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Let $\psi(x)$ denote the digamma function $$ \psi(x)=\Gamma(x)\frac{\partial}{\partial x} \Gamma(x). $$ Consider $x=x_1 +x_2+\dots +x_m$, where $x_j>0$, for $j=1, \ldots,m$. Is there any formula to decompose $\psi(x)$ in terms of $\psi(x_1),\ldots,\psi(x_m)$?

I know that in the very special case of $m=2$ and $x_1=x_2=x/2$, with $x>0$, Legendre duplication formula allows to claim $$ \psi(x_1+x_2)=\log 2 +\frac{1}{2}\left( \psi(x_1) + \psi(x_2+1/2) \right) $$ and I was wondering whether something more general than that is known in the literature.

Jack London
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1 Answers1

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I'm pretty sure the answer is no. In the case that $x_1=...=x_m=z$ however there is the nice formula $$\psi^{[n]}(mz)=\delta_{n,0}\ln m +\frac{1}{m^{n+1}}\sum_{k=0}^{m-1}\psi^{[n]}\left(z+\frac{k}{m}\right)$$

K.defaoite
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  • Ok, many thanks. Could you point me to a reference? – Jack London Oct 12 '20 at 09:32
  • It's on Wolfram Mathworld although I'm unsure of the original source, as Mathworld does not state which reference it comes from. The list of references is at the bottom of the article, but you'll have to do your own digging to find which one it came from. I unfortunately could not find a proof for the general polygamma, but Flammable Maths has a proof for the Gamma function. Perhaps some of the methods used there will prove useful. – K.defaoite Oct 12 '20 at 11:54
  • I suspect that this identity comes from Abramowitz and Stegun's Handbook of Mathematical Functions although there are several other references cited. – K.defaoite Oct 12 '20 at 11:58
  • Many thanks, I'll do the search. – Jack London Oct 12 '20 at 12:03
  • @Jack London I managed to come up with a proof of this formula for $m,n \in \mathbb{N}$, will write it up as soon as I can. – K.defaoite Oct 12 '20 at 12:44
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    Sure, thanks. Meanwhile, I've found the formula in Abramowitz and Stegun's book, under the name "Multiplication Formula" on page 260. – Jack London Oct 12 '20 at 13:42
  • @JackLondon I checked over my proof and unfortunately it is flawed. Maybe the proof of this has been discussed elsewhere on this site, but I'm not sure. EDIT: found a proof, but personally I don't really understand it as it involves finite fields and p-adic theory and so on which I know almost nothing about. Here's a link: https://math.stackexchange.com/questions/597867/the-multiplication-formula-for-the-hurwitz-zeta-function – K.defaoite Oct 13 '20 at 10:15