Let $z \in C$ and consider the following integral equation: $$\frac{\Gamma(a)}{\Gamma(b)}\frac{1}{ z \dfrac{\mathrm{d}}{\mathrm{d}z} {_{1}F_{1}}(b,ba;z)}= \int_{0}^{+ \infty}x^{z1}K(x) \mathrm{d}x$$ I would like to find the kernel $K(x)$ and I would also like to write it as an infinite product: have tried to search on many books but I didn't find the desired result (I looked especially in the books "Tables of integral transforms" and "Tables of Mellin transforms"). Thank you and best regards.

$\begingroup$ $ _{1}F_{1}(b,ba;z)$ is the hypergeometric function $\endgroup$– Adam HammamApr 11, 2021 at 12:01
1 Answer
Because of the identity $$\frac{d}{dz}\, _1F_1(b;ba;z)=\frac{b }{ba}\, _1F_1(b+1;b+1a;z)$$ the function $K(x)$ is given by $$K(x)=\frac{(ba)\Gamma(a)}{b\Gamma(b)}f(a,b,x)$$ with $f(a,b,x)$ the inverse Mellin transform of $\frac{1}{z\;_1F_1(b+1,b+1a;z)}$.
For integer $a$ this has a closed form expression as a power series in $x$, for example, $$f(1,b,x)=\theta(1ex)(1(ex)^b),\;\;b>0.$$



$\begingroup$ $\theta(x)$ is the unit step function (equal to unity if $x>0$, equal to zero otherwise), $e$ is the base of the natural logarithm. $\endgroup$ Apr 11, 2021 at 19:47

$\begingroup$ Is $f(a,b,x)$ equal to $\theta(a−ex)(a−(ex)^{b})$? $\endgroup$ Apr 12, 2021 at 18:06

$\begingroup$ And can we express the hypergeometric function as an infinite product? $\endgroup$ Apr 12, 2021 at 19:14