  We derive asymptotic expansions of the Kummer functions $M(a,b,z)$ and
$U(a,b+1,z)$ for large positive values of $a$ and $b$, with $z$ fixed. For both
functions we consider $b/a\le 1$ and $b/a\ge 1$, with special attention for the
case $a\sim b$. We use a uniform method to handle all cases of these
parameters.
