# relation to Kummer equation

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##### 1: 13.14 Definitions and Basic Properties
###### Whittaker’s Equation
Standard solutions are: …
##### 2: 13.2 Definitions and Basic Properties
13.2.1 $z\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}z}^{2}}+(b-z)\frac{\mathrm{d}w}{\mathrm{d% }z}-aw=0.$
##### 4: 13.3 Recurrence Relations and Derivatives
###### §13.3(i) Recurrence Relations
Kummer’s differential equation (13.2.1) is equivalent to
13.3.14 $(a+1)zU\left(a+2,b+2,z\right)+(z-b)U\left(a+1,b+1,z\right)-U\left(a,b,z\right)% =0.$
##### 7: Errata
• Equations (10.22.37), (10.22.38), (14.17.6)–(14.17.9)

The Kronecker delta symbols have been moved furthest to the right, as is common convention for orthogonality relations.

• Equation (9.7.2)

Following a suggestion from James McTavish on 2017-04-06, the recurrence relation $u_{k}=\frac{(6k-5)(6k-3)(6k-1)}{(2k-1)216k}u_{k-1}$ was added to Equation (9.7.2).

• A number of additions and changes have been made to the metadata to reflect new and changed references as well as to how some equations have been derived.

• Equation (13.2.7)
13.2.7 $U\left(-m,b,z\right)=(-1)^{m}{\left(b\right)_{m}}M\left(-m,b,z\right)=(-1)^{m}% \sum_{s=0}^{m}\genfrac{(}{)}{0.0pt}{}{m}{s}{\left(b+s\right)_{m-s}}(-z)^{s}$

The equality $U\left(-m,b,z\right)=(-1)^{m}{\left(b\right)_{m}}M\left(-m,b,z\right)$ has been added to the original equation to express an explicit connection between the two standard solutions of Kummer’s equation. Note also that the notation $a=-n$ has been changed to $a=-m$.

Reported 2015-02-10 by Adri Olde Daalhuis.

• Equation (13.2.8)
13.2.8 $U\left(a,a+n+1,z\right)=\frac{(-1)^{n}{\left(1-a-n\right)_{n}}}{z^{a+n}}\*M% \left(-n,1-a-n,z\right)=z^{-a}\sum_{s=0}^{n}\genfrac{(}{)}{0.0pt}{}{n}{s}{% \left(a\right)_{s}}z^{-s}$

The equality $U\left(a,a+n+1,z\right)=\frac{(-1)^{n}{\left(1-a-n\right)_{n}}}{z^{a+n}}\*M% \left(-n,1-a-n,z\right)$ has been added to the original equation to express an explicit connection between the two standard solutions of Kummer’s equation.

Reported 2015-02-10 by Adri Olde Daalhuis.

• ##### 8: 13.29 Methods of Computation
The integral representations (13.4.1) and (13.4.4) can be used to compute the Kummer functions, and (13.16.1) and (13.16.5) for the Whittaker functions. …
###### §13.29(iv) Recurrence Relations
The recurrence relations in §§13.3(i) and 13.15(i) can be used to compute the confluent hypergeometric functions in an efficient way. … normalizing relationnormalizing relation
##### 9: 31.3 Basic Solutions
###### §31.3(iii) Equivalent Expressions
Each is related to the solution (31.3.1) by one of the automorphisms of §31.2(v). … The full set of 192 local solutions of (31.2.1), equivalent in 8 sets of 24, resembles Kummer’s set of 24 local solutions of the hypergeometric equation, which are equivalent in 4 sets of 6 solutions (§15.10(ii)); see Maier (2007).
##### 10: 7.11 Relations to Other Functions
###### Confluent Hypergeometric Functions
7.11.4 $\operatorname{erf}z=\frac{2z}{\sqrt{\pi}}M\left(\tfrac{1}{2},\tfrac{3}{2},-z^{% 2}\right)=\frac{2z}{\sqrt{\pi}}e^{-z^{2}}M\left(1,\tfrac{3}{2},z^{2}\right),$