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Newman's identities, lucas sequences and congruences for certain partition functions
- Part of
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- Journal:
- Proceedings of the Edinburgh Mathematical Society / Volume 63 / Issue 3 / August 2020
- Published online by Cambridge University Press:
- 08 October 2020, pp. 709-736
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- Article
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Let r be an integer with 2 ≤ r ≤ 24 and let pr(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 pr(n) by using some identities on pr(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,
and for k ≥ 0,
\[ \textrm{spt}\bigg( \frac{1991n(3n+1) }{2} +83\bigg) \equiv \textrm{spt}\bigg(\frac{1991n(3n+5)}{2} +2074\bigg) \equiv 0\ (\textrm{mod} \ 11), \]where spt(n) denotes Andrews's smallest parts function.
\[ \textrm{spt}\bigg( \frac{143\times 5^{6k} +1 }{24}\bigg)\equiv 2^{k+2} \ (\textrm{mod}\ 11), \]
