Download Basic Hypergeometric Series and Applications by Nathan J. Fine PDF

By Nathan J. Fine

The speculation of walls, based by way of Euler, has led in a traditional option to the belief of uncomplicated hypergeometric sequence, often referred to as Eulerian sequence. those sequence have been first studied systematically by means of Heine, yet many early effects are attributed to Euler, Gauss, and Jacobi. at the present time, study in $q$-hypergeometric sequence is particularly lively, and there are actually significant interactions with Lie algebras, combinatorics, particular capabilities, and quantity concept. even though, the speculation has been constructed to such an volume and with this type of large quantity of robust and basic effects that the topic can look rather daunting to the uninitiated. via supplying an easy method of easy hypergeometric sequence, this publication offers an exceptional easy creation to the topic. the start line is an easy functionality of numerous variables pleasurable a few $q$-difference equations. the writer offers an uncomplicated technique for utilizing those equations to procure adjustments of the unique functionality. A bilateral sequence, shaped from this functionality, is summed as an unlimited product, thereby offering a chic and fruitful end result which works again to Ramanujan. through exploiting a different case, the writer is ready to overview the coefficients of a number of sessions of endless items when it comes to divisor sums. He additionally touches on common transformation concept for uncomplicated sequence in lots of variables and the elemental multinomial, that is a generalization of a finite sum. those advancements lead clearly to the mathematics domain names of partition thought, theorems of Liouville variety, and sums of squares. touch is additionally made with the mock theta-functions of Ramanujan, that are associated with the rank of walls. the writer provides a couple of examples of modular services with multiplicative coefficients, in addition to the beginnings of an common optimistic method of the sector of modular equations. Requiring in simple terms an undergraduate historical past in arithmetic, this booklet presents a quick access into the sector. scholars of walls, simple sequence, theta-functions, and modular equations, in addition to study mathematicians attracted to an user-friendly method of those components, will locate this booklet priceless and enlightening. end result of the simplicity of its process and its accessibility, this paintings may perhaps turn out necessary as a textbook.

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Alternatively, we can expand the left side in powers of a and compare coefficients. 4) is established in full generality. It is an example of a bi-basic identity. 32 FUNDAMENTAL PROPERTIES OF BASIC HYPERGEOMETRIC SERIES The methods of §20 can be pushed a little further. 4). 9) M o o y (2/n+1;y)oo(feyn;g)ootn fay)™ ~0 {cyn;y)oc{ayn;q)oo = (b/a;q)oo \p (qn+1;q)oc{ctyn;y)oo^ {q',q)oo ^ 0 ( ^ n / a ; ^ ) o o ( ^ n ; 2 / ) o o a ' Notes §1. The most natural question that occurs in studying this book is: Why does have so much structure and yield such diverse and interesting results in such a natural way?

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. 74) kjinyK ) • m 23. Partitions with odd parts and with distinct parts. 5), put b = i. 7). Multiply HS = F + G by (1 — i), equate real parts, and replace

42) *(*) = £ ( - ! ) = X ) (l - (<7*+1)oo) = £ fc>0 fc>0 ( ! - I I (! - 9") ) • V n>/c / 13. Iteration of (a,t) —> (aq,tq). 2) 6 ^ (6g) n (ag/6) n+1 l *j ' valid for \t\ < 1. 3) (1 - *)F(a,0; *) = £ ) ^ ( - a ^ 2 ) - ( l - a^ 2 n + 1 )(/( 3 n 2 + n )/ 2 . 4) ^ ( - a ) " ( l - ag 2 » + 1 ) 9 ( 3 " 2 + n >/ 2 . 5) {a0 ag (a^)n(a<7/6)2n+ i(^r. *) u-)" = £ft)"—EC"*1)*". n>0 n>0 where we have set z = ail — a). 14. ). i) (. *"'><«)V*. This identity contains much information about the function F(a, 6; £), and many of our earlier results are easily derived from it.

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