Appell's two-variable functions $F_1, F_2, F_3$ and $F_4$ are known to have numerous uses in applied mathematics, notably mathematical Physics. I am looking for generalized Laplace transforms (if they exist) of these functions relative to one of the two variables (I found at least two references that give some of them relative to a linear combination of the two variables). In more formal words, the question is about finding closed forms for:
$$ I_1 = \int_0^\infty x^\alpha e^{-s \, x} F_{1}(a, b, c, d; x, y) \, dx$$
$$ I_2 = \int_0^\infty x^\alpha e^{-s \, x} F_{2}(a, b, c, d, e; x, y) \, dx$$
$$ I_3 = \int_0^\infty x^\alpha e^{-s \, x} F_{3}(a, b, c, d, e; x, y) \, dx$$
$$ I_4 = \int_0^\infty x^\alpha e^{-s \, x} F_{4}(a, b, c, d; x, y) \, dx,$$
where $\alpha, s, a, b, c, d, e$ are the parameters and $x, y$ the variables.
The only reliable hint I could find is in Harold Exton's handbook ([2]), where Exton gives a method for finding the Laplace transform of $F_1$ using Mellin-Barnes integral representations of $F_1$.
Unfortunately, he did not complete the argument:
"page 99: (…) We make use of the double Barnes integral for the Appell function F1 given by Appell and Kampé de Fériet (1926p page 40. This is
$$F_1(a,b,b',c;x,y) = \frac{\Gamma(c)}{(2 \pi i)^2 \Gamma(a) \Gamma(b) \Gamma(b')} \int_{-i \infty}^{i \infty}\int_{-i \infty}^{i \infty} \frac{\Gamma(a+u+v)\Gamma(b+u)\Gamma(b'+v)\Gamma(-u)\Gamma(-v)(-x)^u (-y )^v}{\Gamma(c+u+v)} \, du \, dv$$ (5.2.4.26)
Some indication is now given as to the evaluation of the Laplace integral of the function $F_1$
[Exton proceeds, interchanging the order of integration and evaluating the inner integral as a gamma function, which gives:]
$$P = \frac{(2 \pi i)^2 \Gamma(c) \Gamma(d) \Gamma(d')}{\Gamma(f)} \int_0^{\infty} e^{-s\, t} t^{a-1} F_1(c, d, d', f; xt, yt) \, dt \, \\= \frac{\Gamma(a)}{s^a} \ \int_{-i \infty}^{i \infty}\int_{-i \infty}^{i \infty} \frac{\Gamma(c+u+v) \Gamma(a+u+v) \Gamma(d+u) \Gamma(d'+v) \Gamma(-u) \Gamma(-v) (-x)^u (-y )^v}{\Gamma(f+u+v)} (-\frac{x}{s})^u (-\frac{y}{s})^v\, du \, dv$$ (5.2.4.28)
In order to obtain a representation of this last result in terms of convergent series, the above integral may be written as an integral of Barnes type of a [Meijer] G-function of one variable. Some rather lengthy manipulation eventually leads to the sum of six double hypergeometric series of higher order with argument $s/x$ and $s/y$ .
Exton did not give the six series alluded to but only special cases of interest expressed using Kampé de Feriet functions or generalized hypergeometric functions of one variable.
He also only considered the case in which both function variables $x$ and $y$ are linked to the Laplace transform by the same integration variable $t$, but his method is obviously valid if only one variable is linked (so with no $t$ as a multiplier of $y$ in $F_1(c, d, d', f; xt, y)$ for example).
The method looks fine and I have looked around for whether there is any paper giving the forms (probably in Meijer-G representations), without success so far.
In any case, series representations should not be used, except for heuristic hints: their domain of convergence is too limited for the Laplace transforms to be defined.
But Euler-type integral representations and, as in Exton's book, Mellin-Barnes integral representations could be considered (as analytic continuations giving meaning to the Laplace transforms) or possibly expressions as a series of Gauss hypergeometric functions, since these can have Laplace transforms (see here).
Again, these may well have been published, but I failed to find anything that neatly answers the question.
[2]: Exton, Harold, Handbook of hypergeometric integrals. Theory, applications, tables, computer programs, Mathematics & its Applications. Chichester: Ellis Horwood Limited Publishers. 316 p. (1978). ZBL0377.33001.
Best Answer
A straight forward method is the following.
Using $$ F_{1}(a; b, c; d; x, y) = \sum_{n=0}^{\infty} \sum_{m=0}^{\infty} \frac{(a)_{n+m} \, (b)_{n} \, (c)_{m}}{n! \, m! \, (d)_{n+m}} \, x^n \, y^m $$ then \begin{align} I_{1} &= \int_{0}^{\infty} e^{-s t} \, t^{\alpha} \, F_{1}(a; b, c; d; x t, y) \, dt \\ &= \sum_{n,m} A_{n,m} \, x^n \, y^m \, \int_{0}^{\infty} e^{-s t} \, t^{\alpha + n} \, dt \\ &= \sum_{n,m} A_{n,m} \, \frac{\Gamma(n+\alpha+1) \, x^n \, y^m}{s^{n+\alpha+1}} \\ &= \frac{\Gamma(\alpha+1)}{s^{\alpha+1}} \, \sum_{n,m} \frac{(a)_{n+m} \, (b)_{n} \, (\alpha+1)_{n} \, (c)_{m}}{n! \, m! \, (d)_{n+m}} \, \left(\frac{x}{s}\right)^{n} \, y^m \\ &= \frac{\Gamma(\alpha+1)}{s^{\alpha+1}} \, \large{F}_{1:0,0;}^{1:2,1;}\left(\frac{x}{s}, y \right), \end{align} where the last function is the Kampe de Feriet function, which in this example, is defined by $$ \large{F}_{1:0,0;}^{1:2,1;}\left(\frac{x}{s}, y \right) = \sum_{n,m} \frac{(a)_{n+m} \, (b_{1})_{n} \, (b_{2})_{n} \, (c_{1})_{m}}{n! \, m! \, (d)_{n+m}} \, x^{n} \, y^m $$ Note: the full notation of the Kampe de Feriet function is not typed here.
In a similar manor $$ \int_{0}^{\infty} e^{-s t} \, t^{\alpha} \, F_{1}(a; b, c; d; x t, y t) \, dt = \frac{\Gamma(\alpha+1)}{s^{\alpha+1}} \, \large{F}_{1:0,0;}^{2:1,1;}\left(\frac{x}{s}, \frac{y}{s} \right),$$ where $$ \large{F}_{1:0,0;}^{2:1,1;}\left(x, y \right) = \sum_{n,m} \frac{(a_{1})_{n+m} \, (a_{2})_{n+m} \, (b_{1})_{n} \, (c_{1})_{m}}{n! \, m! \, (d)_{n+m}} \, x^{n} \, y^m $$ and $$ \int_{0}^{\infty} e^{-s t} \, t^{\alpha} \, F_{1}(a; b, c; d; x, y t) \, dt = \frac{\Gamma(\alpha+1)}{s^{\alpha+1}} \, \large{F}_{1:0,0;}^{1:1,2;}\left(x, \frac{y}{s} \right),$$ where $$ \large{F}_{1:0,0;}^{1:1,2;}\left(x, y \right) = \sum_{n,m} \frac{(a)_{n+m} \, (b_{1})_{n} \, (c_{1})_{m} \, (c_{2})_{m}}{n! \, m! \, (d)_{n+m}} \, x^{n} \, y^m. $$
The same would apply for the Horn functions (all 34), triple variable functions, Lauricella functions and so on.
There are other results, but they seem, in some cases, not efficient and straight forward.