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.

  • $\begingroup$ $ _{1}F_{1}(b,b-a;z)$ is the hypergeometric function $\endgroup$ – 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, $$f(1,b,x)=\theta(1-e|x|)(1-(ex)^b),\;\;b>0.$$

  • $\begingroup$ Thank you very much. $\endgroup$ – Adam Hammam Apr 11 at 19:39
  • $\begingroup$ What is $\theta$ and e? $\endgroup$ – Adam Hammam Apr 11 at 19:40
  • $\begingroup$ $\theta(x)$ is the unit step function (equal to unity if $x>0$, equal to zero otherwise), $e$ is the base of the natural logarithm. $\endgroup$ – Carlo Beenakker Apr 11 at 19:47
  • $\begingroup$ Is $f(a,b,x)$ equal to $\theta(a−e|x|)(a−(ex)^{b})$? $\endgroup$ – Adam Hammam Apr 12 at 18:06
  • $\begingroup$ And can we express the hypergeometric function as an infinite product? $\endgroup$ – Adam Hammam Apr 12 at 19:14

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