We introduce a new tool, the factorial polynomials, to study rook equivalence of Ferrers boards. We provide a set of invariants for rook equivalence as well as a very simple algorithm for deciding rook equivalence of Ferrers boards. We then count the number of Ferrers boards rook equivalent to a given Ferrers board.

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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.

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Let / be a real-valued measurable function on a measure space (S, ©, p.) and let

(f(s)), sES, is also measurable. Conversely, if 4> is not Borel measurable, then there exists a measurable function/ on some measure space such that of is not measurable. We summarize these two remarks in the statement that a function preserves measurability under composition if and only if it is Borel measurable. The purpose of this note is to characterize functions 4> which preserve convergence (in various senses) of measurable functions. After these results were obtained it was pointed out that Theorems 5 and 6 follow from theorems of P. R. Halmos [2]. Theorem 7 was proved by