An integral hidden in Gradshteyn and Ryzhik. (English) Zbl 0939.33007
The authors provide a closed-form expression for the integral
\[
\begin{aligned} N_{0,4}(a;m) & := \int^\infty_0 {dz\over (z^4+ 2az^2+ 1)^{m+1}},\quad \text{where }m\in \mathbb{N}\text{ and }a\in (-1,\infty):\\N_{0,4}(a;m) &= {\pi\over 2^{m+{3\over 2}}(a+ 1)^{m+{1\over 2}}} P^{(m+{1\over 2},-m-{1\over 2})}(a)\\ & = {\pi\over 2^{3m+{3\over 2}}(a+ 1)^{m+{1\over 2}}} \sum^m_{k= 0} 2^k {2m-2k\choose m-k}{m+k\choose m}(a+ 1)^k.\end{aligned}
\]
They have given the CPU table of \(N_{0,4}(4,m)\) by comparing CPU times of the direct calculation with the CPU time of their formula. There is a significance difference.
Reviewer: R.S.Dahiya (Ames)
MSC:
33C45 | Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.) |
Online Encyclopedia of Integer Sequences:
Triangle T(n,k) = d(n-k,n), 0 <= k <= n, where d(l,m) = Sum_{k=l..m} 2^k * binomial(2*m-2*k, m-k) * binomial(m+k, m) * binomial(k, l).Coefficients of a polynomial representation of the integral of 1/(x^4 + 2*a*x^2 + 1)^(n+1) from x = 0 to infinity.
Triangle T(n,k), read by rows, giving the numerator of the coefficient of x^k in the Boros-Moll polynomial P_n(x) for n >= 0 and 0 <= k <=n.
Triangle T(n,k), read by rows, giving the denominator of the coefficient of x^k in the Boros-Moll polynomial P_n(x) for n >= 0 and 0 <= k <= n.
References:
[1] | G. Boros, V. Moll, Landen Transformations and the Integration of Rational Functions, preprint.; G. Boros, V. Moll, Landen Transformations and the Integration of Rational Functions, preprint. · Zbl 0988.33009 |
[2] | Gradshteyn, I. S.; Ryzhik, I. M., (Jeffrey, A., Table of Integrals, Series and Products (1994), Academic Press: Academic Press New York) · Zbl 0918.65002 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.