Integration of confluent hypergeometric function of two variables Is it possible to integrate confluent hypergeometric function of two variables?
I am trying to solve the integral
$\int_0^tx^{n-1}e^{px}\Phi_3(a,b,cx,mx)~dx$ 
where $n$ is a positive integer, $a,b,c$ and $m$ are positive constants. 
 A: $\int_0^tx^{n-1}e^{px}\Phi_3(a,b,cx,mx)~dx$
$=\int_0^tx^{n-1}e^{px}\sum\limits_{r=0}^\infty\sum\limits_{s=0}^\infty\dfrac{(a)_r(cx)^r(mx)^s}{(b)_{r+s}r!s!}dx$ (accrodindg to http://en.wikipedia.org/wiki/Humbert_series)
$=\int_0^t\sum\limits_{r=0}^\infty\sum\limits_{s=0}^\infty\dfrac{(a)_rc^rm^sx^{n+r+s-1}e^{px}}{(b)_{r+s}r!s!}dx$
$=\left[\sum\limits_{r=0}^\infty\sum\limits_{s=0}^\infty\sum\limits_{u=0}^{n+r+s-1}\dfrac{(-1)^{n+r+s-u-1}(n+r+s-1)!(a)_rc^rm^sx^ue^{px}}{(b)_{r+s}p^{n+r+s-u}r!s!u!}\right]_0^t$ (according to http://en.wikipedia.org/wiki/List_of_integrals_of_exponential_functions)
$=\sum\limits_{r=0}^\infty\sum\limits_{s=0}^\infty\sum\limits_{u=0}^{n+r+s-1}\dfrac{(-1)^{n+r+s-u-1}(n+r+s-1)!(a)_rc^rm^st^ue^{pt}}{(b)_{r+s}p^{n+r+s-u}r!s!u!}-\sum\limits_{r=0}^\infty\sum\limits_{s=0}^\infty\dfrac{(-1)^{n+r+s-1}(n+r+s-1)!(a)_rc^rm^s}{(b)_{r+s}p^{n+r+s}r!s!}$
A: BTW: somebody should check this carefully and erase ths comment; I don't know what I would do with my life if I didn't spend 3/4 of the time correcting mistakes
There are alternate ways to express the answer. For any $f(x)$ having
a taylor expansion around zero we have:
${\displaystyle \intop_{0}^{t}e^{-px}f(}x)dx={\displaystyle \intop_{0}^{t}e^{-px}\sum_{n=0}^{\infty}a_{n}x^{n}dx={\displaystyle \sum_{n=0}^{\infty}a_{n}\gamma\left(n+1,p\cdot t\right)/p^{n+1}}}$ 
or the simpler:
$${\displaystyle \intop_{0}^{t}e^{-x}f(}p\cdot x)dx={\displaystyle \intop_{0}^{t}e^{-x}\sum_{n=0}^{\infty}a_{n}p^{n}\cdot x^{n}dx={\displaystyle \sum_{n=0}^{\infty}a_{n}\gamma\left(n+1,t\right)\cdot p^{n}}}$$ 
Whenever the summation/intgration can be switched.
While the last just switches a function for the infinite series variable;
an alternate expansion in $t$ yields a sequence of ascending order
polynomials $p_{n}$; $n$ is the order.
$\sum_{n=0}^{\infty}a_{n}\cdot p_{n}(p)\cdot t^{n}$
This is obvious when the answer is thrown in a coefficient matrix
form (double Taylor expansion) and the properties of $\gamma\left(n,t\right)$
are examined.  In addition the coefficients of the polynomials are simple to write down.
A third representation has been redacted for editing due to errors.
rrogers
