Basic Hypergeometric Series, Second Edition (Encyclopedia of by George Gasper, Mizan Rahman

By George Gasper, Mizan Rahman

This up to date variation will proceed to satisfy the wishes for an authoritative accomplished research of the quickly growing to be box of easy hypergeometric sequence, or q-series. It contains deductive proofs, routines, and worthwhile appendices. 3 new chapters were further to this variation masking q-series in and extra variables: linear- and bilinear-generating services for easy orthogonal polynomials; and summation and transformation formulation for elliptic hypergeometric sequence. furthermore, the textual content and bibliography were elevated to mirror contemporary advancements. First version Hb (1990): 0-521-35049-2

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Mr ) 1< 1, Icql < 1, show that A.. r+ 2 'f'r+ = I [ a ' b,I b qml , ... , br qmr . a-II-(m1+···+mr) ] b b b ' q, q cq, I,···, r (bq/a, cq; q)oo (b l /b; q)ml ... (br/b; q)mr bm1+ ... + 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]).

Iii) Show that r+ 1'. q, qz] " [a1'b ... ' ab r+ 1 'Pr 1, ... , 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: .

3 The q-binomial theorem One of the most important summation formulas for hypergeometric series is given by the binomial theorem: 2FI (a, C; C; z) where Izl = IFo(a;-; z) = f (a),n zn n=O n. 1) < 1. We shall show that this formula has the following q-analogue rI-. ( . _ . 3 The q-binomial theorem 9 which was derived by Cauchy [1843]' Heine [1847] and by other mathematicians. See Askey [1980a], which also cites the books by Rothe [1811] and Schweins [1820]' and the remark on p. 491 of Andrews, Askey, and Roy [1999] concerning the terminating form of the q-binomial theorem in Rothe [1811].

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