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)*Delivered-to*: l-mathgroup@mail-archive0.wolfram.com*Delivered-to*: l-mathgroup@wolfram.com*Delivered-to*: mathgroup-newout@smc.vnet.net*Delivered-to*: mathgroup-newsend@smc.vnet.net

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