梅森 - 勒尔奇插值值与阿波斯托尔 - 梅森多项式族
Mersenne-Lerch Interpolation Values and Apostol-Mersenne Polynomial Families
浏览论文内容
中文总结 AI 辅助
该论文研究与阿波斯托尔 - 梅森多项式族相关的梅森 - 勒尔奇插值族，基于梅森平移多项式构造，得到多项式显式公式及相关性质，与\(q\) - 演算比较得出一些关系，如令\(q = 2\)可得相关量。
中文摘要 AI 辅助
本文介绍了与阿波斯托尔 - 梅森多项式族相关的伯努利型和欧拉型梅森 - 勒尔奇插值族。其构造基于梅森平移多项式\(P_{n,M}(x;m)\)，给出了这些多项式的显式公式，包括用第二类M - 斯特林数展开。在非正整数处，所得插值值恢复了\(r\)阶阿波斯托尔 - 梅森 - 伯努利和阿波斯托尔 - 梅森 - 欧拉多项式，还得到了这些值的导数、积分、加法和差分公式。与\(q\) - 演算比较表明，通过在相应\(q\) - 演算表达式中令\(q = 2\)可得到M - 阶乘、M - 二项式系数、M - 导数和M - 指数。
英文摘要
This paper introduces Bernoulli-type and Euler-type Mersenne-Lerch interpolation families associated with Apostol-Mersenne polynomial families. Their construction is based on the Mersenne translation polynomials \(P_{n,M}(x;m)\), defined by \[ e_M^{xt}(e_M^t)^m = \sum_{n=0}^{\infty}P_{n,M}(x;m)\frac{t^n}{M_n!}. \] Explicit formulas for these polynomials are derived, including an expansion in terms of M-Stirling numbers of the second kind. At nonpositive integers, the resulting interpolation values recover the Apostol-Mersenne-Bernoulli and Apostol-Mersenne-Euler polynomials of order \(r\). Derivative, integral, addition, and difference formulas are also obtained for these values. A comparison with \(q\)-calculus shows that the M-factorial, M-binomial coefficient, M-derivative, and M-exponential are obtained by setting \(q=2\) in the corresponding \(q\)-calculus expressions.
