New weighted Rogers-Ramanujan partition theorems and their implications

Authors:
Krishnaswami Alladi and Alexander Berkovich

Journal:
Trans. Amer. Math. Soc. **354** (2002), 2557-2577

MSC (2000):
Primary 11P83, 11P81; Secondary 05A19

Published electronically:
March 11, 2002

MathSciNet review:
1895193

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Abstract: This paper has a two-fold purpose. First, by considering a reformulation of a deep theorem of Göllnitz, we obtain a new weighted partition identity involving the Rogers-Ramanujan partitions, namely, partitions into parts differing by at least two. Consequences of this include Jacobi's celebrated triple product identity for theta functions, Sylvester's famous refinement of Euler's theorem, as well as certain weighted partition identities. Next, by studying partitions with prescribed bounds on successive ranks and replacing these with weighted Rogers-Ramanujan partitions, we obtain two new sets of theorems - a set of three theorems involving partitions into parts (mod 6), and a set of three theorems involving partitions into parts (mod 7), .

**1.**Krishnaswami Alladi,*A combinatorial correspondence related to Göllnitz’ (big) partition theorem and applications*, Trans. Amer. Math. Soc.**349**(1997), no. 7, 2721–2735. MR**1422593**, 10.1090/S0002-9947-97-01944-2**2.**Krishnaswami Alladi,*Partition identities involving gaps and weights*, Trans. Amer. Math. Soc.**349**(1997), no. 12, 5001–5019. MR**1401759**, 10.1090/S0002-9947-97-01831-X**3.**Krishnaswami Alladi,*Partition identities involving gaps and weights. II*, Ramanujan J.**2**(1998), no. 1-2, 21–37. Paul Erdős (1913–1996). MR**1642869**, 10.1023/A:1009749606406**4.**Krishnaswami Alladi,*On a partition theorem of Göllnitz and quartic transformations*, J. Number Theory**69**(1998), no. 2, 153–180. With an appendix by Basil Gordon. MR**1617309**, 10.1006/jnth.1997.2215**5.**Krishnaswami Alladi, George E. Andrews, and Basil Gordon,*Generalizations and refinements of a partition theorem of Göllnitz*, J. Reine Angew. Math.**460**(1995), 165–188. MR**1316576****6.**K. Alladi and A. Berkovich,*A double bounded key identity for Göllnitz's (Big) partition theorem*, in Symbolic Computation, Number Theory, Special Functions, Physics, and Combinatorics, Frank Garvan and Mourad Ismail, Eds., Developments in Mathematics Vol. 4, Kluwer Academic Publishers, Dordrecht (2001), 13-32.(see also CO/0007001)**7.**K. Alladi and A. Berkovich,*New finite versions of Jacobi's triple product, Sylvester, and Lebesgue identities (in preparation)*.**8.**George E. Andrews,*Sieves in the theory of partitions*, Amer. J. Math.**94**(1972), 1214–1230. MR**0319883****9.**George E. Andrews,*The theory of partitions*, Addison-Wesley Publishing Co., Reading, Mass.-London-Amsterdam, 1976. Encyclopedia of Mathematics and its Applications, Vol. 2. MR**0557013****10.**George E. Andrews, R. J. Baxter, D. M. Bressoud, W. H. Burge, P. J. Forrester, and G. Viennot,*Partitions with prescribed hook differences*, European J. Combin.**8**(1987), no. 4, 341–350. MR**930170**, 10.1016/S0195-6698(87)80041-0**11.**A. O. L. Atkin,*A note on ranks and conjugacy of partitions*, Quart. J. Math. Oxford Ser. (2)**17**(1966), 335–338. MR**0202688****12.**A. O. L. Atkin and P. Swinnerton-Dyer,*Some properties of partitions*, Proc. London Math. Soc. (3)**4**(1954), 84–106. MR**0060535****13.**David M. Bressoud,*Extension of the partition sieve*, J. Number Theory**12**(1980), no. 1, 87–100. MR**566873**, 10.1016/0022-314X(80)90077-3**14.**F. J. Dyson,*Some guesses in the theory of partitions*, Eureka**8**(1944), 10-15.**15.**H. Göllnitz,*Partitionen mit Differenzenbedingungen*, J. Reine Angew. Math.**225**(1967), 154–190 (German). MR**0211973****16.**Basil Gordon,*A combinatorial generalization of the Rogers-Ramanujan identities*, Amer. J. Math.**83**(1961), 393–399. MR**0123484****17.**Ronald L. Graham, Donald E. Knuth, and Oren Patashnik,*Concrete mathematics*, 2nd ed., Addison-Wesley Publishing Company, Reading, MA, 1994. A foundation for computer science. MR**1397498****18.**J. J. Sylvester,*A constructive theory of partitions in three Acts--an Interact, and an Exodium*, Amer. J. Math.**5**(1882), 251-330.

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Additional Information

**Krishnaswami Alladi**

Affiliation:
Department of Mathematics, University of Florida, Gainesville, Florida 32611

Email:
alladi@math.ufl.edu

**Alexander Berkovich**

Affiliation:
Department of Mathematics, University of Florida, Gainesville, Florida 32611

Email:
alexb@math.ufl.edu

DOI:
https://doi.org/10.1090/S0002-9947-02-02977-X

Keywords:
G\"{o}llnitz theorem,
Rogers-Ramanujan partitions,
method of weighted words,
Jacobi triple product identity,
Sylvester's theorem,
weighted partition identities,
successive ranks

Received by editor(s):
September 1, 2001

Published electronically:
March 11, 2002

Additional Notes:
Research of the first author supported in part by the National Science Foundation Grant DMS 0088975

Research of the second author supported in part by a University of Florida CLAS Research Award

Article copyright:
© Copyright 2002
American Mathematical Society