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Newman's identities, lucas sequences and congruences for certain partition functions
Proceedings of the Edinburgh Mathematical Society ( IF 0.7 ) Pub Date : 2020-10-08 , DOI: 10.1017/s0013091520000115 Ernest X.W. Xia
Proceedings of the Edinburgh Mathematical Society ( IF 0.7 ) Pub Date : 2020-10-08 , DOI: 10.1017/s0013091520000115 Ernest X.W. Xia
Let r be an integer with 2 ≤ r ≤ 24 and let p r (n ) be defined by $\sum _{n=0}^\infty p_r(n) q^n = \prod _{k=1}^\infty (1-q^k)^r$ . In this paper, we provide uniform methods for discovering infinite families of congruences and strange congruences for p r (n ) by using some identities on p r (n ) due to Newman. As applications, we establish many infinite families of congruences and strange congruences for certain partition functions, such as Andrews's smallest parts function, the coefficients of Ramanujan's ϕ function and p -regular partition functions. For example, we prove that for n ≥ 0, \[ \textrm{spt}\bigg( \frac{1991n(3n+1) }{2} +83\bigg) \equiv \textrm{spt}\bigg(\frac{1991n(3n+5)}{2} +2074\bigg) \equiv 0\ (\textrm{mod} \ 11), \] and for k ≥ 0, \[ \textrm{spt}\bigg( \frac{143\times 5^{6k} +1 }{24}\bigg)\equiv 2^{k+2} \ (\textrm{mod}\ 11), \] where spt(n ) denotes Andrews's smallest parts function.
中文翻译:
纽曼恒等式、卢卡斯序列和某些配分函数的同余
让r 为 2 ≤ 的整数r ≤ 24 并让p r (n ) 定义为$\sum _{n=0}^\infty p_r(n) q^n = \prod _{k=1}^\infty (1-q^k)^r$ . 在本文中,我们提供了发现无限同余族和奇异同余的统一方法p r (n ) 通过使用一些身份p r (n ) 由于纽曼。作为应用,我们为某些配分函数建立了许多无限的同余族和奇异同余族,例如 Andrews 的最小部分函数、Ramanujan 的系数φ 功能和p - 正则分区函数。例如,我们证明对于n ≥0,\[ \textrm{spt}\bigg( \frac{1991n(3n+1) }{2} +83\bigg) \equiv \textrm{spt}\bigg(\frac{1991n(3n+5)}{2 } +2074\bigg) \equiv 0\ (\textrm{mod} \ 11), \] 并且对于ķ ≥0,\[ \textrm{spt}\bigg( \frac{143\times 5^{6k} +1 }{24}\bigg)\equiv 2^{k+2} \ (\textrm{mod}\ 11), \] 其中 spt(n ) 表示安德鲁斯的最小部分函数。
更新日期:2020-10-08
中文翻译:
纽曼恒等式、卢卡斯序列和某些配分函数的同余
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