论文标题
关于“外来积分”的注释
A note on "exotic integrals"
论文作者
论文摘要
我们考虑伯努利（Bernoulli）的测量$μ_p$在间隔$ [0,1] $上。对于标准Lebesgue，数字$ 0 $和$ 1 $的实际数字表示，同等概率$ 1/2 $。对于Bernoulli措施，数字$ 0 $和$ 1 $的概率分别为$ P $和$ 1-P $。我们为各种$μ_p$ integrals提供明确的表达式。特别是，多项式的积分是根据特殊Hessenberg矩阵的决定因素表示的，而特殊的Hessenberg矩阵的决定因素，而这些矩阵又是由二项式系数的Pascal矩阵构造的。这使我们可以在Legendre多项式基础上找到$μ_p$的傅立叶系数的封闭形式表达式。同时，三角傅立叶系数是一些特殊的整个功能的值，该功能接收了显式的无限乘积扩展，并满足了有趣的属性，包括与Stirling数字和Polygarithm的连接。
We consider Bernoulli measures $μ_p$ on the interval $[0,1]$. For the standard Lebesgue measure the digits $0$ and $1$ in the binary representation of real numbers appear with an equal probability $1/2$. For the Bernoulli measures, the digits $0$ and $1$ appear with probabilities $p$ and $1-p$, respectively. We provide explicit expressions for various $μ_p$-integrals. In particular, integrals of polynomials are expressed in terms of the determinants of special Hessenberg matrices, which, in turn, are constructed from the Pascal matrices of binomial coefficients. This allows us to find closed-form expressions for the Fourier coefficients of $μ_p$ in the Legendre polynomial basis. At the same time, the trigonometric Fourier coefficients are values of some special entire function, which admits an explicit infinite product expansion and satisfies interesting properties, including connections with the Stirling numbers and the polylogarithm.
