The Generalized Incomplete Gamma Function as sum over Modified Bessel Functions of the First Kind

We represent the Generalized Incomplete Gamma Function Γ(α,x;b)=∫x∞tα−1exp(−t−bt)dt,α∈R,x≥0,b≥0,but not both x=b=0, if α≤0, as a sum of Modified Bessel Functions valid for non-integer αα. For integer values of αα we derive the corresponding limit case. Moreover, we discuss numerical techniques to evaluate this function.