论文标题
分配给算术函数的多项式系数的公式
Formulas for coefficients of polynomials assigned to arithmetic functions
论文作者
论文摘要
我们附着在归一化的（非变化）算术函数$ g $和$ h $递归定义的多项式功能上。令$ p_0^{g，h}（x）：= 1 $。然后\ begin {qore} p_n^{g，h}（x）：= \ frac {x} {h（n）} \ sum_ {k = 1}^{n}^{n} g（k）\，p_ {n-k}对于特殊的$ g $和$ h $，我们获得了d'arcais polyenmials，\ end {equation}等于dedekind $η$ - 功能的$ -z $ th powers的系数，并且也由nekrasov and okounkov作为挂钩长度公式。示例由Pochhammer多项式，第二类的Chebyshev多项式以及相关的Laguerre多项式提供。我们提出了$ p_n^{g，h}（x）$系数的显式公式和身份，该系数将$ g $和$ h $的影响分开。最后，我们提供了多个应用程序。
We attach to normalized (non-vanishing) arithmetic functions $g$ and $h$ recursively defined polynomials. Let $P_0^{g,h}(x):=1$. Then \begin{equation} P_n^{g,h}(x) := \frac{x}{h(n)} \sum_{k=1}^{n} g(k) \, P_{n-k}^{g,h}(x). \end{equation} For special $g$ and $h$, we obtain the D'Arcais polynomials, which are equal to the coefficients of the $-z$th powers of the Dedekind $η$-function and are also given by Nekrasov and Okounkov as a hook length formula. Examples are offered by Pochhammer polynomials, Chebyshev polynomials of the second kind, and associated Laguerre polynomials. We present explicit formulas and identities for the coefficients of $P_n^{g,h}(x)$ which separate the impact of $g$ and $h$. Finally, we provide several applications.
