A Rial Generalization Of The Rogers-ramanujan Identities

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MR0123484 (23 #A809) 05.10 (10.48) Gordon, Basil A combinatorial generalization of the Rogers-Ramanujan identities. Amer. J. Math. 83 1961 393–399 The Rogers-Ramanujan identities [G. H. Hardy and E. M. Wright, An introduction to the theory of numbers, Clarendon, Oxford, 1960, p. 290; for review of 3rd ed. of 1954, see MR0067125 (16,673c)]: 2 ∞ ∞ Y X xn 5n+1 −1 5n+4 −1 , (1 − x ) (1 − x ) = 2 ) · · · (1 − xn ) (1 − x)(1 − x n=0 n=0 ∞ ∞ Y X 5n+2 −1 5n+3 −1 (1 − x ) (1 − x ) = 0

n=0

2

xn +n (1 − x)(1 − x2 ) · · · (1 − xn )

can be interpreted combinatorially as follows. For the first identity, the number of partitions of any integer N into parts not congruent to 0, ±2 (mod 5) is equal to the number of partitions of N = N1 + · · · + Nk with Ni ≥ Ni+1 + 2. For the second identity, the number of partitions of N into parts not congruent to 0, ±1 (mod 5) is equal to the number of partitions N = N1 + · · · + Nk with Ni ≥ Ni+1 + 2, Nk ≥ 2. In the present paper the following extension is proved. The number of partitions of N into parts not congruent to 0, ±t (mod 2d + 1), where 1 ≤ t ≤ d, is equal to the number of partitions of the form N = N1 + · · · + Nk with Ni ≥ Ni+1 , Ni ≥ Ni+d−1 + 2 and Nk−t+1 ≥ 2. The method of proof is similar to Schur’s proof of the Rogers-Ramanujan identities [I. Schur, S.-B. Deutsch. Akad. Wiss. Berlin 1917, 302–321]. Reviewed by L. Carlitz c Copyright American Mathematical Society 1962, 2007