In my research, I often face the Humbert series in the form of:
$\Phi_2(a,b;a-b+1;-\alpha x, -\beta x$), where $a,b\in\mathbb Z$, $a<b$ and $\alpha,\beta\in \mathbb R$ with $0<\alpha,\beta \le1$, and $0\le x\in \mathbb R$. Note that $\alpha\ne\beta$.
(For the definition of the Humbert series, $\Phi_2$, refer to https://en.wikipedia.org/wiki/Humbert_series)
and I've been looking for any method (approximation and others) to reduce this function, but couldn't find any proper ones.
I hope the function can be simplified into any simpler forms or perhaps, $_1 F_1(m,n,\gamma x)$, which is a confluent hypergeometric function. The latter conjecture is due to the fact that the two independent variables in $\Phi_2$ are just $x$ with different scales $\alpha, \beta$.
Any help will be appreciated!
Some identies I've tried are as follows:
- A. K. Rathie, "On a representation of Humbert's double hypergeometric series $\Phi_2$ in a series of Gauss's _2F_1 function"
- $\Phi_2(a,b;c;x,y)=\sum\limits_{m = 0}^\infty {\frac{{\left( a \right)_m }}{{\left( c \right)_m }}{}_2F_1 \left( { - m,b;1 - a - m;\frac{y}{x}} \right)\frac{{x^m }}{{m!}}} $
- Some reduction formulas for $x=y$ or $y=-x$ are known in the literature, but they not my case due to $\alpha\ne\beta$.
- S. Wald, "On integral representations and asymptotics of some hypergeometric functions in two variables"
- $\Phi _2 \left( {\beta ,\beta ';\gamma ; - tx, - ty} \right) = \frac{{\Gamma \left( \gamma \right)t^{1 - \gamma } }}{{\Gamma \left( {\gamma - \varepsilon } \right)\Gamma \left( \varepsilon \right)}} \int_0^t {v^{\gamma-\varepsilon-1} {}_1F_1(\beta;\gamma-\varepsilon;-xv)\times{}_1F_1(\beta';\varepsilon;y(v-t))}(t-v)^{\varepsilon-1}dv$, where $0<\varepsilon<\gamma$ is a fixed constant.