Solution for Moment problem

Question

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)?

Answer 1

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