Numerical investigation of certain asymptotic results in the theory of partitions

Authors:
M. S. Cheema and W. E. Conway

Journal:
Math. Comp. **26** (1972), 999-1005

MSC:
Primary 10A45; Secondary 10-04, 10J20

DOI:
https://doi.org/10.1090/S0025-5718-1972-0314756-0

MathSciNet review:
0314756

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: A numerical investigation of some of the asymptotic formulas in partitions is made. Comparisons with actual computed values show that in certain cases only the relative error tends to zero and the errors are significant. Only in the case of is it found that only a few terms of the asymptotic series are sufficient to obtain the exact value.

**[1]**S. K. Agarwal & J. M. Gandhi, ``Generalization of a certain function related to the partition function.'' Private communication.**[2]**M. S. Cheema & C. T. Haskell, ``Multirestricted and rowed partitions,''*Duke Math. J.*, v. 34, 1967, pp. 443-451. MR**36**#125. MR**0217030 (36:125)****[3]**J. M. Gandhi, ``Generalization of a certain function related to the partition function,''*Mathematica (Cluj)*, v. 11 (34), 1969, pp. 245-251. MR**41**#5316. MR**0260692 (41:5316)****[4]**B. Gordon & L. Houten, ``Notes on plane partitions. III,''*Duke Math. J.*, v. 36, 1969, pp. 801-824. MR**40**#1358. MR**0248104 (40:1358)****[5]**B. Gordon & L. Houten, ``Notes on plane partitions,''*J. Combinatorial Theory*, v. 4, 1968, pp. 72-99. MR**36**#1339. MR**0218252 (36:1339)****[6]**G. H. Hardy & S. Ramanujan, ``Asymptotic formulae in combinatorial analysis,''*Proc. London Math. Soc.*, v. 17, 1918, pp. 75-115.**[7]**P. A. MacMahon,*Combinatory Analysis*. Vols. I, II, Chelsea, New York, 1960. MR**25**#5003. MR**0141605 (25:5003)****[8]**D. H. Lehmer, ``The series for the partition function,''*Trans. Amer. Math. Soc.*, v. 43, 1938, pp. 271-295. MR**1501943****[9]**Hans Rademacher, ``On the Selberg formula for ,''*J. Indian Math. Soc.*, v. 21, 1957, pp. 41-55. MR**19**, 1163. MR**0092818 (19:1163a)****[10]**A. L. Whiteman, ``A sum connected with the series for the partition function,''*Pacific J. Math.*, v. 6, 1956, pp. 159-176. MR**18**, 195. MR**0080122 (18:195b)****[11]**A. L. Whiteman, ``A sum connected with the partition function,''*Bull. Amer. Math. Soc.*, v. 53, 1947, pp. 598-603. MR**8**, 567. MR**0020590 (8:567a)****[12]**E. M. Wright, ``Asymptotic partition formula. I. Plane partitions,''*Quart. J. Math.*, v. 2, 1931, pp. 177-189.

Retrieve articles in *Mathematics of Computation*
with MSC:
10A45,
10-04,
10J20

Retrieve articles in all journals with MSC: 10A45, 10-04, 10J20

Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1972-0314756-0

Keywords:
Generating functios,
-line partitions,
plane partitions,
asymptotic formulas,
modular functions,
relative errors,
Bernoulli numbers

Article copyright:
© Copyright 1972
American Mathematical Society