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Newman's identities, lucas sequences and congruences for certain partition functions
- Source :
- Proceedings of the Edinburgh Mathematical Society. 63:709-736
- Publication Year :
- 2020
- Publisher :
- Cambridge University Press (CUP), 2020.
-
Abstract
- 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, \[ \textrm{spt}\bigg( \frac{1991n(3n+1) }{2} +83\bigg) \equiv \textrm{spt}\bigg(\frac{1991n(3n+5)}{2} +2074\bigg) \equiv 0\ (\textrm{mod} \ 11), \] 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.
- Subjects :
- Lucas sequence
General Mathematics
010102 general mathematics
0102 computer and information sciences
Function (mathematics)
Congruence relation
01 natural sciences
Ramanujan's sum
Combinatorics
symbols.namesake
Integer
010201 computation theory & mathematics
symbols
Rank (graph theory)
0101 mathematics
Mathematics
Subjects
Details
- ISSN :
- 14643839 and 00130915
- Volume :
- 63
- Database :
- OpenAIRE
- Journal :
- Proceedings of the Edinburgh Mathematical Society
- Accession number :
- edsair.doi...........8b41f0dda7a3828282513074742c8e29