Asymptotic expansions of Kummer hypergeometric functions for large values of the parameters

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 $b\mathrm{/}a\le 1$ and $b\mathrm{/}a\ge 1$, with special attention for the case $a\sim b$. We use a uniform method to handle all cases of these parameters.