G. Gasper, Summation, transformation, and expansion formulas for bibasic series, Trans. Amer. Math. Soc. 312(1989), 257 - 277  Home/Search   Document Not in Database   Summary   Related Articles   Check

....they will never contribute to b x =a x and b y =a y . Thus, b x =a x and b y =a y are in any case integer powers of q and p, respectively, coming from ffl P= P . Therefore, they do not take influence on the computation of fl and ffi at all. 4. 1 Bibasic Summation Formulas In 1989, Gasper [3] derived the indefinite bibasic summation formula (1 Gamma a) 1 Gamma b) a; b; p) k (c; a=bc; q) k (q; aq=b; q) k (ap=c; bcp; p) k = ap; bp; p) n (cq; aq=bc; q) n (q; aq=b; q) n (ap=c; bcp; p) n = gn (21) by showing that g k is a bibasic hypergeometric solution of the ....

....(21) is the case d = 1, m = 0 of (22) Since the output of qTelescope for identity (22) is quite lengthy, here we shall consider only the case m = Gamma1 after dividing the summand by the constant fraction on the right hand side. Of course, the algorithm works for symbolic m as well. In[3]: qTelescope[ 1 a d pk qk) 1 b d pk qk) qfac[a,p,k] qfac[b,p,k] qfac[c,q,k] qfac[a d2 b c,q,k] qk d (1 c d) 1 a d b c) qfac[d q,q,k] qfac[a d q b,q,k] qfac[a d p c,p,k] qfac[b c p d,p,k] 1 a) 1 b) 1 c) 1 a d2 b c) k, 1, n] 2 a d q Out[3] 1 (qfac[a p, p, n] qfac[b p, p, ....

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G. Gasper, Summation, transformation, and expansion formulas for bibasic series, Trans. Amer. Math. Soc., 312 (1989), 257--277.

....natural generalization of Bressoud s matrix inverse (1.3) such as (1.4) is the natural generalization of Gould s and Hsu s (1.1) and Carlitz s (1.2) A special case of (1.5) namely (4. 3) which involves rising q factorials with two different bases, has been crucial in papers by Gasper and Rahman [8, 19]. They used inversion together with an indefinite bibasic sum to derive numerous beautiful bibasic, cubic, and quartic summation formulas for basic hypergeometric series (see section 3 for hypergeometric definitions) They also extended this method to obtain bibasic, cubic, and quartic ....

....of the second application we obtain a basic hypergeometric transformation formula (identity (4.12) The former contains an infinite family of summation formulas for very well poised hypergeometric series. Besides, we use the opportunity to clearly demonstrate that what Gasper and Rahman do in [8, 19] is indeed inversion, though in disguise. This fact does not seem to be as accepted as it should be. Their extension in [9, 20; 10, sec. 3.6] has an explanation in terms of partial inversion . Finally, in section 5 we apply other special cases of (1.4) to obtain curious identities ( 5.5) ....

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G. Gasper, Summation, transformation, and expansion formulas for bibasic series, Trans. Amer. Soc. 312 (1989), 257--278.

....and the Bailey [1941] and Daum [1942] q Kummer summation formula 2 OE 1 (a; b; aq=b; q; Gammaq=b) Gammaq; q) 1 (aq; aq 2 =b 2 ; q 2 ) 1 (aq=b; Gammaq=b; q) 1 ; jq=bj 1; 4:3) derived in x1.8 of BHS. Formula (4. 2) can also be derived by the finite difference method employed in Gasper [1989] and Gasper and Rahman [1990a] to derive bibasic extensions of (4.2) and by the method pointed out in Rahman [1990] Using (4.2) we obtain the expansion formula u 0 = X k0 u k ffi k;0 = X k0 u k k X j=0 (a; qa 1 2 ; Gammaqa 1 2 ; q Gammak ; q) j (q; a 1 2 ; Gammaa 1 2 ; ....

Gasper, G. [1989] Summation, transformation, and expansion formulas for bibasic series, Trans. Amer. Math. Soc., 312, 257--277.

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G. Gasper, Summation, transformation, and expansion formulas for bibasic series, Trans. Amer. Math. Soc. 312(1989), 257 - 277

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G. Gasper, Summation, transformation, and expansion formulas for bibasic series, Trans. Amer. Math. Soc. 312(1989), 257 - 277

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