How to express a hypergeometric confluent function in the pFq form?

I would like to implement the following function due to Erdeliy and need it as $_pF_q^{(\alpha)}(a,b;c,d)$

$\Phi_3=\Sigma_{m} \Sigma_{n} \frac{(\beta)_m}{(\gamma)_{m+n}m!n!}x^{m}y^{n}$, where the sums run to infinity.

I have a package that implements hypergeometric functions of all kinds in MATLAB due to Plamen and Koev.

However, how can I express $\Phi_3$ as $_pF_q^{(\alpha)}(a,b;c,d)$? I know the basic definition of a hypergeometric function and suspect that here $p=q=1$ but am not quite sure how to proceed.

I would greatly appreciate your help!

• Nice, thanks for the upvote. Sadly people tend to avoid this topic like a pest. It is however so useful and important. – Hirek Mar 11 '15 at 19:22

Humbert-type series and Generalized hypergeometric function come from different generalization approaches, so it does not guarantee that they can interchange each other.

For Kampé de Fériet function, according to http://www.iosrjournals.org/iosr-jm/papers/Vol8-issue6/L0866770.pdf?id=7287, this special case can reduce to Generalized hypergeometric function:

$\mathrm{F}^{p:0;...;0}_{q:0;...;0}\Bigg(\begin{matrix}a_1,...,a_p&:&-&;&...&;&-&\\b_1,...,b_q&:&-&;&...&;&-&\end{matrix}\Bigg|x_1,...,x_n\Bigg)={_pF_q}(a_1,...,a_p;b_1,...,b_q;x_1+...+x_n)$

But I don't know whether the $\Phi_3$ function (also a special case of Kampé de Fériet function) can reduce to Generalized hypergeometric function or not.

But I know that the $\Phi_3$ function in some cases can express to certain definte integral of Generalized hypergeometric function.

$\because\sum\limits_{m=0}^\infty\sum\limits_{n=0}^\infty\dfrac{x^my^n}{(\gamma)_{m+n}m!n!}={_0F_1(-;\gamma;x+y)}$

$\therefore\Phi_3(\beta,\gamma;x,y)=\sum\limits_{m=0}^\infty\sum\limits_{n=0}^\infty\dfrac{(\beta)_mx^my^n}{(\gamma)_{m+n}m!n!}=\dfrac{1}{\Gamma(\beta)}\int_0^\infty\sum\limits_{m=0}^\infty\sum\limits_{n=0}^\infty\dfrac{t^{\beta+m-1}e^{-t}x^my^n}{(\gamma)_{m+n}m!n!}dt=\dfrac{1}{\Gamma(\beta)}\int_0^\infty t^{\beta-1}e^{-t}{_0F_1(-;\gamma;xt+y)}~dt$ when $\beta>0$

• Thanks @Harry Peter so you're saying that there is no way to expressi $\Phi_3$ as a confluent hypergeometric function? – Hirek Mar 15 '15 at 11:56