I am currently doing some inspection on the orthogonal polynomials with respect to the lognormal distribution. Does anyone already work on that or know some cool references?

All the best,


  • $\begingroup$ Please note that the orthogonal polynomials for the lognormal distribution are not dense in $L_2$ equipped with the measure induced by the lognormal (see the article by Ernst et. al. dx.doi.org/10.1051/m2an/2011045). $\endgroup$ – Elmar Zander Jun 10 '16 at 9:23
  • $\begingroup$ Yes, I know see proposition 1.1 in pierre-olivier.goffard.me/Publications/… $\endgroup$ – LaGauffreBZH Jun 11 '16 at 13:01
  • $\begingroup$ Maybe less technical than what is proposed in the article you make reference to. I will definitely have a look at it even though I am not sure to understand. Thanks for the comment! $\endgroup$ – LaGauffreBZH Jun 11 '16 at 13:04
  • $\begingroup$ Interesting article. I've never seen the approach in approximating density ratios. We only do expansions of random variables in orthogonal polynomials and I frequently have to warn people (students/engineers) not to use lognormals as basis random variables in their expansions. So maybe it's good that the comment is there to warn other people. $\endgroup$ – Elmar Zander Jun 15 '16 at 10:52
  • $\begingroup$ Expansion of random variables? Is it related to Malliavin calculus? $\endgroup$ – LaGauffreBZH Jun 16 '16 at 12:13

Orthogonal polynomials with respect to the lognormal distribution go by the name of Stieltjes-􏰈Wigert polynomials. Two recent studies of their properties:

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  • $\begingroup$ Thank you, let see if this factor 1/x in the lognormal pdf is not an issue. Cheers. $\endgroup$ – LaGauffreBZH Aug 27 '15 at 7:47

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