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 ∞ t α − 1 exp ( − t − b t ) d t , 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 α . For integer values of α we derive the corresponding limit case. Moreover, we discuss numerical techniques to evaluate this function.