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

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We explain combinatorially the occurrence of certain classical sequences of orthogonal polynomials as sequences of rook polynomials, and we give some new examples related to general stairstep boards.is defined to be the vector r (B)=(rt,rf,rf, ... ), where rt = 1. Note that for sufficiently large i, r/ = O. Two boards are rook equivalent if they have the same