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We are able to calculate the value of the sum

$\sum_{k=0}^\infty \frac{(a_1)_k(a_2)_k\dots(a_p)_k}{(b_1)_k(b_2)_k\dots(b_{p-1})_k}\cdot\frac{x^k}{k!}$, which equals the generalised hypergeometric function

${}_pF_q\left[\begin{matrix} a_1,a_2,\dots ,a_p \\ b_1,b_2,\dots ,b_{p-1} \end{matrix}\quad;x\right]$.

Is it possible to calculate (or bound) $\sum_{k=0}^\infty \frac{(a_1)_k(a_2)_k\dots(a_p)_k}{(b_1)_k(b_2)_k\dots(b_{p-1})_k}\cdot\frac{x^k}{k!}\cdot k$ ?

Thank you very much for any suggestions!

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$$S=\sum_{k\geq0}\frac{(a_1)_k\cdots(a_{p})_k}{(b_1)_k\cdots(b_{q})_k}\frac{kx^k}{k!}$$ $$S=\frac{(a_1)_0\cdots(a_{p})_0}{(b_1)_0\cdots(b_{q})_0}\frac{0x^0}{0!}+\sum_{k\geq1}\frac{(a_1)_k\cdots(a_{p})_k}{(b_1)_k\cdots(b_{q})_k}\frac{kx^k}{k!}$$ $$S=\sum_{k\geq1}\frac{(a_1)_k\cdots(a_{p})_k}{(b_1)_k\cdots(b_{q})_k}\frac{kx^k}{k!}$$ $$S=x\sum_{k\geq1}\frac{(a_1)_k\cdots(a_{p})_k}{(b_1)_k\cdots(b_{q})_k}\frac{x^{k-1}}{(k-1)!}$$ $$S=x\sum_{k\geq0}\frac{(a_1)_{k+1}\cdots(a_{p})_{k+1}}{(b_1)_{k+1}\cdots(b_{q})_{k+1}}\frac{x^{k}}{k!}$$ Note that $(a)_{n+1}=a\cdot(a+1)_n$. Thus, $$S=x\sum_{k\geq0}\frac{a_1\cdots a_{p}}{b_1\cdots b_{q}}\frac{(a_1+1)_k\cdots(a_{p}+1)_k}{(b_1+1)_k\cdots(b_{q}+1)_k}\frac{x^{k}}{k!}$$ $$S=x\frac{a_1\cdots a_{p}}{b_1\cdots b_{q}}\sum_{k\geq0}\frac{(a_1+1)_k\cdots(a_{p}+1)_k}{(b_1+1)_k\cdots(b_{q}+1)_k}\frac{x^{k}}{k!}$$ $$S=x\frac{a_1\cdots a_{p}}{b_1\cdots b_{q}}\;_pF_q\big(a_1+1,\dots,a_{p}+1;b_1+1,\dots,b_q+1;x\big)$$ And there you go.

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  • $\begingroup$ Thank you so very much! You saved my day! So very nice of you. If some literature would be on your mind for me to get in this calculations better, I would be also most thankful. $\endgroup$ – Rafael Oct 30 '18 at 17:21
  • $\begingroup$ @Rafael You're very welcome $\endgroup$ – clathratus Oct 30 '18 at 17:22
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    $\begingroup$ Would you know please some literature to improve in this calculations? $\endgroup$ – Rafael Nov 2 '18 at 11:11
  • $\begingroup$ @Rafael Not really. I just started by looking at the series and going from there. Of course Wikipedia is always a good place to start. And then I would suggest Wolfram Function's site, and Wolfram Mathworld might have some good information. Just search "hypergeometric functions" on all of them. I'll tell you if I find anything good though. $\endgroup$ – clathratus Nov 2 '18 at 15:23
  • $\begingroup$ Thank you very much. I searched also Google Scholar, Web of Science and Scopus, but was not satisfied with the results. If I include your previous solution in my PhD thesis, you will be cited as appropriate. $\endgroup$ – Rafael Nov 2 '18 at 15:32

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