History of Secondary polynomials of the Legendre polynomials $q_n$

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when	who	what
2026-09-10 09:20	bmatschke	the differential operator is upright; d is not a variable	current reviewed
2026-09-09 09:14	bmatschke	say it in sentences rather than in dashes
2026-09-04 14:24	bmatschke	who asked for a table is not a fact about the mathematics; the issue is answered in the issue
2026-09-02 20:36	bmatschke	restore the Laguerre clause, which went with the sentence about how a Hermite table ought to be normalised and left the comment ending on a comma
2026-09-02 16:53	bmatschke	the weights are the values divided by P_n there, not the values; and three remarks about this website rather than about the mathematics
2026-09-02 13:10	bmatschke	a comment states a fact about the mathematics; how strong a search hit would be is a remark about this website
2026-09-02 13:07	bmatschke	drop the sentence I added: the comment already derived the degree and the rationality, two clauses earlier
2026-09-02 13:07	bmatschke	the definition says what the object is; that it has degree n-1 and rational coefficients is a consequence, and belongs with the reasoning that gets there
2026-09-02 10:34	zeta3 with claude (agent run 20260902	secondary polynomials of the Legendre polynomials for n <= 50, from Bonnet's recurrence at q_0 = 0, q_1 = 2, each checked against the defining integral and against DLMF 14.7.3 before being sent
2026-09-02 10:34	zeta3	checking that this table can be written to
2026-09-02 10:33	zeta3 with claude (agent run 20260902T101602Z)	draft: secondary polynomials of the Legendre polynomials, proposal 6 of BATCH-2026-09-02

What changed between 2026-09-09 09:14 and 2026-09-10 09:20

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 Title: Secondary polynomials of the Legendre polynomials $q_n$ Definition: For $n\geq 0$ the secondary polynomial of the Legendre polynomial $P_n$-  is $q_n(x)=\int_{-1}^{1}\frac{P_n(t)-P_n(x)}{t-x}\,dt$ CITE{Wiki}, the integral-  taken against the density $1$ on $[-1,1]$.+  is $q_n(x)=\int_{-1}^{1}\frac{P_n(t)-P_n(x)}{t-x}\,\mathrm{d}t$ CITE{Wiki}, the+  integral taken against the density $1$ on $[-1,1]$. Parameters:   n:

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     degree $n$ with $P_n(1)=1$, orthogonal on $[-1,1]$ for the density $\rho(t)=1$.     The quotient $(P_n(t)-P_n(x))/(t-x)$ is a polynomial in $t$ and $x$, so $q_n$-    is a combination of the moments $\int_{-1}^{1}t^i\,dt$ and is rational; it has-    degree $n-1$ because $P_n$ has degree $n$.+    is a combination of the moments $\int_{-1}^{1}t^i\,\mathrm{d}t$ and is rational;+    it has degree $n-1$ because $P_n$ has degree $n$.   comment-density: CITE{Wiki} defines the secondary polynomials for "a density" and     fixes none. Taking $\rho=1$, the inner product the table of Legendre polynomials

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     for the secondary measure $d\mu=\dfrac{du}{\pi^2+\ln^2\frac{1+u}{1-u}}$ of the     density $1$ CITE{Wiki-measure} (given there on $[0,1]$ as $1/(\pi^2+\ln^2\frac{x}{1-x})$),-    and $\int_{-1}^{1}q_n^2\,d\mu=\frac{2}{2n+1}=\int_{-1}^{1}P_n^2\,dt$.+    and $\int_{-1}^{1}q_n^2\,\mathrm{d}\mu=\frac{2}{2n+1}=\int_{-1}^{1}P_n^2\,\mathrm{d}t$.   comment-names: 'Up to the index and a factor $2$ these are the associated polynomials     of order one, or numerator polynomials, of the Legendre family: $q_{n+1}/2$ has

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   formula-sum: $q_n(x)=2\sum_{k=1}^{n}\frac{1}{k}\,P_{k-1}(x)\,P_{n-k}(x)$ CITE{DLMF}.   formula-moments: If $P_n(x)=\sum_{j=0}^{n}c_j x^j$ then $q_n(x)=\sum_{j=1}^{n}c_j\sum_{i=0}^{j-1}m_i\,x^{j-1-i}$-    with $m_i=\int_{-1}^{1}t^i\,dt=\frac{1+(-1)^i}{i+1}$.+    with $m_i=\int_{-1}^{1}t^i\,\mathrm{d}t=\frac{1+(-1)^i}{i+1}$.   formula-second-kind: $Q_n(x)=\frac12P_n(x)\ln\frac{1+x}{1-x}-\frac12q_n(x)$ for     $-1<x<1$, and $Q_n(x)=\frac12P_n(x)\ln\frac{x+1}{x-1}-\frac12q_n(x)$ for $x>1$,