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Re: reduction/simplification of hypergeometric-function-related
*To*: mathgroup at smc.vnet.net
*Subject*: [mg127898] Re: reduction/simplification of hypergeometric-function-related
*From*: Alexei Boulbitch <Alexei.Boulbitch at iee.lu>
*Date*: Fri, 31 Aug 2012 03:56:04 -0400 (EDT)
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Note: In the previous posting of this query, the variable "alpha" was employed, which led to now-apparent problems in the mailing with the presentation of the formula in plaintext. I've replaced "alpha" by "a", so hopefully now the formula below will be usable as intended.
> 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 a)
Gamma[5/2 + 3 a] Gamma[
2 + 5 a] ((-54 +
a (39 +
5 a (628 + 25 a (161 + 2 a (-581 + 740 a))))) HypergeometricPFQ[{1,
2/5 + a, 3/5 + a, 4/5 + a, 5/6 + a, 7/6 + a, 6/5 + a}, {13/10 + a,
3/2 + a, 17/10 + a, 19/10 + a, 2 + a, 21/10 + a}, 27/
64] + (347274 +
5 a (-312019 +
25 a (22255 + 8 a (-2431 + 925 a)))) HypergeometricPFQ[{2, 2/5 + a,
3/5 + a, 4/5 + a, 5/6 + a, 7/6 + a, 6/5 + a}, {13/10 + a, 3/2 + a,
17/10 + a, 19/10 + a, 2 + a, 21/10 + a}, 27/64] +
10 ((-769797 +
25 a (66227 + 4 a (-12843 + 3700 a))) HypergeometricPFQ[{3,
2/5 + a, 3/5 + a, 4/5 + a, 5/6 + a, 7/6 + a, 6/5 + a}, {13/10 +
a,
3/2 + a, 17/10 + a, 19/10 + a, 2 + a, 21/10 + a}, 27/64] +
75 ((44133 + 8 a (-6131 + 1850 a)) HypergeometricPFQ[{4, 2/5 + a,
3/5 + a, 4/5 + a, 5/6 + a, 7/6 + a, 6/5 + a}, {13/10 + a,
3/2 + a, 17/10 + a, 19/10 + a, 2 + a, 21/10 + a}, 27/64] +
8 ((-7981 + 3700 a) HypergeometricPFQ[{5, 2/5 + a, 3/5 + a,
4/5 + a, 5/6 + a, 7/6 + a, 6/5 + a}, {13/10 + a, 3/2 + a,
17/10 + a, 19/10 + a, 2 + a, 21/10 + a}, 27/64] +
3700 HypergeometricPFQ[{6, 2/5 + a, 3/5 + a, 4/5 + a, 5/6 + a ,
7/6 + a, 6/5 + a}, {13/10 + a, 3/2 + a, 17/10 + a, 19/10 + a,
2 + a, 21/10 + a}, 27/64])))))/(3 Gamma[1 + a] Gamma[
3 + 2 a] Gamma[13/2 + 5 a])
Hi, Paul,
I cannot tell, if it is possible to simplify it further. Does not seem so. However, its plot looks rather simple. Try, for example, to build it on the interval 0<a<=1. This is often so with hypergeometric functions. If you have no special reasons to keep the result in the exact form, I would recommend to approximate this behaviour by some simple analytic function, like a polynomial, and further work with this one.
Hope it helps. Have fun, Alexei
Alexei BOULBITCH, Dr., habil.
IEE S.A.
ZAE Weiergewan,
11, rue Edmond Reuter,
L-5326 Contern, LUXEMBOURG
Office phone : +352-2454-2566
Office fax: +352-2454-3566
mobile phone: +49 151 52 40 66 44
e-mail: alexei.boulbitch at iee.lu
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