frac {\mathrm {d} ^{2}w}{\mathrm {d} z^{2}}}+(b-z){\frac {\mathrm {d} w}{\mathrm {d} z}}-aw=0}$ ,

${\displaystyle M(a,b,z)=\,{}_{1}F_{1}(a;b;z)=\sum _{n=0}^{\infty }{\frac {(a)^{(n)}}{(b)^{(n)}}}{\frac {z^{n}}{n!}}}$

Kummer 函数是高斯超几何函数的极限情形[1]

${\displaystyle {}_{1}F_{1}(a;c;z)=\lim _{b\rightarrow \infty }\,{}_{2}F_{1}(a,b;c;{\tfrac {z}{b}})}$

${\displaystyle z^{1-c}\,_{2}F_{1}(1+a-c,1+b-c;2-c;z)}$

${\displaystyle z^{1-b}\,_{1}F_{1}(1+a-b;2-b;z)}$

${\displaystyle U(a,b,z)={\frac {\Gamma (1-b)}{\Gamma (a-b+1)}}M(a,b,z)+{\frac {\Gamma (b-1)}{\Gamma (a)}}z^{1-b}M(a-b+1,2-b,z).}$

${\displaystyle U(a,b,z)=z^{-a}\cdot {}_{2}F_{0}(a,a-b+1;;-z^{-1})}$

${\displaystyle U(a,b,z)\approx z^{-a}\cdot {}_{2}F_{0}(a,a-b+1;;-z^{-1}),\quad z\rightarrow \infty ,|\arg z|<{\frac {3\pi }{2}}}$

Kummer 函数是整函数，而 Tricomi 函数一般有奇点 0。

### 可转化为 Kummer 方程的二阶线性常微分方程

${\displaystyle (A+Bz){\frac {d^{2}w}{dz^{2}}}+(C+Dz){\frac {dw}{dz}}+(E+Fz)w=0}$

${\displaystyle z{\frac {d^{2}w}{dz^{2}}}+(C+Dz){\frac {dw}{dz}}+(E+Fz)w=0}$

${\displaystyle z{\frac {d^{2}w}{dz^{2}}}+\left(C+{\frac {D}{\sqrt {D^{2}-4F}}}z\right){\frac {dw}{dz}}+\left({\frac {E}{\sqrt {D^{2}-4F}}}+{\frac {F}{D^{2}-4F}}z\right)w=0}$

${\displaystyle w(z)=\exp[-(1+{\tfrac {D}{\sqrt {D^{2}-4F}}}){\tfrac {z}{2}}]f(z),\quad f(z)=k_{1}M(a,C,z)+k_{2}U(a,C,z),\quad a=(1+{\tfrac {D}{\sqrt {D^{2}-4F}}}){\tfrac {C}{2}}-{\tfrac {E}{\sqrt {D^{2}-4F}}},k_{1},k_{2}\in \mathbb {C} }$

### 李代数参数与惠泰克方程

Kummer 方程的李代数参数[注 1][3]定义为

${\displaystyle \alpha =b-1,\theta =2a-b,}$

${\displaystyle F_{\alpha ,\theta }(z){\text{ and }}z^{-\alpha }F_{-\alpha ,\theta }(z)}$

${\displaystyle {\frac {\mathrm {d} ^{2}w}{\mathrm {d} z^{2}}}+\left(-{\frac {1}{4}}+{\frac {\kappa }{z}}+{\frac {1/4-\mu ^{2}}{z^{2}}}\right)w=0.}$

${\displaystyle M_{\kappa ,\mu }\left(z\right)=\exp \left(-z/2\right)z^{\mu +{\tfrac {1}{2}}}M\left(\mu -\kappa +{\frac {1}{2}},1+2\mu ;z\right)}$
${\displaystyle W_{\kappa ,\mu }\left(z\right)=\exp \left(-z/2\right)z^{\mu +{\tfrac {1}{2}}}U\left(\mu -\kappa +{\frac {1}{2}},1+2\mu ;z\right)}$

