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I want to invert a sequence of moments and find a function f(x) satisfying: $$m_r=\int x^{r}f(x) dx=\int x^{r} dF(x)$$

The sequence of moments is given by:

$m_{2s+1}=0$

$m_{2s}=\sum_{k=1}^{s}\binom{2s-k}{s}\frac{k}{2s-k}d^{k}\frac{(d-1)^{s+1-k}}{d-c}\left(1-\left(\frac{c-1}{d-1}\right)^{s-k+1}\right),$

for each $s\geq0$. Here $d > c\geq3$ are fixed integers.

I found for this problem that $\omega:=\sup\left\{ |x|:0<F(x)<1\right\} =2\sqrt{d-1},$ so we can replace $\int x^{r}f(x) dx$ by $\int^{\omega}_{-\omega } x^{r}f(x) dx$

I am trying to expand f(x) using Chebyshev polynomials in order to find its coefficients, but I was unsuccessful simplifying the expressions.

Do you know how I could get a closed form for f(x)?

LuHell
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  • Step 1: see if you can find some "closed form" for the moments. Step 2: hope to apply http://mathoverflow.net/questions/79868/what-does-mellin-inversion-really-mean/79925#79925 – T. Amdeberhan Dec 28 '16 at 17:26
  • Any linear recurrence for the moments may also be of use – Pietro Majer Dec 28 '16 at 17:49
  • @PietroMajer The recurrence is nonlinear. Any suggestions on how to proceed? – LuHell Dec 29 '16 at 18:30
  • Try to deduce an equation for $f$ from the recurrence; if you add it I'll see. – Pietro Majer Dec 29 '16 at 19:03
  • Thanks @PietroMajer. I put the reduced problem with a recurrence below. – LuHell Dec 29 '16 at 19:25
  • what is the interval where the measure is supported? is it the above $[-\omega,\omega]$? (a good starting point would be to get rid of some constants, and renormalize e.g. in $[-1,1]$ ) – Pietro Majer Dec 29 '16 at 21:06
  • Yes that is the support. I normalized to [-1,1], however the expressions just don't simplify. – LuHell Dec 29 '16 at 21:13
  • The expressions appear in this question [http://mathoverflow.net/questions/258232/closed-expression-for-the-sum] – LuHell Dec 29 '16 at 21:14

1 Answers1

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I solved it partially, but it remains to solve for $m_{2s}=\sum_{k=1}^s k {2\,s-k-1\choose s-1}\left({\frac{d}{c-1}} \right) ^{k-1},$ where $d>c\geq3$ are integers.

Besides, $m_{2s}$ satisfy the recurrence relation $\left( {d}^{2}+{d}^{2}s \right) m_{2s} + \left( {c}^{2}s- 2\,s c +s- d s c + d s \right) m_{2(s+1)} ={\frac { \left( c-1 \right) ^{2}{4}^{s}\Gamma \left( s+1/2 \right) }{\sqrt {\pi }\Gamma \left( s \right) }}$

LuHell
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