Asymptotics and closed form of a generalized incomplete gamma function

AbstractWe consider the asymptotic behavior of the function Γ(α,x;b) = ∫x∞ tα−1e−t−btdt, x > 0, α > 0, b ⩾ 0, as x tends to infinity. We give several expansions for the case that α and b are fixed and we give a uniform expansion in which α and b may range through unbounded intervals. We give a closed form for half-integer values of α

Computation of solutions to linear difference and differential equations with a prescribed asymptotic behavior

Linear differential equations usually arise from mathematical modeling of physical experiments and real-world problems. In most applications these equations are linked to initial or boundary conditions. But sometimes the solution under consideration is characterized by its asymptotic behavior, which leads to the question how to infer from the asymptotic growth of a solution to its initial values. In this paper we show that under some mild conditions the initial values of the desired solution can be computed by means of a continuous-time analogue of a modified matrix continued fraction. For numerical applications we develop forward and backward algorithms which behave well in most situations. The topic is closely related to the theory of special functions and its extension to higher-dimensional problems. Our investigations result in a powerful tool for solving some classical mathematical problems. To demonstrate the efficiency of our method we apply it to Poincaré type and Kneser’s differential equation