论文标题
在与部分权力序列相关的正交laurent多项式上
On orthogonal Laurent polynomials related to the partial sums of power series
论文作者
论文摘要
令$ f（z）= \ sum_ {k = 0}^\ infty d_k z^k $，$ d_k \ in \ mathbb {c} \ backslash \ {0 \} $，$ d_0 = 1 $，成为具有非零radius convergence $ covergence $ p $ρ$ρ$：$ 0 <$：$ 0 <<peq \ peq +peq +pec \ pe fe。用$ f_n（z）$表示$ f $的n-th-thth部分总和，$ r_ {2n}（z）= \ frac {f_ {f_ {2n}（z）} {z^n} $，$ r_ {2n+1}（z）（z）（z）= \ frac = \ frac {f _ {f_ {f_ {2n+1}} $ n = 0,1,2，... $。由于亨德里克森（Hendriksen）和范·罗苏姆（Van Rossum）的结果，在laurent polyenmials上存在线性函数$ \ mathbf {l} $，因此$ \ mathbf {l}（r_n r_m）= 0 $，当$ n \ not = m $时，而$ \ \ \ \ \ \ \ \ \ m m ive v {l_n^2）（r_n^2）我们在部分总和的上述情况下为$ \ mathbf {l} $提出了一个明确的积分表示形式。我们使用生成函数理论中的方法。也研究了这种lurent多项式的有限系统。
Let $f(z) = \sum_{k=0}^\infty d_k z^k$, $d_k\in\mathbb{C}\backslash\{ 0 \}$, $d_0=1$, be a power series with a non-zero radius of convergence $ρ$: $0 <ρ\leq +\infty$. Denote by $f_n(z)$ the n-th partial sum of $f$, and $R_{2n}(z) = \frac{ f_{2n}(z) }{ z^n }$, $R_{2n+1}(z) = \frac{ f_{2n+1}(z) }{ z^{n+1} }$, $n=0,1,2,...$. By the result of Hendriksen and Van Rossum there exists a linear functional $\mathbf{L}$ on Laurent polynomials, such that $\mathbf{L}(R_n R_m) = 0$, when $n\not= m$, while $\mathbf{L}(R_n^2)\not= 0$. We present an explicit integral representation for $\mathbf{L}$ in the above case of the partial sums. We use methods from the theory of generating functions. The case of finite systems of such Laurent polynomials is studied as well.
