# Download A Century of mathematics in America (History of Mathematics, by Richard Askey, Uta C. Merzbach PDF

By Richard Askey, Uta C. Merzbach

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In Sections 3-6 we shall describe how some of the discoveries in Sylvester's magnum opus continue to influence current research. It is my firm belief that Sylvester's paper still deserves study. To emphasize this point, Section 7 is devoted to a new proof of one of Sylvester's identities. Section 8 considers a generalization of this result which leads to a new proof of the Rogers-Ramanujan identities. 3. THE DURFEE SQUARE COMBINATORIALLY Act II of [28] starts off with Durfee's observation that the number of selfconjugate partitions of an integer n equals the number of partitions of n into distinct odd parts.

136] to give a combinatorial proof of an identity of N. J. Fine. 6. COMBINATORICS OF JACOBI'S TRIPLE PRODUCT IDENTITY One of the most important identities in the theory of elliptic theta functions is Jacobi's Triple Product Identity [7; p. 2) that W(z) = E00 zmAm, where Am = gmAm-1. J. J. SYLVESTER, JOHNS HOPKINS AND PARTITIONS 31 Consequently by iteration (z) = A0 O(z) 00 zmq(m+l/2). E m=-00 The Exodion of [28] (partly due to A. S. Hathaway) provides a very nice means of seeing that A0 is just H,,, I (I- q')'.

Functions with a finite number of zeros and infinities (algebraic functions). 2. Striped functions (trigonometric functions). In these the stripes may be equal, or may vary progressively, or periodically. The stripes may be simple, or themselves compounded of stripes. Thus, sin(a sin z) will be composed of stripes each consisting of a bundle of parallel stripes (infinite in number) folded over onto itself. 3. Chequered functions (elliptic functions). 4. Functions whose patterns are central or spiral.