Re: confluent hypergeometric function

[Karim Belabas <Karim.Belabas@math.u-bordeaux1.fr>]

* David Joyner [2006-10-29 15:15]:
> I have a question about hyperu(alpha,beta,x), documented in
> http://pari.math.u-bordeaux.fr/dochtml/html.stable/Transcendental_functions.html#hyperu
> 
> Is this the same as $_1F_1(alpha,beta,x)$? 

No, although there is a "simple" relation between the 2, and
_1F_1(a,b,z) and U(a,b,z) they are independant solutions to the same 2nd
order differential equation:

  z f'' + (b-z) f' - af = 0    [ ' = d/dz ]

See Abramowitz & Stegun, p 504. _1F_1 is Kummer's M function, and hyperu
is Kummer's U.

A few values explicit values are computed on p.512  (Ex 11 and 12, loc. cit.).
I just checked that they are consistent with hyperu's answers.

Cheers,

    K.B.
-- 
Karim Belabas                  Tel: (+33) (0)5 40 00 26 17
Universite Bordeaux 1          Fax: (+33) (0)5 40 00 69 50
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