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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
    Commented Oct 5, 2014 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
    Commented Oct 5, 2014 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
    Commented Oct 5, 2014 at 19:37
  • $\begingroup$ Sorry, I certainly assume that the degree of $p_{n}$ is equal to $n$. $\endgroup$
    – Twi
    Commented Oct 5, 2014 at 20:38
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    $\begingroup$ doi.org/10.3842/SIGMA.2018.056 looks related $\endgroup$
    – Fedor Petrov
    Commented Mar 22, 2021 at 7:35

2 Answers 2

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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
    Commented Oct 6, 2014 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
    Commented Oct 6, 2014 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
    Commented Oct 6, 2014 at 12:32
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There has been recent advances in the study of orthogonal polynomials with respect to logarithmic weights of the form $$w(x)=\log\frac{2k}{(1-x)}~\text{on}~(-1,1),\qquad k>1,$$ in particular the asymptotics of their recurrence coefficients, in

T.O. Conway, P. Deift, Percy, Asymptotics of polynomials orthogonal with respect to a logarithmic weight. SIGMA Symmetry Integrability Geom. Methods Appl. 14 (2018), (available here, also mentioned in the comment by F. Petrov).

The proof is based on Riemann-Hilbert/steepest-descent methods and one of the main ingredient is a comparison with the Legendre orthogonal polynomials. The results verify a conjecture of A. Magnus for the recurrence coefficients.

From the paper :

``The weight $-\log x$ on $[0,1]$ corresponds to the case $k=1$ for which our analysis is not yet complete. The vanishing of the weight $\log\frac{2}{1-x}$ at the point $-1$ corresponds to a Fisher-Hartwig singularity for the related problem on the unit circle. This paper is in the line of questioning concerning the effect that singularities and zeroes in the measure have on the asymptotic behavior of orthogonal polynomials. The logarithmic singularities explored in this paper are of practical interest in both physics and mathematics.''

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