By Nathan J. Fine
The idea of walls, based by way of Euler, has led in a common approach to the assumption of uncomplicated hypergeometric sequence, sometimes called Eulerian sequence. those sequence have been first studied systematically through Heine, yet many early effects are attributed to Euler, Gauss, and Jacobi. at the present time, learn in $q$-hypergeometric sequence is especially lively, and there at the moment are significant interactions with Lie algebras, combinatorics, specified services, and quantity idea. despite the fact that, the speculation has been built to such an volume and with one of these large quantity of robust and common effects that the topic can seem really daunting to the uninitiated. via delivering an easy method of simple hypergeometric sequence, this publication offers an exceptional common advent to the topic. the start line is an easy functionality of a number of variables pleasurable a couple of $q$-difference equations. the writer provides an undemanding procedure for utilizing those equations to acquire differences of the unique functionality. A bilateral sequence, shaped from this functionality, is summed as an unlimited product, thereby supplying a chic and fruitful outcome which matches again to Ramanujan. via exploiting a unique case, the writer is ready to assessment the coefficients of a number of sessions of endless items by way of divisor sums. He additionally touches on basic transformation idea for simple sequence in lots of variables and the fundamental multinomial, that's a generalization of a finite sum. those advancements lead evidently to the mathematics domain names of partition thought, theorems of Liouville kind, and sums of squares. touch is usually made with the mock theta-functions of Ramanujan, that are associated with the rank of walls. the writer offers a few examples of modular services with multiplicative coefficients, besides the beginnings of an trouble-free positive method of the sphere of modular equations. Requiring merely an undergraduate historical past in arithmetic, this e-book presents a swift access into the sector. scholars of walls, simple sequence, theta-functions, and modular equations, in addition to learn mathematicians attracted to an trouble-free method of those parts, will locate this booklet worthwhile and enlightening. due to the simplicity of its strategy and its accessibility, this paintings may perhaps end up valuable as a textbook.
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72) ir(n) j>l 7r(n) That is, the total number of frequencies > i in all partitions of n is equal to the total number of times that the part i occurs. This generalizes Example 14. EXAMPLE 2 3 . Let w(k) = k(k + l)/2. 73) E E M * i + l)/2 = np(n). 7r(n)i>l In Theorem 7, take h(j) = 6(j\j0) (Kronecker delta) and let w(k) = A([fc/m]), where m is a positive integer and [ ] denotes the greatest integer function. Then we obtain, after dropping the subscript in j 0 , EXAMPLE 24 . E „ ( n ) A([V™1) = £*(») * ( W This holds even when A(0) / 0.
91), by summing separately over even and odd M, and by taking into account the fact that the number of parts has the same parity as N when all the parts are odd. 24. Continuation. 1) £>Y%. 11) 1 + (1 + a)qF(-a-\ 0; aq) = J2 a V l 4 . 11) can be interpreted arithmetically. 2) is a q N = n + 1 + pi + p 2 + Vpu t n N = a~ a q , + • • •. where n + 1 > Pi > Vi > • • • > Pt > 0. For t, n, and TV fixed we obtain all the partitions of TV into t + 1 distinct parts of which n -j-1 is the largest; thus k = (n -f 1) — (t -f 1) is the difference between the largest part and the number of parts.
2) is a q N = n + 1 + pi + p 2 + Vpu t n N = a~ a q , + • • •. where n + 1 > Pi > Vi > • • • > Pt > 0. For t, n, and TV fixed we obtain all the partitions of TV into t + 1 distinct parts of which n -j-1 is the largest; thus k = (n -f 1) — (t -f 1) is the difference between the largest part and the number of parts. We are therefore led to the definition, due to Dyson: (OA 'W ^ e ran^ °f a Partiti°n Z5 ^ e difference between the largest part and the number of parts. THE RANK OF A PARTITION 47 For arbitrary partitions the rank may be positive, negative, or zero; for partitions into distinct parts the rank cannot be negative.