# Kummer equation

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##### 2: 13.3 Recurrence Relations and Derivatives
13.3.7 $U\left(a-1,b,z\right)+(b-2a-z)U\left(a,b,z\right)+a(a-b+1)U\left(a+1,b,z\right% )=0,$
13.3.9 $U\left(a,b,z\right)-aU\left(a+1,b,z\right)-U\left(a,b-1,z\right)=0,$
Kummer’s differential equation (13.2.1) is equivalent to …
13.3.14 $(a+1)zU\left(a+2,b+2,z\right)+(z-b)U\left(a+1,b+1,z\right)-U\left(a,b,z\right)% =0.$
##### 3: 13.14 Definitions and Basic Properties
###### Whittaker’s Equation
This equation is obtained from Kummer’s equation (13.2.1) via the substitutions $W=e^{-\frac{1}{2}z}z^{\frac{1}{2}+\mu}w$, $\kappa=\tfrac{1}{2}b-a$, and $\mu=\tfrac{1}{2}b-\tfrac{1}{2}$. … Standard solutions are: …
13.14.5 $U\left(a,b,z\right)=e^{\frac{1}{2}z}z^{-\frac{1}{2}b}W_{\frac{1}{2}b-a,\frac{1% }{2}b-\frac{1}{2}}\left(z\right).$
The principal branches correspond to the principal branches of the functions $z^{\frac{1}{2}+\mu}$ and $U\left(\tfrac{1}{2}+\mu-\kappa,1+2\mu,z\right)$ on the right-hand sides of the equations (13.14.2) and (13.14.3); compare §4.2(i). …
##### 4: 13.6 Relations to Other Functions
13.6.2 $M\left(1,2,2z\right)=\frac{e^{z}}{z}\sinh z,$
13.6.3 $M\left(0,b,z\right)=U\left(0,b,z\right)=1,$
13.6.18 $U\left(\tfrac{1}{2}-\tfrac{1}{2}n,\tfrac{3}{2},z^{2}\right)=2^{-n}z^{-1}H_{n}% \left(z\right).$
##### 5: 13.12 Products
13.12.1 $M\left(a,b,z\right)M\left(-a,-b,-z\right)+\frac{a(a-b)z^{2}}{b^{2}(1-b^{2})}M% \left(1+a,2+b,z\right)M\left(1-a,2-b,-z\right)=1.$
##### 6: 13.4 Integral Representations
13.4.2 ${\mathbf{M}}\left(a,b,z\right)=\frac{1}{\Gamma\left(b-c\right)}\int_{0}^{1}{% \mathbf{M}}\left(a,c,zt\right)t^{c-1}(1-t)^{b-c-1}\,\mathrm{d}t,$ $\Re b>\Re c>0$,
13.4.3 ${\mathbf{M}}\left(a,b,-z\right)=\frac{z^{\frac{1}{2}-\frac{1}{2}b}}{\Gamma% \left(a\right)}\int_{0}^{\infty}e^{-t}t^{a-\frac{1}{2}b-\frac{1}{2}}J_{b-1}% \left(2\sqrt{zt}\right)\,\mathrm{d}t,$ $\Re a>0$.
13.4.4 $U\left(a,b,z\right)=\frac{1}{\Gamma\left(a\right)}\int_{0}^{\infty}e^{-zt}t^{a% -1}(1+t)^{b-a-1}\,\mathrm{d}t,$ $\Re a>0$, $|\operatorname{ph}{z}|<\frac{1}{2}\pi$,
13.4.13 ${\mathbf{M}}\left(a,b,z\right)=\frac{z^{1-b}}{2\pi\mathrm{i}}\int_{-\infty}^{(% 0+,1+)}e^{zt}t^{-b}\!\left(1-\frac{1}{t}\right)^{-a}\,\mathrm{d}t,$ $|\operatorname{ph}{z}|<\frac{1}{2}\pi$.
##### 7: 13.11 Series
13.11.1 $M\left(a,b,z\right)=\Gamma\left(a-\tfrac{1}{2}\right)e^{\frac{1}{2}z}\left(% \tfrac{1}{4}z\right)^{\frac{1}{2}-a}\*\sum_{s=0}^{\infty}\frac{{\left(2a-1% \right)_{s}}{\left(2a-b\right)_{s}}}{{\left(b\right)_{s}}s!}\*\left(a-\tfrac{1% }{2}+s\right)\*I_{a-\frac{1}{2}+s}\left(\tfrac{1}{2}z\right),$ $a+\frac{1}{2},b\neq 0,-1,-2,\dots$,
13.11.2 $M\left(a,b,z\right)=\Gamma\left(b-a-\tfrac{1}{2}\right){\mathrm{e}}^{\frac{1}{% 2}z}\left(\tfrac{1}{4}z\right)^{a-b+\frac{1}{2}}\sum_{s=0}^{\infty}(-1)^{s}% \frac{{\left(2b-2a-1\right)_{s}}{\left(b-2a\right)_{s}}(b-a-\frac{1}{2}+s)}{{% \left(b\right)_{s}}s!}I_{b-a-\frac{1}{2}+s}\left(\tfrac{1}{2}z\right),$ $b-a+\frac{1}{2},b\neq 0,-1,-2,\dots$.
