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Linear quintuple-product identities

Author(s): Richard Blecksmith; John Brillhart.
Journal: Math. Comp. 72 (2003), 1019-1033.
MSC (2000): Primary 11F11
Posted: August 14, 2002
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Abstract: In the first part of this paper, series and product representations of four single-variable triple products $T_0$, $T_1$, $T_2$, $T_3$ and four single-variable quintuple products $Q_0$, $Q_1$, $Q_2$, $Q_3$ are defined. Reduced forms and reduction formulas for these eight functions are given, along with formulas which connect them. The second part of the paper contains a systematic computer search for linear trinomial $Q$ identities. The complete set of such families is found to consist of two 2-parameter families, which are proved using the formulas in the first part of the paper.

References:

1.
C. Adiga, B. C. Berndt, S. Bhargava, and G. N. Watson, Chapter $16$ of Ramanujan's second notebook: theta-functions and q-series, Mem. Amer. Math. Soc., 53 (1985), Providence, RI. MR 86e:33004

2.
R. Blecksmith, J. Brillhart, and I. Gerst, Some infinite product identities, Math. Comp., 51 (1988), 301-314. MR 89f:05017

3.
R. Blecksmith, J. Brillhart, and I. Gerst, On a certain (mod 2) identity and a method of proof by expansion, Math. Comp., 56 (1991), 775-794. MR 91j:11087

4.
R. Blecksmith, J. Brillhart, and I. Gerst, New proofs for two infinite product identities, Rocky Mountain J. Math., 22 (1992), 819-823. MR 93h:11114

5.
R. Blecksmith, J. Brillhart, and I. Gerst, A fundamental modular identity and some applications, Math. Comp., 61 (1993), 83-95. MR 94c:11100

6.
R. Blecksmith, J. Brillhart, and I. Gerst, Four identities involving the single-variable quintuple product, Abstracts AMS, 14 (1993), 584.

7.
J. M. Borwein and P. B. Borwein, Pi and the AGM, Wiley & Sons, New York, 1987. MR 89a:11134

8.
G. H. Hardy and E. M. Wright, An introduction to the theory of numbers, fourth ed., Clarendon Press, Oxford, 1960.

9.
L. Slater, Further identities of the Rogers-Ramanujan type, Proc. London Math. Soc. $(2)$, 54 (1952), 147-167. MR 14:138e

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Additional Information:

Richard Blecksmith
Affiliation: Department of Mathematical Sciences, Northern Illinois University, DeKalb, Illinois 60115
Email: richard@math.niu.edu

John Brillhart
Affiliation: Department of Mathematics, University of Arizona, Tucson, Arizona 85721
Email: jdb@math.arizona.edu

DOI: 10.1090/S0025-5718-02-01461-8
PII: S 0025-5718(02)01461-8
Keywords: Triple product, quintuple product, linear identity, search algorithm
Received by editor(s): August 29, 2001
Posted: August 14, 2002
Additional Notes: Research was supported in part by Northern Illinois University Research and Artistry grant
Dedicated: Dedicated to our longtime friend John Selfridge
Copyright of article: Copyright 2002, American Mathematical Society

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