How to obtain the symbolic expression for the integral?

I need to evaluate the following expression,

$$\frac{1}{\Gamma(1-\alpha)} \frac{d}{d x} \int_{0}^{x}(x-t)^{-\alpha}(\exp(\lambda t)) d t, 0<\alpha<1, \lambda\in \mathbb{C}$$

using Mathematica to obtain $$\sum_{k=0}^{\infty} \frac{\lambda^{k} x^{k-\alpha}}{\Gamma(k+1-\alpha)}$$.

So I used the following code

Assuming[
Re[x] > 0 && Im[x] == 0 && Re[- α] > -1,
1/Gamma[1 - α] D[Integrate[(x - t)^(-α) Exp[λ t], {t, 0, x}], x]
]


And obtain the following output

(λ^α (x λ)^-α + E^(x λ) λ^α (Gamma[1 - α] - Gamma[1 - α, x λ]))/Gamma[1 - α]


$$\sum_{k=0}^{\infty} \frac{\lambda^{k} x^{k-\alpha}}{\Gamma(k+1-\alpha)}$$

• A definite integral of a function $f(t)$ with respect to $t$ in the limits $0\le t\le x$ cannot depend on $t$ Oct 11, 2019 at 8:16
• if you do this to your result Map[FullSimplify, Series[%1, {x, 0, 5}, Assumptions -> x > 0] // Normal // Expand] you can check that it works Oct 13, 2019 at 15:52

I guess, you can take the n-th term of the expansion of the exponent (which is as you know, (l*t)^n/n! where I put l instead of lambda. Then integrate it:

Assuming[
n > 0 && n \[Element] Integers && Re[x] > 0 && Im[x] == 0 &&
Re[-\[Alpha]] > -1,
1/Gamma[1 - \[Alpha]] D[
Integrate[(x - t)^(-\[Alpha])*(l*t)^n/n!, {t, 0, x}], x]]

(*  (l^n x^(n - \[Alpha]) (1 + n - \[Alpha]))/Gamma[
2 + n - \[Alpha]]   *)


and here you get it.

Have fun!