13.11.3 ${\mathbf{M}}\left(a,b,z\right)={\mathrm{e}}^{\frac{1}{2}z}\sum_{s=0}^{\infty}A% _{s}\left(b-2a\right)^{\frac{1}{2}(1-b-s)}\left(\tfrac{1}{2}z\right)^{\frac{1}% {2}(1-b+s)}J_{b-1+s}\left(\sqrt{2z(b-2a)}\right),$
##### 8: 13.8 Asymptotic Approximations for Large Parameters
13.8.5 $U\left(a,b,z\right)\sim b^{-\frac{1}{2}a}e^{\frac{1}{4}\zeta^{2}b}\left(% \lambda\left(\frac{\lambda-1}{\zeta}\right)^{a-1}U\left(a-\tfrac{1}{2},\zeta% \sqrt{b}\right)-\left(\lambda\left(\frac{\lambda-1}{\zeta}\right)^{a-1}-\left(% \frac{\zeta}{\lambda-1}\right)^{a}\right)\frac{U\left(a-\tfrac{3}{2},\zeta% \sqrt{b}\right)}{\zeta\sqrt{b}}\right)$
13.8.6 $M\left(a,b,b\right)=\sqrt{\pi}\left(\frac{b}{2}\right)^{\frac{1}{2}a}\left(% \frac{1}{\Gamma\left(\frac{1}{2}(a+1)\right)}+\frac{(a+1)\sqrt{8/b}}{3\Gamma% \left(\frac{1}{2}a\right)}+O\left(\frac{1}{b}\right)\right),$
13.8.7 $U\left(a,b,b\right)=\sqrt{\pi}\left(2b\right)^{-\frac{1}{2}a}\left(\frac{1}{% \Gamma\left(\frac{1}{2}(a+1)\right)}-\frac{(a+1)\sqrt{8/b}}{3\Gamma\left(\frac% {1}{2}a\right)}+O\left(\frac{1}{b}\right)\right).$
13.8.17 $M\left(a,b,z\right)={\mathrm{e}}^{\nu z}\frac{\Gamma^{*}(b)}{\Gamma^{*}(a)}% \left(1+\frac{(1-\nu)(1+6\nu^{2}z^{2})}{12a}+O\left(\frac{1}{\min(a^{2},b^{2})% }\right)\right),$
13.8.18 $U\left(a,b+1,z\right)=z^{-b}{\mathrm{e}}^{(1-\nu)z}\frac{\Gamma\left(b\right)}% {\Gamma\left(a\right)}\left(1+\frac{\nu z(1-\nu)(2-\nu z)}{2a}+O\left(\frac{1}% {\min(a^{2},b^{2})}\right)\right),$ $\Re z>0$,
##### 9: 15.10 Hypergeometric Differential Equation
###### §15.10(ii) Kummer’s 24 Solutions and Connection Formulas
15.10.17 $w_{3}(z)=\frac{\Gamma\left(1-c\right)\Gamma\left(a+b-c+1\right)}{\Gamma\left(a% -c+1\right)\Gamma\left(b-c+1\right)}w_{1}(z)+\frac{\Gamma\left(c-1\right)% \Gamma\left(a+b-c+1\right)}{\Gamma\left(a\right)\Gamma\left(b\right)}w_{2}(z),$
15.10.18 $w_{4}(z)=\frac{\Gamma\left(1-c\right)\Gamma\left(c-a-b+1\right)}{\Gamma\left(1% -a\right)\Gamma\left(1-b\right)}w_{1}(z)+\frac{\Gamma\left(c-1\right)\Gamma% \left(c-a-b+1\right)}{\Gamma\left(c-a\right)\Gamma\left(c-b\right)}w_{2}(z),$
15.10.21 $w_{1}(z)=\frac{\Gamma\left(c\right)\Gamma\left(c-a-b\right)}{\Gamma\left(c-a% \right)\Gamma\left(c-b\right)}w_{3}(z)+\frac{\Gamma\left(c\right)\Gamma\left(a% +b-c\right)}{\Gamma\left(a\right)\Gamma\left(b\right)}w_{4}(z),$
15.10.25 $w_{1}(z)=\frac{\Gamma\left(c\right)\Gamma\left(b-a\right)}{\Gamma\left(b\right% )\Gamma\left(c-a\right)}w_{5}(z)+\frac{\Gamma\left(c\right)\Gamma\left(a-b% \right)}{\Gamma\left(a\right)\Gamma\left(c-b\right)}w_{6}(z),$
##### 10: 13.29 Methods of Computation
13.29.6 $w(n)={\left(a\right)_{n}}U\left(n+a,b,z\right),$