Geodesic
Indetermined moment problems related to $q$-functional equations
Teoriâ veroâtnostej i ee primeneniâ, Tome 65 (2020) no. 3, pp. 617-633

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For $q\in (0,1)$, $p_1,p_2,p\in \mathbb{R}_+$, we characterize all the solutions of the $q$-functional equations $(1+p_2q^{1/2}x)f(qx)=q^{\beta-1/2}(x+p_1q^{-1/2})f(x)$ and $f(qx)=q^{\beta- 1}(x^2+p^2q^{-1})f(x)$, $x>0$, $\beta\in \mathbb{R}$, and we also show that these solutions solve corresponding indetermined moment problems.
Keywords: moment problems, $q$-functional equations, log-normal density. 
@article{TVP_2020_65_3_a8,
     author = {M. L\'opez-Garc{\'\i}a},
     title = {Indetermined moment problems related to $q$-functional equations},
     journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
     pages = {617--633},
     publisher = {mathdoc},
     volume = {65},
     number = {3},
     year = {2020},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TVP_2020_65_3_a8/}
}
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M. López-García. Indetermined moment problems related to $q$-functional equations. Teoriâ veroâtnostej i ee primeneniâ, Tome 65 (2020) no. 3, pp. 617-633. http://geodesic.mathdoc.fr/item/TVP_2020_65_3_a8/