Mathematica 9 is now available
Services & Resources / Wolfram Forums / MathGroup Archive
-----

MathGroup Archive 2012

[Date Index] [Thread Index] [Author Index]

Search the Archive

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
  • Delivered-to: mathgroup-newout@smc.vnet.net
  • Delivered-to: mathgroup-newsend@smc.vnet.net

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])



  • Prev by Date: Re: Group and Replace itens sequence in a list
  • Next by Date: Locator Bug
  • Previous by thread: Re: Problems obtaining a free c compiler...
  • Next by thread: reduction/simplification of hypergeometric-function-related formula