The confluent hypergeometric functions M(a,b;z)M(a,b;z) and U(a,b;z)U(a,b;z) for large bb and zz

We obtain new and complete asymptotic expansions of the confluent hypergeometric functions M(a,b;z)M(a,b;z) and U(a,b;z)U(a,b;z) for large bb and zz. The expansions are different in the three different regions: z+a+1−b>0z+a+1−b>0, z+a+1−b<0z+a+1−b<0 and z+a+1−b=0z+a+1−b=0. The expansions are not of Poincaré type and we give explicit expressions for the terms of the expansions. In some cases, the expansions are valid for complex values of the variables too. We give numerical examples which show the accuracy of the expansions.