By George Gasper

ISBN-10: 0511526253

ISBN-13: 9780511526251

ISBN-10: 0521833574

ISBN-13: 9780521833578

A fantastic reference at the topic. fabric on generalized hypergeometric features (starting with Gauss' hypergeometric functionality) is gifted via the q analogy's. the fabric is complicated and is easily written with a good and readable typeface. The creation to q sequence will fulfill the newbie. The record of approximately 500 references protecting the total topic is definitely worth the rate alone.

Lorenz H. Menke, Jr.

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**Example text**

18] while working on multivariable orthogonal polynomials. 11 Also see Jackson [1917, 1951] and, for fractional q-integrals and q-derivatives, AI-Salam [1966] and Agarwal [1969b]. Toeplitz [1963, pp. 53-55] pointed out that around 1650 Fermat used a q-integral type Riemann sum to evaluate the integral of xk on the interval [0, b]. AI-Salam and Ismail [1994] evaluated a q-beta integral on the unit circle and found corresponding systems of biorthogonal rational functions. 2 The q-binomial coefficient [~L' which is also called the Gaus- sian binomial coefficient, counts the number of k dimensional subspaces of an n dimensional vector space over a field GF(q), q a prime power (Goldman and Rota [1970]), and it is the generating function, in powers of q, for partitions into at most k parts not exceeding n - k (Sylvester [1882]).

Mr (bcq, q/ a; q)oo (b l ; q)ml ... (b r ; q)mr X c- I , b, bq/bl , ... ,bq/br ] r+ 2 ¢r+ I [ bq/ a, bqI -ml /b I , ... , bqI -m r /b r ; q, cq . 35 Use Ex. 2(v) to prove that if x and yare indeterminates such that xy = qyx, q commutes with x and y, and the associative law holds, then (See Cigler [1979], Feinsilver [1982]' Koornwinder [1989], Potter[1950]' Schiitzenberger [1953], and Yang [1991]). 36 Verify that if x and yare indeterminates satisfying the conditions in Ex. 35, then (i) (ii) eq(y)eq(x) = eq(x + y), eq(x)eq(y) = eq(x + y - yx); Eq(x)Eq(y) = Eq(x + y), Eq(y)Eq(x) = Eq(x + y + yx).

R t (al, ... • ,br ;q)00}o t ° when < q terminate. (qt, b1t, ... , brt; q)oo d (al t , ... 5 Show that (e,bqn;q)rn _ (b/e;q)n ~ (q-n,e;q)kqk ( k. q2inq-(~), (i) 2¢I(q-n,ql-n;qb2;q2,q2) = ,q n " (a; e; q, e /) (e/a; )q)oo , (11.. ) 2¢1 (a, b; e;p,p ) = (1/ b/. -1) 00 , p > 1. 8 Show that, when 2¢1 = lal < 1 and (a 2qz; q)oo ( ) . 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).

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

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