We study rather general multiple zeta functions whose denominators are given by polynomials. The main aim is to prove explicit formulas for the values of those multiple zeta functions at non-positive integer points. We first treat the case when the polynomials are power sums, and observe that some “trivial zeros” exist. We also prove that special values are sometimes transcendental. Then we proceed to the general case, and show an explicit expression of special values at non-positive integer points which involves certain period integrals. We give examples of transcendental values of those special values or period integrals. We also mention certain relations among Bernoulli numbers which can be deduced from our explicit formulas. Our proof of explicit formulas are based on the Euler–Maclaurin summation formula, Mahler’s theorem, and a Raabe-type lemma due to Friedman and Pereira.

中文翻译：

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多项式分母为非正整数的多个 zeta 函数的值

我们研究了相当一般的多重 zeta 函数，其分母由多项式给出。主要目的是证明这些多个 zeta 函数在非正整数点处的值的明确公式。我们首先处理多项式是幂和的情况，并观察到存在一些“平凡零”。我们还证明了特殊价值有时是超越的。然后我们继续进行一般情况，并显示在非正整数点处特殊值的显式表达式，其中涉及某些周期积分。我们给出了那些特殊值或周期积分的超越值的例子。我们还提到了伯努利数之间的某些关系，这些关系可以从我们的显式公式中推导出来。我们对显式公式的证明基于欧拉-麦克劳林求和公式，马勒定理，