# Download Basic Hypergeometric Series (Encyclopedia of Mathematics and by George Gasper PDF

By George Gasper

A superb 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 record of approximately 500 references protecting the complete topic is definitely worth the rate alone.

Lorenz H. Menke, Jr.

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**Example text**

137]. 3 Cigler [1979] derived an operator form ofthe q-binomial theorem. MacMahon [1916, Arts. 105-107] showed that if a multiset is permuted, then 36 Basic hypergeometric series the generating function for inversions is the q-multinomial coefficient. Also see Carlitz [1963a], Kadell [1985a], and Knuth [1973, p. 33, Ex. 16]. Gasper derived the q-multinomial theorem in part (ii) several years ago by using the q-binomial theorem and mathematical induction. 1)] and the q-Vandermonde sum. Some sums of q-multinomial coefficients are considered in Bressoud [1978, 1981c].

Q! 2) (Zq2; q)oo. 8) n= 1 in terms of infinite products. 11) n= 1 and 00 194 (X, q) = II (1- q2n)(1_ 2q2n-1 cos2x + q4n-2). 12) n= 1 It is common to write 19 k (x) for 19 k (x, q), k = 1, ... ,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 .

By an extension of the ratio test (Bromwich [1959, p. 241]), it converges absolutely for Izl = 1 if r = s + 1 and Re [b 1 + ... + bs (a 1 + ... + ar )] > o. If r > s + 1 and z i=- 0 or r = s + 1 and Iz I > 1, then this series diverges, unless it terminates. If 0 < Iql < 1, the r¢s series converges absolutely for all z if r ::; sand for Izl < 1 if r = s + 1. This series also converges absolutely if Iql > 1 and Izl < Ib 1b2·· ·bsql/lala2·· ·arl· It diverges for z i=- 0 if 0 < Iql < 1 and r > s + 1, and if Iql > 1 and Izl > Ib1 b2·· ·bsql/lala2· ··arl, unless it terminates.