${\displaystyle M_{\kappa ,\mu }\left(z\right)=\exp \left(-z/2\right)z^{\mu +{\tfrac {1}{2}}}F_{2\mu ,-2\kappa }(z)}$

## 积分表示

${\displaystyle \mathrm {B} (a,c-a)\,{}_{2}F_{1}(a,b;c;{\tfrac {z}{b}})=\int _{1}^{\infty }t^{b-c}(t-1)^{c-a-1}(t-{\tfrac {z}{b}})^{-b}\mathrm {d} t=\int _{1}^{\infty }t^{-c}(t-1)^{c-a-1}(1-{\tfrac {z}{bt}})^{-b}\mathrm {d} t,\Re (c)>\Re (a)>0,|\arg(1-{\tfrac {z}{b}})|<\pi }$

${\displaystyle \mathrm {B} (a,c-a)\,{}_{1}F_{1}(a;c;z)=\int _{1}^{\infty }t^{-c}(t-1)^{c-a-1}e^{\tfrac {z}{t}}\mathrm {d} t,\Re (c)>\Re (a)>0}$

${\displaystyle \Gamma (a)U(a,b,z)=\int _{0}^{\infty }e^{-zt}t^{a-1}(1+t)^{b-a-1}\,dt,\quad \Re (a)>0}$

## 变换公式

${\displaystyle {}_{2}F_{1}(a,b;c;{\tfrac {z}{b}})=(1-{\tfrac {z}{b}})^{-b}\,{}_{2}F_{1}(c-a,b;c;{\tfrac {1}{b}}{\tfrac {bz}{z-b}}),\quad |\arg(1-{\tfrac {z}{b}})|<\pi }$

${\displaystyle {}_{1}F_{1}(a;c;z)=e^{z}\,{}_{1}F_{1}(c-a;c;-z)}$

${\displaystyle U(a,b,z)=z^{1-b}U\left(1+a-b,2-b,z\right)}$ .

## 特殊情形

### 柱函数

${\displaystyle I_{\nu }(z)={\frac {z^{\nu }}{2^{\nu }e^{z}\Gamma (\nu +1)}}M(\nu +{\frac {1}{2}},2\nu +1,z)}$
${\displaystyle K_{\nu }(z)={\sqrt {\pi }}(2z)^{\nu }e^{-z}U(\nu +{\frac {1}{2}},2\nu +1,z)}$

### Γ, 误差函数

${\displaystyle \gamma (a,z)={\frac {z^{a}}{a}}M(a,a+1,-z),\quad a\notin \mathbb {Z} _{0}^{-}}$
${\displaystyle \Gamma (a,z)=e^{-z}U(1-a,1-a,z)}$

${\displaystyle \operatorname {erf} (z)={\frac {2z}{\sqrt {\pi }}}M({\frac {1}{2}},{\frac {3}{2}},-z^{2})}$

### 正交多项式及相关函数

${\displaystyle L_{n}^{(\alpha )}(z)={n+\alpha \choose n}M(-n,\alpha +1,z),\quad \alpha \notin \mathbb {Z} ^{-}}$

（物理学上的）厄米多项式可以表示为[1]

${\displaystyle H_{n}(z)=2^{n}U(-{\frac {n}{2}},{\frac {1}{2}},z^{2}),\quad n\in \mathbb {Z} _{0}^{+},\Re (z)>0}$

## 注

1. 关于李代数参数，详见超几何函数

## 参考文献

1. 吴崇试. 17. 数学物理方法（第二版）. 北京大学出版社. [2003]. ISBN 9787301068199.
2. Daalhuis, Adri B. Olde, 合流超几何函数, Olver, Frank W. J.; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W. (编), NIST Handbook of Mathematical Functions, Cambridge University Press, 2010, ISBN 978-0521192255, MR2723248
3. Dereziński, Jan. Hypergeometric type functions and their symmetries. arXiv:1305.3113.