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 $$F_1(a,b,b',c;x,y) = \frac{\Gamma(c)}{(2 \pi i)^2 \Gamma(a) \Gamma(b) \Gamma(b')} \int_{-i \infty}^{i \infty}\int_{-i \infty}^{i \infty} \frac{\Gamma(a+u+v)\Gamma(b+u)\Gamma(b'+v)\Gamma(-u)\Gamma(-v)(-x)^u (-y )^v}{\Gamma(c+u+v)} \, du \, dv$$ (
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:]

$$P = \frac{(2 \pi i)^2 \Gamma(c) \Gamma(d) \Gamma(d')}{\Gamma(f)} \int_0^{\infty} e^{-s\, t} t^{a-1} F_1(c, d, d', f; xt, yt) \, dt \, \\= \frac{\Gamma(a)}{s^a} \ \int_{-i \infty}^{i \infty}\int_{-i \infty}^{i \infty} \frac{\Gamma(c+u+v) \Gamma(a+u+v) \Gamma(d+u) \Gamma(d'+v) \Gamma(-u) \Gamma(-v) (-x)^u (-y )^v}{\Gamma(f+u+v)} (-\frac{x}{s})^u (-\frac{y}{s})^v\, du \, dv$$ ( 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.


2 Answers 2


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.

  • $\begingroup$ I for sure considered such an argument before posting the question. At least for $F_2 $ and possibly other functions (I have not checked yet) this cannot work. Neither your series nor the associated KdF function (defined as a series) converge. Check references in Appel and KdF 1926, Exton 1976 or Srivastava 1985. It may well work with Mellin-Barnes, not with series. Convergence domain is zero. This is why Exton did not use series on the cited passages. There is probably no other way out than with Meijer-G functions resulting from MB integrals $\endgroup$ Nov 19, 2022 at 17:49
  • 2
    $\begingroup$ I confirm my above comment: the Laplace-type integrals $\int_0^\infty e^{-st} t^\alpha F_i(...;x, yt) \,dt, \, i=1,...,4 $ have no meaning at all if you use series representations. It was shown by Horn(1889), Appell & KdF(1926:397-398) that the convergence domain of Appell functions defined as series is included in a square. So you cannot integrate them to infinity. $\endgroup$ Nov 21, 2022 at 0:46
  • $\begingroup$ Would you happen to know of any Kampe de Feriet function (with numerical parameters) that is the root of a sextic equation? I've been trying to find one for ages. Or better, can any of the Appell functions (since they are also two-variable) be a root of a sextic? $\endgroup$ Jun 23, 2023 at 16:18

I took me more than a year to come by a correct solution. It is inspired by published work on the Laplace transform bearing on both variables $x, y$ by Manilal Shah in the 60's [1], of a more general kind as it addressed the Laplace transform of the bivariate Meijer-G function (of which Appell's functions are particular cases). Oddly enough, I have not come by published solutions for the easier Laplace transform bearing on just one of the two variables, although it is an altogether simpler problem.

For what follows the main reference is [2].

As indicated in the question, using Mellin-Barnes integrals (MBI) can hardly be escaped (the other option is Euler-Laplace integrals and it is by no way simpler). Let us first notice that, for example, Appell's $F_2$ series can be analytically continued using the following MBI:

\begin{align}F_2(a, b, c, d, e; x, y) = \frac{\Gamma(d)\Gamma(e)}{\Gamma(a)\Gamma(b)\Gamma(c) (2\pi i)^2} \int_L \frac{\Gamma(a+t+s)\Gamma(b+s)\Gamma(c+t)}{\Gamma(d+s)\Gamma(e+t)} \Gamma(-s)\Gamma(-t)(-x)^s (-y)^t \; ds\,dt \\ &\tag{1} \end{align}

where $L=i\mathbb{R}^2,$ (from [3: 5.8.3, p. 232]).

