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arXiv 2408.152122024-08-27math.CA

$x^m (-\log x)^l$ 在区间 $0\le x \le 1$ 内的 Chebyshev 逼近

Chebyshev approximation of $x^m (-\log x)^l$ in the interval $0\le x \le 1$

Richard J. Mathar

备注：: 9 pages, no figures. Integrals table in the anc directory

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AI总结：

本文研究了 $x^m (-\log x)^l$ 关于平移 Chebyshev 多项式的级数展开，通过分部积分将所需积分族化简为已知可由 polygamma 函数有限和表示的 $m=0$ 积分，解决了相关计算问题。

摘要：

$x^m (-\log x)^l$ 关于平移 Chebyshev 多项式 $T_n^*(x)$ 的级数展开需计算积分族 $\int_0^1 x^m (-\log x)^l dx / \sqrt{x-x^2}$。我们证明可通过分部积分将其化为指数 $m=0$ 的积分和，后者已知可表示为 polygamma 函数的有限和。

英文摘要：

The series expansion of $x^m (-\log x)^l$ in terms of the shifted Chebyshev Polynomials $T_n^*(x)$ requires evaluation of the integral family $\int_0^1 x^m (-\log x)^l dx / \sqrt{x-x^2}$. We demonstrate that these can be reduced by partial integration to sums over integrals with exponent $m=0$ which have known representations as finite sums over polygamma functions.

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