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By George Gasper
An effective reference at the topic. fabric on generalized hypergeometric capabilities (starting with Gauss' hypergeometric functionality) is gifted via the q analogy's. the fabric is complex 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 masking the whole topic is well worth the fee alone.
Lorenz H. Menke, Jr.
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Additional info for Basic Hypergeometric Series (Encyclopedia of Mathematics and its Applications)
50) , sin( 7ru) for noninteger values of u and view [a; u] as a trigonometric deformation of a since lima-to [a; u] = a. The corresponding rts trigonometric hypergeometric series can be defined by rts(al, a2, ... , ar; bl , ... , bs ; u, z) = f [al,a2, ... [bl , ... 51 ) Basic hypergeometric series 8 where n [n; u]! 54) where q = e 2nia , it follows that (qa;q)n n(l-a)/2-n(n-I)/4 [a,. 55) rts(al' a2,···, ar ; bl ,···, bs; u, z) rI-. (a1 -_ r'l's q ,qa2 , ... , qar •,q b1 , ... 49). 6, and the corresponding elliptic (and theta) hypergeometric series and their summation and transformation formulas are considered in Chapter 11.
13 Show that u(z) = 2¢1 (a, b; c; q, z) satisfies (for Izl < 1 and in the formal power series sense) the second order q-differential equation z(c _ abqz)V2 u + [1 - c q 1-q (l-a)(l-b) (1 _ q)2 u = 0, + (1 - a)(l - b) - (1 - abq) z] V u 1-q q where Vq is defined as in Ex. 12. 14 Let Ixl = 2F1 (a, b; c; z), where Izl < 1. 3. Define . Slllq (x) COS q = eq(ix) - eq( -ix) 2. ) . n= 0 q, q 2n Also define S. () lllq x = Eq(ix) - Eq( -ix) 2i ' Show that eq(ix) = cosq(x) + i sinq(x), Eq(ix) = Cosq(x) + iSinq(x), sinq(x)Sinq(x) + cosq(x)Cosq(x) = 1, sinq(x)Cosq(x) - Sinq(x) cosq(x) = 0.
Rahman and Suslov [1996a] used the method of first order linear difference equations to prove the q-binomial and q-Gauss formulas. Bender  used partitions to derive an extension of the q-Vandermonde Notes 35 sum in the form of a generalized q-binomial Vandermonde convolution. 33) appeared in Atakishiyev and Suslov [1992a], but without any explicit reference to the q-exponential function. Also see Suslov [1998-2003] and the q-convolutions in Carnovale , Carnovale and Koornwinder , and Rogov .
Basic Hypergeometric Series (Encyclopedia of Mathematics and its Applications) by George Gasper