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Recently, I have encountered the family of orthogonal polynomials $p_{n}(x)$ which is orthogonal w.r.t. the function $-\ln(x)$ on $(0,1)$. This means we have $$\int_{0}^{1}p_{n}(x)p_{m}(x)\ln(1/x)dx=\delta_{mn}, \quad \forall m,n\geq0.$$

Most likely, this family of orthogonal polynomials have been studied in past since the weight function is very simple. Nevertheless, I found no really useful information about them till now. Primarily, I am interested in the coefficients from the three-term recurrence relation but any other properties related to $p_{n}(x)$ would be valuable. It seems this family does to match any of the well known and described family of orthogonal polynomials (belonging to the Askey scheme).

Any relevant information on literature is desirable. Thank you.

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  • $\begingroup$ The family of orthogonal polynomials with respect to a weight is not unique unless you make some restricting choices. Do you assume, for example, that $p_n$ has degree $n+1$? Note that all of your polynomials must vanish at zero. $\endgroup$ – Joonas Ilmavirta Oct 5 '14 at 16:44
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    $\begingroup$ @JoonasIlmavirta: There is a very standard procedure that gives unique $p_n$'s: you run Gram-Schmidt on $1,x,x^2,\ldots$ and make the leading coefficient positive. Also, there is no reason why the $p_n$ would have to vanish at zero. $\endgroup$ – Christian Remling Oct 5 '14 at 19:26
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    $\begingroup$ @ChristianRemling, that is what I thought, but I could imagine that some other choice would come in an application. Vanishing at zero was a silly miscalculation; that would happen if the weight was not integrable near zero. $\endgroup$ – Joonas Ilmavirta Oct 5 '14 at 19:37
  • $\begingroup$ Sorry, I certainly assume that the degree of $p_{n}$ is equal to $n$. $\endgroup$ – Twi Oct 5 '14 at 20:38
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As you perhaps know already, there are several systematic methods that in principle recover the recursion coefficients from the orthogonality measure $d\mu(x)=-\chi_{(0,1)}(x)\ln x\, dx$. However, it is usually safe to assume that this will not give a very explicit answer.

Here, you could for example compute the moments $$ m_n = \int x^n\, d\mu(x) = \frac{1}{(n+1)^2} , $$ and then there are formulae that express $a_n,b_n$ in terms of determinants of Hankel type matrices built from these moments. See for example Teschl's book on Jacobi matrices; see formulae (2.109), (2.113), and (2.118). As expected, these seem to be getting out of hand quickly.

W. Gautschi, On generating orthogonal polynomials, SIAM J. Sci. Statist. Comput. 3 (1982), 289–317 presents some numerical work on this question for several weight functions, including the one $w(x)=-\ln x$ you are interested in.

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  • $\begingroup$ The connection you mention between the Hankel matrix of moments $m_{n}$ with the family of orthogonal polynomials associated with the weight $-\chi_{(0,1)}\ln x$ is exactly the way I arrived at the polynomials whose properties I am asking to. However, I wasn't able to evaluate the Hankel determinants by myself. Thanks for the referrence on Gautschi's paper which I didn't know. $\endgroup$ – Twi Oct 6 '14 at 6:40
  • $\begingroup$ @Twi: not to criticize you, but recall that posting a question, one should kindly provide all relevant information to it. Otherwise, the effort of the people who answers may resolve in some waste of time, to find again facts that are already known to the questioner... $\endgroup$ – Pietro Majer Oct 6 '14 at 6:58
  • $\begingroup$ @PietroMajer Sure, you're right. I wanted to be brief and address the problem directly, however, I should mention that I known the moments of the measure in question and I am familiarized with the theory of the moment problem and orthogonal polynomials on the real line (as it is demonstrated, for instance, in the great Akhiezer's book: The Classical Moment Problem and Some Related Questions in Analysis). $\endgroup$ – Twi Oct 6 '14 at 12:32

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