# Orthogonal polynomials with respect to the lognormal distribution

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,

Pierre-O.

• 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). – Elmar Zander Jun 10 '16 at 9:23
• Yes, I know see proposition 1.1 in pierre-olivier.goffard.me/Publications/… – LaGauffreBZH Jun 11 '16 at 13:01
• 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! – LaGauffreBZH Jun 11 '16 at 13:04
• 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. – Elmar Zander Jun 15 '16 at 10:52
• Expansion of random variables? Is it related to Malliavin calculus? – 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:

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