Hypergeometric Forms for Ising-Class Integrals
We apply experimental-mathematical principles to analyzecertain integrals relevant to the Ising theory of solid-state physics. Wefind representations of the these integrals in terms of MeijerG-functions and nested-Barnes integrals. Our investigations began bycomputing 500-digit numerical values of Cn,k,namely a 2-D array of Isingintegrals for all integers n, k where n is in [2,12]and k is in [0,25].We found that some Cn,k enjoy exact evaluations involving DirichletL-functions or the Riemann zeta function. In theprocess of analyzinghypergeometric representations, we found -- experimentally and strikingly-- that the Cn,k almost certainly satisfy certain inter-indicialrelations including discrete k-recursions. Using generating functions,differential theory, complex analysis, and Wilf-Zeilberger algorithms weare able to prove some central cases of these relations.
- Research Organization:
- Lawrence Berkeley National Lab. (LBNL), Berkeley, CA (United States)
- Sponsoring Organization:
- USDOE
- DOE Contract Number:
- DE-AC02-05CH11231
- OSTI ID:
- 928484
- Report Number(s):
- LBNL-61037; R&D Project: KC6714; BnR: YN0100000; TRN: US200811%%269
- Journal Information:
- Experimental Mathematics, Vol. 16, Issue 3; Related Information: Journal Publication Date: 2007
- Country of Publication:
- United States
- Language:
- English
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