1, because 1 _ qa lim - - = a. 12) is usually called Heine's series or, in view of the base q, the basic hypergeometric series or q-hypergeometric series. Analogous to Gauss' notation, Heine used the notation ¢(a, b, c, q, z) for his series. 14) n= 0 q; q n c; q n where (a; q)n = { ~i _ a)(l - aq) ... (1 _ aqn-I), n=O, n= 1,2, ... 15) is the q-shifted factorial and it is assumed that c i- q-rn for m = 0,1, ....

Iv) Prove the q-Leibniz formula V;[J(z)g(z)] = ~ [~] qV;-k J(zqk)V;g(z). 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.

Ar; bl , ... , bs ; z) == rFs [alb' a2,·· ·b' ar ; z] I, ... 19) where a dash is used to indicate the absence of either numerator (when r = 0) or denominator (when s = 0) parameters. 21 ) IFI ( -n; a + 1; x ) . n. Generalizing Heine's series, we shall define an r¢s basic hypergeometric series by Lna (x) A. ( A. [aI, a2, ... , a r r'f's al,a2,···,ar ; bI,···, bs;q,z ) -= r'f's b b ;q,z ] = f n= 0 1, ... 22) with (~) = n(n - 1)/2, where q -=I 0 when r > s + 1. 22) it is assumed that the parameters bl , ...

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