By George Gasper

ISBN-10: 0511526253

ISBN-13: 9780511526251

ISBN-10: 0521833574

ISBN-13: 9780521833578

An effective reference at the topic. fabric on generalized hypergeometric capabilities (starting with Gauss' hypergeometric functionality) is gifted by way of the q analogy's. the cloth is complicated and is definitely written with a decent and readable typeface. The advent to q sequence will fulfill the newbie. The checklist of approximately 500 references protecting the total topic is well worth the expense alone.

Lorenz H. Menke, Jr.

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Extra info for Basic Hypergeometric Series (Encyclopedia of Mathematics and its Applications)

Sample text

5) when restricted. 14) that 1 f is suitably t x-I ((tq; q)oo ) dqt, R ex> 0 , y r--I- 0 ,- 1,- 2, ... , tqY;qoo which clearly approaches the beta function integral B q(x, y ) = 1 o B(x, y) = 11 t x - 1 (1 - t)y-l dt, Re x, Re y > 0, (1117) .. 8) 24 Basic hypergeometric series as q ----+ 1-. 1) in the q-integral form 2¢1 ) _ fq(c) ( a b. c. 10) where Iarg(l- z)1 < 7r and Re c> Re b > O. The q-integral notation is, as we shall see later, quite useful in simplifying and manipulating various formulas involving sums of series.

15 Prove the transformation formulas A. (i) 2'/'1 A. (ii) 2'/'1 [q-n,b. C' q, Z ] _ (bzq-n/c;q)oo (bz/c; q)oo [q-n,c/b,o. ] c, cq/bz ,q, q , [q-n,b. ] _ (c/b;q)nbn A. [q-n,b,q/z. /] C ,q,z - (C;q)n 3'/'1 bq1-n/C ,q,z C q-n,b. ) C, q n (iii) A. 3 '/'2 31;2 , ° [q-n,b,bzq-n/C. ] bq 1-n/ C, ,q,q . 18 Show that (i) (ii) (iii) (c;q)oo 1¢I(a;c;q,z) = (z;q)oo 1¢I(az/c;z;q,c), and deduce that 1¢1(-bq;O;q,-q) = (-bq2;q2)00/(q;q2)00, (z;q)oo 2¢I(a,0;c;q,z) = (az;q)oo 1¢2(a;c,az;q,cz), f n= 0 ((a;2~)n) q, a ,q qG)(at/z)n 2¢I(q-n,a;ql-n/a;q, qz2/a) n Iztl < 1.

14) This series expansion will be used in the next section to derive a q-integral representation for Bq(x, y). 4) The bilateral q-integral is defined by 1 00 f(t) dqt = (1 - q) f n= -00 [f(qn) + f( _qn)] qn. 5) when restricted. 14) that 1 f is suitably t x-I ((tq; q)oo ) dqt, R ex> 0 , y r--I- 0 ,- 1,- 2, ... , tqY;qoo which clearly approaches the beta function integral B q(x, y ) = 1 o B(x, y) = 11 t x - 1 (1 - t)y-l dt, Re x, Re y > 0, (1117) .. 8) 24 Basic hypergeometric series as q ----+ 1-.

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