The Ramanujan Journal ( IF 0.7 ) Pub Date : 2020-07-22 , DOI: 10.1007/s11139-020-00275-w James Mc Laughlin
Motivated by results of Hirschhorn, Tang, and Baruah and Kaur on vanishing coefficients (in arithmetic progressions) in a new class of infinite product which have appeared recently, we further examine such infinite products, and find that many such results on vanishing coefficients may be grouped into families. For example, one result proved in the present paper is that if \(b\in \{1,2,\dots , 9, 10\}\) and the sequence \(\{r_n\}\) is defined by
$$\begin{aligned} (q^{8b},q^{11-8b};q^{11})_{\infty }^3 (q^{11-b},q^{11+b};q^{22})_{\infty } =:\sum _{n=-756}^{\infty }r_nq^n, \end{aligned}$$then \(r_{11n+6b^2+b}=0\) for all n. Further, if \(b\in \{1,3,5,7,9\}\), then \(r_{11n+4b^2+b}=0\) for all n also. Each particular value of b gives a specific result, such as the following (for \(b=1\)): if the sequences \(\{a_n\}\) is defined by
$$\begin{aligned} \sum _{n=0}^{\infty } a_n q^n := (q^{3},q^{8} ;q^{11} )_{\infty }^3 (q^{10},q^{12} ;q^{22})_{\infty }, \end{aligned}$$then \(a_{11n+5}=a_{11n+7}=0\).
中文翻译:
系数消失的新的无限q乘积展开
根据最近出现的一类新型无限产品的Hirschhorn,Tang,Baruah和Kaur关于消失系数的结果(以算术级数表示),我们进一步研究了这种无限积,发现许多关于消失系数的结果可能是归为一类。例如,本文证明的一个结果是,如果\(b \ in \ {1,2,\ dots,9,10 \} \)和序列\(\ {r_n \} \)由
$$ \ begin {aligned}(q ^ {8b},q ^ {11-8b}; q ^ {11})_ {\ infty} ^ 3(q ^ {11-b},q ^ {11 + b }; q ^ {22})_ {\ infty} =:\ sum _ {n = -756} ^ {\ infty} r_nq ^ n,\ end {aligned} $$那么\(r_ {11n + 6b ^ 2 + b} = 0 \)对于所有n。此外,如果\(b \ in \ {1,3,5,7,9 \} \)中的\ n,则\(r_ {11n + 4b ^ 2 + b} = 0 \)也为所有n。b的每个特定值都会给出特定的结果,例如以下内容(对于\(b = 1 \)):如果序列\(\ {a_n \} \)由以下项定义
$$ \ begin {aligned} \ sum _ {n = 0} ^ {\ infty} a_n q ^ n:=(q ^ {3},q ^ {8}; q ^ {11})_ {\ infty} ^ 3(q ^ {10},q ^ {12}; q ^ {22})_ {\ infty},\ end {aligned} $$然后\(a_ {11n + 5} = a_ {11n + 7} = 0 \)。