Some new observations on the Göllnitz-Gordon and Rogers-Ramanujan identities

Alladi, Krishnaswami

doi:10.1090/S0002-9947-1995-1284910-4

Some new observations on the Göllnitz-Gordon and Rogers-Ramanujan identities
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Trans. Amer. Math. Soc. 347 (1995), 897-914 Request permission

Abstract:

Two new, short and elementary proofs of the Göllnitz-Gordon identities are presented by considering the odd-even split of the Euler Pentagonal Series and the Triangular Series of Gauss. Using this approach the equality of certain shifted partition functions are established. Next, the odd and even parts of the famous Rogers-Ramanujan series are shown to have interesting product representations ($\bmod 80$). From this, new shifted partition identities are derived.

References

Krishnaswami Alladi, On the modified convergence of some continued fractions of Rogers-Ramanujan type, J. Combin. Theory Ser. A 65 (1994), no. 2, 214–245. MR 1268339, DOI 10.1016/0097-3165(94)90021-3
Krishnaswami Alladi, The quintuple product identity and shifted partition functions, J. Comput. Appl. Math. 68 (1996), no. 1-2, 3–13. MR 1418747, DOI 10.1016/0377-0427(95)00251-0
George E. Andrews, Partition identities, Advances in Math. 9 (1972), 10–51. MR 306105, DOI 10.1016/0001-8708(72)90028-X
George E. Andrews, The theory of partitions, Encyclopedia of Mathematics and its Applications, Vol. 2, Addison-Wesley Publishing Co., Reading, Mass.-London-Amsterdam, 1976. MR 0557013
George E. Andrews, Unsolved Problems: Further Problems on Partitions, Amer. Math. Monthly 94 (1987), no. 5, 437–439. MR 1541095, DOI 10.2307/2322727
George E. Andrews, Euler’s “exemplum memorabile inductionis fallacis” and $q$-trinomial coefficients, J. Amer. Math. Soc. 3 (1990), no. 3, 653–669. MR 1040390, DOI 10.1090/S0894-0347-1990-1040390-4
H. Göllnitz, Partitionen mit Differenzenbedingungen, J. Reine Angew. Math. 225 (1967), 154–190 (German). MR 211973, DOI 10.1515/crll.1967.225.154
Basil Gordon, Some continued fractions of the Rogers-Ramanujan type, Duke Math. J. 32 (1965), 741–748. MR 184001
Anjali Kalvade, Equality of shifted partition functions, J. Indian Math. Soc. (N.S.) 54 (1989), no. 1-4, 155–164. MR 1069268

Second memoir on the expansion of certain infinite products

25

L. J. Slater, Further identities of the Rogers-Ramanujan type, Proc. London Math. Soc. (2) 54 (1952), 147–167. MR 49225, DOI 10.1112/plms/s2-54.2.147

Proof of certain identities in combinatory analysis

20