By George Gasper

ISBN-10: 0511526253

ISBN-13: 9780511526251

ISBN-10: 0521833574

ISBN-13: 9780521833578

An excellent reference at the topic. fabric on generalized hypergeometric features (starting with Gauss' hypergeometric functionality) is gifted via the q analogy's. the cloth is complicated and is easily written with a good and readable typeface. The advent to q sequence will fulfill the newbie. The checklist of approximately 500 references masking the full topic is definitely worth the cost alone.

Lorenz H. Menke, Jr.

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

**Sample text**

Mn and it is the a = q-Cml+ ... 9) when n = ml + ... +mr . 10 The q-gamma and q- beta functions 21 was introduced by Thomae [1869] and later by Jackson [1904e]. Heine [1847] gave an equivalent definition, but without the factor (l-q)l-x. 2) which clearly approaches n! as q ---+ 1-. Hence f q(n + 1) tends to f( n + 1) = n! as q ---+ 1-. 1). 3) q---+I we shall give a simple, formal proof due to Gosper; see Andrews [1986]. 11]) and the well-known functional equation for the gamma function r(x + 1) = xr(x), r(1) = 1.

Z;q 00 Ibq/a 2 < 1, 1 (a 2, a2/b; b; l, bq/a2) (a2,q;f)~ [(b/~;q)oo + (-b/~;q)oo] . 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 [1847] (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]¢.

4. 13) one can think of the theta functions 19 1 (x, q) and 192 (x, q) as one-parameter deformations (generalizations) of the trigonometric functions sin x and cos x, 1. 7 q-Saalschutz formula 17 respectively. 14) is never zero. 9) it is clear that [a; CT, T] is well-defined, [-a; CT, T] = -[a;CT,TJ, [1;CT,T] = 1, and . 15) hm [a; CT, T] = . ( ) = [a;CT]. III T-tOO sIn 1[CT ° Hence, the elliptic number [a; CT, T] is a one-parameter deformation of the trigonometric number [a; CT] and a two-parameter deformation of the number a.

### Basic Hypergeometric Series (Encyclopedia of Mathematics and its Applications) by George Gasper

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