Partitions of large multipartites with congruence conditions. I
HTML articles powered by AMS MathViewer
- by M. M. Robertson and D. Spencer PDF
- Trans. Amer. Math. Soc. 219 (1976), 299-322 Request permission
Abstract:
Let $p({n_1}, \ldots ,{n_j}:{A_1}, \ldots ,{A_j})$ be the number of partitions of $({n_1}, \ldots ,{n_j})$ where, for $1 \leqslant l \leqslant j$, the lth component of each part belongs to the set ${A_l} = \bigcup \nolimits _{h(l) = 1}^{q(l)} {\{ {a_{lh(l)}} + Mv :v = 0,1,2, \ldots \} }$ and $M,q(l)$ and the ${a_{lh(l)}}$ are positive integers such that $0 < {a_{l1}} < \cdots < {a_{lq(l)}} \leqslant M$. Asymptotic expansions for $p({n_1}, \ldots ,{n_j}:{A_1}, \ldots ,{A_j})$ are derived, when the ${n_l} \to \infty$ subject to the restriction that ${n_1} \cdots {n_j} \leqslant n_l^{j + 1 - \in }$ for all l, where $\in$ is any fixed positive number. The case $M = 1$ and arbitrary j was investigated by Robertson [10] while several authors between 1940 and 1960 investigated the case $j = 1$ for different values of M.References
- Emil Grosswald, Some theorems concerning partitions, Trans. Amer. Math. Soc. 89 (1958), 113โ128. MR 97371, DOI 10.1090/S0002-9947-1958-0097371-5
- Peter Hagis Jr., A problem on partitions with a prime modulus $p\geq 3$, Trans. Amer. Math. Soc. 102 (1962), 30โ62. MR 146166, DOI 10.1090/S0002-9947-1962-0146166-3
- Shรด Iseki, A partition function with some congruence condition, Amer. J. Math. 81 (1959), 939โ961. MR 108473, DOI 10.2307/2372997 K. Knopp, Theory and application of infinite series, Blackie, London and Glasgow, 1928.
- Joseph Lehner, A partition function connected with the modulus five, Duke Math. J. 8 (1941), 631โ655. MR 5523
- John Livingood, A partition function with the prime modulus $P>3$, Amer. J. Math. 67 (1945), 194โ208. MR 12101, DOI 10.2307/2371722
- Ivan Niven, On a certain partition function, Amer. J. Math. 62 (1940), 353โ364. MR 1235, DOI 10.2307/2371459
- Harsh Anand Passi, An asymptotic formula in partition theory, Duke Math. J. 38 (1971), 327โ337. MR 285504 H. Rademacher, On the partition function $p(n)$, Proc. London Math. Soc. (2) 43 (1937), 241-254.
- M. M. Robertson, Partitions of large multipartites, Amer. J. Math. 84 (1962), 16โ34. MR 140499, DOI 10.2307/2372803
- M. M. Robertson, The evaluation of a definite integral which occurs in asymptotic partition theory, Proc. Roy. Soc. Edinburgh Sect. A 65 (1960/61), 283โ309 (1960/61). MR 141649
- E. C. Titchmarsh, The Theory of the Riemann Zeta-Function, Oxford, at the Clarendon Press, 1951. MR 0046485
- E. M. Wright, Partitions of large bipartites, Amer. J. Math. 80 (1958), 643โ658. MR 96628, DOI 10.2307/2372777
- E. M. Wright, The asymptotic behaviour of the generating functions of partitions of multi-partites, Quart. J. Math. Oxford Ser. (2) 10 (1959), 60โ69. MR 104633, DOI 10.1093/qmath/10.1.60
Additional Information
- © Copyright 1976 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 219 (1976), 299-322
- MSC: Primary 10J20
- DOI: https://doi.org/10.1090/S0002-9947-1976-0401688-2
- MathSciNet review: 0401688