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Discriminants of classical quasi-orthogonal polynomials with application to Diophantine equations. (English) Zbl 1429.33016

The authors give explicit formulae for the discriminant of classical quasi-orthogonal polynomials, generalizing the result given by K. Dilcher and K. B. Stolarsky for the quasi-orthogonal Chebyshev polynomials of the second kind [Trans. Am. Math. Soc. 357, No. 3, 965–981 (2005; Zbl 1067.12001)].
The main results are:
Theorem 3.1. (quasi-Jacobi polynomials) Let \(c\) be a constant and let \(P_{n,c}^{(\alpha,\beta)}(x) = P_{n}^{(\alpha,\beta)}(x)+cP_{n-1}^{(\alpha,\beta)}(x)\). Then \[ \begin{split} \hbox{disc}(P_{n,c}^{(\alpha,\beta)})=&\frac{(2n+\alpha+\beta)^{2n+1}}{2^{n(n-1)}} \prod_{k=1}^n k^{k-2n+3} \cdot\prod_{k=1}^{n-1}(k+\alpha)^{k-1}(k+\beta)^{k-1}(n+k+\alpha+\beta)^{n-k-1}\\ &\cdot\frac{(-c)^nP_{n,c}^{(\alpha,\beta)}(-(2n(n+\alpha+\beta)c^2 + (\alpha^2-\beta^2)c+2(n+\alpha)(n+\beta))/(2n+\alpha+\beta))^2c)}{(n+\alpha+cn)(n+\beta-cn)}.\end{split} \] Furthermore, \(\hbox{disc}(P_{n,c}^{(\alpha,\beta)}\) is a polynomial in \(c\) of degree \(2(n-1)\).
Theorem 3.6 (quasi-Laguerre polynomials) Let \(c\) be a constant and let \(L_{n,c}^{\alpha)}(x)= L_{n}^{\alpha)}(x)+cL_{n-1}^{\alpha)}(x)\). Then \[ \hbox{disc}(L_{n,c}^{\alpha)}= \frac{1}{n+\alpha+cn}\prod_{k=1}^n (k+\alpha)^{k-1}\\ \cdot (-1)^nL_{n,c}^{\alpha)}\left(\frac{nc^2+(2n+\alpha)c+n+\alpha)}{c}\right). \]
Theorem 3.8 (quasi-Hermite polynomials) Let \(c\) be a constant and let \(H_{n,c}(x)= H_{n}(x)+cH_{n-1}(x)\). Then \[\hbox{disc}(H_{n,c})=2^{n(3n-5)/2}\prod_{k=1}^{n-1} k^k \cdot H_{n,c}\left(-\frac{c^2+2n}{2c}\right).\]
Here the discriminant of a polynomial \(f(x)=a_0x^n+\cdots+a_n\) of degree \(n\), is defined as \[\hbox{disc}(f)=\frac{(-1)^{n(n-1)/2}}{a_0}\hbox{Res}(f,f'),\] where the resultant of two polynomials \(f(x)=a_0x^n+\cdots+a_n\) resp. \(g(x)=b_0x^m+\cdots+b_m\) of fegree \(n\) resp. \(m\) follows from the order \((n+m)\times (n+m)\) determinant where the rows, each with \(n+m\) entries, are given by the coefficients of the polynomials
\[\left|\begin{matrix} a_0 & a_1 & \ldots & a_n & & \cr & \ddots & & & \ddots & \cr & & a_0 & a_1 &\ldots a_n \cr b_0 & b_1 & \ldots & b_m & & \cr & \ddots & & & \ddots & \cr & & b_0 & b_1 & \ldots & b_m \cr \end{matrix}\right|.\] After giving background and an historical overview, the proofs follow. Moreover, the authors give a section of \(8\) pages with the title Applications:
– a generalization of the Hausdorff equations \[\sum_{i=1}^{2r+1}x_iy_i^j=\frac{1}{\sqrt{\pi}}\int_{-\infty}^{\infty}\,t^j e^{-t^2}dt,\ j=0,1,\ldots,2r+1)\] has rational solutions \(x_i,\,y_i\ (1\leq i\leq 2r+1)\),
– Hausdorff type equations for rational solutions,
– results of the solutions of these afore mentioned solutions to be the zeros of quasi-orthogonal polynomials or not,
– existence of rational points on certain curves generated by the discriminant.
An interesting paper, covering quite some ground.

MSC:

33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)
05E99 Algebraic combinatorics
65D32 Numerical quadrature and cubature formulas
12E10 Special polynomials in general fields
11E76 Forms of degree higher than two

Citations:

Zbl 1067.12001

Software:

Magma

References:

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