A treatise on the analytical geometry by Casey J.

By Casey J.

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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 , ... , bs are such that the denominator factors in the terms of the series are never zero.

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]. 2), which can also be found in the books Heine [1878], Bailey [1935, p. 66] and Slater [1966, p. 2). Let us set ..

1) is well-poised and a2 = 1 + ~al' then it is called a very-well-poised series. Dougall's [1907] summation formulas F. [ 7 6 1 a, + ~a, b, c, d, e, -n .

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