I want to calculate the following sum over non-negative integer partitions $$ \sum_{l_1+\cdots +l_n=s} \frac{1}{(l_1!)^2 \cdots (l_n!)^2}. $$ for fixed $n$ and $s.$ I tried to use Vandermonde's identity and induction, but no success so far.


$\newcommand{\bbx}[1]{\,\bbox[8px,border:1px groove navy]{{#1}}\,} \newcommand{\braces}[1]{\left\lbrace\,{#1}\,\right\rbrace} \newcommand{\bracks}[1]{\left\lbrack\,{#1}\,\right\rbrack} \newcommand{\dd}{\mathrm{d}} \newcommand{\ds}[1]{\displaystyle{#1}} \newcommand{\expo}[1]{\,\mathrm{e}^{#1}\,} \newcommand{\ic}{\mathrm{i}} \newcommand{\mc}[1]{\mathcal{#1}} \newcommand{\mrm}[1]{\mathrm{#1}} \newcommand{\pars}[1]{\left(\,{#1}\,\right)} \newcommand{\partiald}[3][]{\frac{\partial^{#1} #2}{\partial #3^{#1}}} \newcommand{\root}[2][]{\,\sqrt[#1]{\,{#2}\,}\,} \newcommand{\totald}[3][]{\frac{\mathrm{d}^{#1} #2}{\mathrm{d} #3^{#1}}} \newcommand{\verts}[1]{\left\vert\,{#1}\,\right\vert}$ \begin{align} &\sum_{\ell_{1}\ +\ \cdots\ +\ \ell_{n}\ =\ s}\hspace{3mm} {1 \over \pars{\ell_{1}!}^{2} \cdots\pars{\ell_{n}!}^{2}} = \sum_{\ell_{1}\ldots\ell_{n}}\hspace{3mm} {1 \over \pars{\ell_{1}!}^{2} \cdots\pars{\ell_{n}!}^{2}} \oint_{\verts{z} = 1^{-}}\,\,\, {1 \over z^{s + 1 - \ell_{1} - \cdots - \ell_{n}}} \,\,\,\,{\dd z \over 2\pi\ic} \\[5mm] = &\ \oint_{\verts{z} = 1^{-}}\,\,\,{1 \over z^{s + 1}} \,\bracks{\sum_{\ell = 0}^{\infty}{z^{\ell} \over \pars{\ell!}^{2}}}^{n} {\dd z \over 2\pi\ic} = \oint_{\verts{z} = 1^{-}}\,\,\,{\mrm{I}_{0}^{n}\pars{2\root{z}} \over z^{s + 1}} {\dd z \over 2\pi\ic} =\ \bbox[15px,#ffe,border:2px dotted navy]{\ds{% \bracks{z^{s}}\mrm{I}_{0}^{n}\pars{2\root{z}}}} \end{align}

$\ds{\mrm{I}_{\nu}\pars{z}}$ is a Modified Bessel Function.

  • $\begingroup$ Thank you for the reply. The problem is that the integral of the power of the Modified Bessel Function is also imposible to calculate or estimate. $\endgroup$ – Hovher Nov 17 '16 at 13:27
  • $\begingroup$ @Hovher Thanks. You're welcome. I know what you mean. Indeed, I was trying to go further but there was not any hope. I look at http://dlmf.nist.gov/10 for some properties which can be helpful too. $\endgroup$ – Felix Marin Nov 17 '16 at 18:12

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.