Bijective proofs of basic hypergeometric series identities

Abstract

Bijections are given which prove the following theorems: the ^-binomial theorem, Heine's 2 Φχ transformation, the g-analogues of Gauss', Rummer's, and Saalschϋtz's theorems, the very well poised 4 Φ 3 and 6 Φ 5 evaluations, and Watson's transformation of an 8 Φ 7 to a 4 Φ 3 . The proofs hold for all values of the parameters. Bijective proofs of the terminating cases follow from the general case. A bijective version of limiting cases of these series is also given. The technique is to mimic the classical proofs, based upon a bijective proof of the ^-binomial theorem and sign-reversing involutions which cancel infinite products. Introduction.In 1969 George Andrews [1] began to develop a calculus for partition functions. His stated goal was to "... translate a sizable portion of the techniques of the elementary theory of basic hypergeometric series into arithmetic terms". Ideally, he wanted to prove any theorem in basic hypergeometric series by a bijection. In this paper we show (under certain requirements) that this can be accomplished.Andrews' main object was to give bijective proofs of partition theorems (such as the Rogers-Ramanujan identities). It was well-known that these theorems were closely related to basic hypergeometric series. If a bijective proof of a partition theorem were desired, could one possibly give a bijective proof of a related basic hypergeometric series? If each step in the manipulation of a basic hypergeometric series could be inteφreted bijectively, the result would be a bijective proof of the partition theorem.Andrews gave a bijective proof of the ^-binomial theorem, which is the cornerstone of basic hypergeometric series. He also showed how to combinatorially interpret cancellations of infinite products, a manipulation of basic hypergeometric series which occurs frequently. However, he used the principle of inclusion-exclusion which, strictly speaking, is not bijective because it cancels objects in clumps. From the recent work of Gessel-Viennot [13] and Garsia-Milne [11] the appropriate bijective replacement for the principle of inclusion-exclusion is a sign-reversing involution. So this part of the theory of basic hypergeometric series can now be done bijectively, which is the main purpose of this paper.

Overpartitions

“…The mapping above (with b = 1) was considered in [18], where it was noted that by summing over the nonnegative integers k, we obtain the q-binomial theorem…”

“…Since the overlined parts form a partition into distinct parts and the non-overlined parts form an ordinary partition, we have the generating function These objects are natural combinatorial structures associated with the q-binomial theorem, Heine's transformation, and Lebesgue's identity (see [20] for a summary with references). In [18], they formed the basis for an algorithmic approach to the combinatorics of basic hypergeometric series. More recently, overpartitions have been found at the heart of bijective proofs of Ramanujan's 1 ψ 1 summation and the q-Gauss summation [14], [15].…”

Rank and Conjugation for the Frobenius Representation of an Overpartition

“…It is originally due to Joichi and Stanton [26] Algorithm I. The generating function for overpartitions with a given number of parts.…”

Rank and Conjugation for a Second Frobenius Representation of an Overpartition

“…For comparison, we record that the overpartition (11,11,10,8,8,7,5, 3, 2, 1, 1) has rank 7, D-rank 0, and M 2 -rank 0, while the overpartition (10,9,7,7,7,5, 1) has rank 7, D-rank 3, and M 2 -rank 2. When the overpartition has neither non-overlined odd parts nor overlined even parts, then the M 2 -rank becomes M 2 -rank for partitions with no repeated odd parts studied in [2].…”