Calculation of an inverse Mellin transform

Let $$z \in C$$ and consider the following integral equation: $$-\frac{\Gamma(a)}{\Gamma(b)}\frac{1}{ z \dfrac{\mathrm{d}}{\mathrm{d}z} {_{1}F_{1}}(b,b-a;z)}= \int_{0}^{+ \infty}x^{z-1}K(x) \mathrm{d}x$$ 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.

• $_{1}F_{1}(b,b-a;z)$ is the hypergeometric function – Adam Hammam Apr 11 at 12:01

Because of the identity $$\frac{d}{dz}\, _1F_1(b;b-a;z)=\frac{b }{b-a}\, _1F_1(b+1;b+1-a;z)$$ the function $$K(x)$$ is given by $$K(x)=-\frac{(b-a)\Gamma(a)}{b\Gamma(b)}f(a,b,x)$$ with $$f(a,b,x)$$ the inverse Mellin transform of $$\frac{1}{z\;_1F_1(b+1,b+1-a;z)}$$.
For integer $$a$$ this has a closed form expression as a power series in $$x$$, for example, $$f(1,b,x)=\theta(1-e|x|)(1-(ex)^b),\;\;b>0.$$
• What is $\theta$ and e? – Adam Hammam Apr 11 at 19:40
• $\theta(x)$ is the unit step function (equal to unity if $x>0$, equal to zero otherwise), $e$ is the base of the natural logarithm. – Carlo Beenakker Apr 11 at 19:47
• Is $f(a,b,x)$ equal to $\theta(a−e|x|)(a−(ex)^{b})$? – Adam Hammam Apr 12 at 18:06