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reduction/simplification of hypergeometric-function-related formula
*To*: mathgroup at smc.vnet.net
*Subject*: [mg127866] reduction/simplification of hypergeometric-function-related formula
*From*: Paul Slater <slater at kitp.ucsb.edu>
*Date*: Tue, 28 Aug 2012 04:52:02 -0400 (EDT)
*Delivered-to*: l-mathgroup@mail-archive0.wolfram.com
*Delivered-to*: l-mathgroup@wolfram.com
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I posted a short preprint
http://arxiv.org/pdf/1203.4498v2.pdf
a few months ago.
The central object in it is the formula in Figure 3--given in plain text at the bottom of this email. (It can be copied-and-pasted into a Mathematica notebook.).
The formula contains a "family" of six 7F6 hypergeometric functions. It seems to have a number of very interesting (quantum-information-theoretic) properties--as indicated in the preprint. (The upper and lower parameters form intriguing sequences, and the argument of all the six functions is (27/64) = (3/4)^3. For non-negative integers and half-integers, it appears to yield rational values.)
It took considerable work to get the formula as "concise" as it is now (LeafCount=530). The original form, generated using the Mathematica FindSequenceFunction command on a sequence of length 32, extended over several pages of output--and also had six (different) hypergeometric functions embedded in it.
I have devoted a considerable amount of effort, unsuccessfully, to see if it can be made more concise/digestible. In particular, I have never been able to derive an equivalent form in which fewer than six independent hypergeometric formulas are present.
Any thoughts?
Thanks!
Paul B. Slater
Formula in question:
(4^(-3 - 2 =CE=B1)
Gamma[5/2 + 3 =CE=B1] Gamma[
2 + 5 =CE=B1] ((-54 +
=CE=B1 (39 +
5 =CE=B1 (628 + 25 =CE=B1 (161 + 2 =CE=B1 (-581 + 740 =CE=B1))))) HypergeometricPFQ[{1,
2/5 + =CE=B1, 3/5 + =CE=B1, 4/5 + =CE=B1, 5/6 + =CE=B1, 7/6 + =CE=B1, 6/5 + =CE=B1}, {13/10 + =CE=B1,
3/2 + =CE=B1, 17/10 + =CE=B1, 19/10 + =CE=B1, 2 + =CE=B1, 21/10 + =CE=B1}, 27/
64] + (347274 +
5 =CE=B1 (-312019 +
25 =CE=B1 (22255 + 8 =CE=B1 (-2431 + 925 =CE=B1)))) =
HypergeometricPFQ[{2, 2/5 + =CE=B1,
3/5 + =CE=B1, 4/5 + =CE=B1, 5/6 + =CE=B1, 7/6 + =CE=B1, 6/5 + =CE=B1}, {13/10 + =CE=B1, 3/2 + =CE=B1,
17/10 + =CE=B1, 19/10 + =CE=B1, 2 + =CE=B1, 21/10 + =CE=B1}, 27/64] +
10 ((-769797 +
25 =CE=B1 (66227 + 4 =CE=B1 (-12843 + 3700 =CE=B1))) HypergeometricPFQ[{3,
2/5 + =CE=B1, 3/5 + =CE=B1, 4/5 + =CE=B1, 5/6 + =CE=B1, 7/6 + =CE=B1, 6/5 + =CE=B1}, {13/10 + =CE=B1,
3/2 + =CE=B1, 17/10 + =CE=B1, 19/10 + =CE=B1, 2 + =CE=B1, 21/10 + =CE=B1}, 27/64] +
75 ((44133 + 8 =CE=B1 (-6131 + 1850 =CE=B1)) HypergeometricPFQ[{4, 2/5 + =CE=B1,
3/5 + =CE=B1, 4/5 + =CE=B1, 5/6 + =CE=B1, 7/6 + =CE=B1, 6/5 + =CE=B1}, {13/10 + =CE=B1,
3/2 + =CE=B1, 17/10 + =CE=B1, 19/10 + =CE=B1, 2 + =CE=B1, 21/10 + =CE=B1}, 27/64] +
8 ((-7981 + 3700 =CE=B1) HypergeometricPFQ[{5, 2/5 + =CE=B1, 3/5 + =CE=B1,
4/5 + =CE=B1, 5/6 + =CE=B1, 7/6 + =CE=B1, 6/5 + =CE=B1}, {13/10 + =CE=B1, 3/2 + =CE=B1,
17/10 + =CE=B1, 19/10 + =CE=B1, 2 + =CE=B1, 21/10 + =CE=B1}, 27/64] +
3700 HypergeometricPFQ[{6, 2/5 + =CE=B1, 3/5 + =CE=B1, 4/5 + =CE=B1, 5/6 + =CE=B1,
7/6 + =CE=B1, 6/5 + =CE=B1}, {13/10 + =CE=B1, 3/2 + =CE=B1, 17/10 + =CE=B1, 19/10 + =CE=B1,
2 + =CE=B1, 21/10 + =CE=B1}, 27/64])))))/(3 Gamma[1 + =CE=B1] Gamma[
3 + 2 =CE=B1] Gamma[13/2 + 5 =CE=B1])
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