By George Gasper
A great reference at the topic. fabric on generalized hypergeometric features (starting with Gauss' hypergeometric functionality) is gifted by means of the q analogy's. the cloth is complex and is easily written with a decent and readable typeface. The advent to q sequence will fulfill the newbie. The checklist of approximately 500 references protecting the whole topic is well worth the cost alone.
Lorenz H. Menke, Jr.
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Additional info for Basic Hypergeometric Series (Encyclopedia of Mathematics and its Applications)
2), ----+ B (x, y) = (1 _ q) (q, qX+ Y; q)oo q (qX,qY;q)oo = (1 - q) (q; q)oo f (qY; q)n qnx (qY;q)oo n=O (q;q)n n+ 1 ) = (1 - q) L (q n+ Y: q) qnx, 00 ( Re x, Re y 00 n=O q ,qoo > O. 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) ..
And q -=I O. Basic analogues of the classical orthogonal polynomials will be considered in Chapter 7 as well as in the exercises at the ends of the chapters. Unless stated otherwise, when dealing with nonterminating basic hypergeometric series we shall assume that Iql < 1 and that the parameters and variables are such that the series converges absolutely. 22) to a similar series in base p with Ipi < 1 (see Ex. 4(i)). The inverted series will have a finite radius of convergence if the original series does.
9 Let ¢(a, b, c) denote the series q-contiguous relations: . (1) ¢(a, b, eq -I 2¢1 (a, b; e; q, z). Verify Heine's  (l-a)(I-b) ) - ¢(a, b, c) = ez ( )( ) ¢(aq, bq, cq), q-e l-e (ii) ¢(aq, b, c) - ¢(a, b, c) = az ~ =~ ¢(aq, bq, eq), Exercises 27 ... (l-b)(l-c/a) 2 (m) ¢(aq, b, cq) - ¢(a, b, c) = az ( )( ) ¢(aq, bq, cq ), 1-c 1-cq . 1 (1- b/aq) (IV) ¢(aq, bq- ,c) - ¢(a, b, c) = az 1 _ c ¢(aq, b, cq). 10 Denoting 2¢1 (a, b; c; q, z), 2¢1 (aq±l, b; c, q, z), 2¢1 (a, bq±l; c; q, z) and 2¢1 (a, b; cq±l; q, z) by ¢, ¢(aq±I), ¢(bq±l) and ¢(cq±I), respectively, show that (i) b(l - a)¢(aq) - a(l - b)¢(bq) = (b - a)¢, (ii) a(l - b/c)¢(bq-l) - b(l - a/c)¢(aq-l) = (a - b)(l - abz/cq)¢, (iii) q(l - a/c)¢(aq-l) + (1 - a)(l - abz/c)¢(aq) = [1 + q - a - aq/c + a2z(l - b/a)/c]¢, (iv) (1 - c)(q - c) (abz - c)¢(cq-l) + (c - a)(c - b)z¢(cq) = (c - l)[c(q - c) + (ca + cb - ab - abq)z]¢.
Basic Hypergeometric Series (Encyclopedia of Mathematics and its Applications) by George Gasper