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

By Nathan J. Fine

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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.