Interchanging order of the (standard) Laplace integral with the Bromwich contour integral is warranted by a well-known theorem of de la Vallée-Poussin so that, provided that the RHS integral absolutely converges, we can write:

\begin{align} \mathscr{L}(x^\alpha F_2(a,b,c,d,e; x, y); p) &= \frac{\Gamma(d)\Gamma(e)}{p^{\alpha+1}\;\Gamma(a)\Gamma(b)\Gamma(c) (2\pi i)^2} \int_L \frac{\Gamma(a+t+s)\Gamma(b+s)\Gamma(c+t)}{\Gamma(d+s)\Gamma(e+t)} \Gamma(-s)\Gamma(-t)\Gamma(\alpha+1+s)\, (-p^{-1})^s (-y)^t\,ds\,dt \\ &= \frac{\Gamma(\alpha+1)}{p^{\alpha+1}} F_{\,0\, :\,1\,;\,1}^{1\,:\,2\,;\,1}\left[^{a\;:\,b, \,\alpha+1\,;\,c}_{\;\;\,:\;\;\;\;d\;\;\;;\,e}\;; \frac{1}{p}, y\right], \tag{2} \end{align}

by [2, eq. (23.15) p.203] (or [4, eqs. p.94]), provided that the Kampé de Fériet function $F$ is well-defined (see below).

The hard bit is now to correctly identify the conditions under which this makes sense. Convergence conditions for two-variable Meijer-G functions have been somewhat controversial from the date of their invention by Sharma in the mid-60's to the mid-eighties. A flurry of papers (mainly from the Indian school of mathematics) did settle the issues and [2] is a comprehensive compendium of solutions brought to these technical difficulties.

As Leucippus conjectured, the solution can actually be represented by a Kampé de Fériet (KdF) function, with the understanding that this function is not defined by its usual series representation, but as a notational variant of a Meijer-G two-variable function (which stands as the analytic continuation of the KdF series). With this understanding, [2: 117-118] gives the general formula:

$$F_{C\,:\,D\,;\,D'}^{A\,:\,B\,;\,B'}\left[^{(a)\,:\,(b)\,;\,(b')}_{(c)\,:\,(d)\,;\,(d')}\;; x, y\right]=\Gamma\left[^{(c), (d), (d')}_{(a),(b)(b')}\right] G^{0,A\,:\,1,B\,;\, 1,B'}_{A,C\,:\,B,D+1\,;\, B',D'+1}\left[^{1-(a)\,:\, 1-(b)\,;\, 1-(b')}_{1-(c)\,:\, 0, 1-(d)\,;\, 0, 1-(d')}; -x, -y\right] \tag{3}$$

where absolute convergence of the Meijer-G function is ensured if:

\begin{cases} |arg(x)| < \frac{\pi}{2}(A+B-C-D) \\ |arg(y)| < \frac{\pi}{2}(A+B'-C-D') \\ |arg(x)-arg(y)| < \frac{\pi}{2}(B+B'+2-D-D') \tag{4} \end{cases}

In the case at hand convergence conditions only come to bear to the phases of $p, x$ as complex numbers. As $A=1, B=2, B'=1; C= 0, D= 1, D'=1$, inserting in $(4)$ yields (the third condition is always satisfied once the first two are):

$$\text{Convergence conditions for } (2): x \notin \mathbb{R}^- \; \land \; \Re{y} > 0, \tag{5}$$

No condition comes to bear on $|p|$ or $|y|$, which is where MBI shine with respect to series representations (see my comments to the answer by Leucippus).

Note that the form of the result is the same as the one that would be obtained by reasoning on the series, so that results by Leucippus are valid although improperly obtained and without the adequate convergence conditions. This comes as a surprise as, at least for $F_2$, it can be shown that the standard convergence conditions ([2, p.103]) for convergence of the KdF series are not satisfied for $F^{1:2;1}_{0:1;1}$, even for $|y|+|1/p| < 1$, so that the series themselves may very well not be defined.

The same line of reasoning can be straightforwardly followed for other Appell hypergeometric functions.


[1] Shah, Manilal "On generalized Meijer function of two variables sans some applications" Comment. Math. Univ. St. Pauli XIX-2, 1970

[2] Hai, Nguyen Thanh & Yakubovich, S.B. The Double Mellin-Barnes Type Integrals and their Applications to Convolution Theory, World Scientific, 1992

[3] Erdelyi, A., Magnus, W., Oberhettinger, F. and Tricomi, F.G. Higher Transcendental Functions, Vol. I, McGraw-Hill, New York.

[4]Exton, Harold, Handbook of hypergeometric integrals. Theory, applications, tables, computer programs, Mathematics & its Applications, Chichester: Ellis Horwood Limited Publishers. 316 p. (1978).


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