论文标题
关于算术进程的Borwein型多项式系数的总和
On sums of coefficients of Borwein type polynomials over arithmetic progressions
论文作者
论文摘要
我们获得了在形式多项式系数的算术上的总和的渐近公式 $$ \ prod_ {j = 1}^n \ prod_ {k = 1}^{p-1}（1-q^{pj-k}）^s，$$ 其中$ p $是一个奇怪的素数，而$ n，s $是正整数。让我们用$ a_i $表示上述多项式中$ q^i $的系数，并假设$ b $是整数。我们证明了这一点 $ \ big | \ sum_ {i \ equiv b \ \ text {mod} \ 2pn} a_i- \ frac {v（b）p^{sn}}} {2pn} {2pn} \ big | \ big | \ leq p^{sn/2}，$$ 其中$ v（b）= p-1 $如果$ b $由$ p $和$ v（b）= -1 $除外。这改善了Goswami和Pantangi的最新结果。
We obtain asymptotic formulas for sums over arithmetic progressions of coefficients of polynomials of the form $$\prod_{j=1}^n\prod_{k=1}^{p-1}(1-q^{pj-k})^s,$$ where $p$ is an odd prime and $n, s$ are positive integers. Let us denote by $a_i$ the coefficient of $q^i$ in the above polynomial and suppose that $b$ is an integer. We prove that $$\Big|\sum_{i\equiv b\ \text{mod}\ 2pn}a_i-\frac{v(b)p^{sn}}{2pn}\Big|\leq p^{sn/2},$$ where $v(b)=p-1$ if $b$ divisible by $p$ and $v(b)=-1$ otherwise. This improves a recent result of Goswami and Pantangi.
