Services & Resources / Wolfram Forums / MathGroup Archive
-----

MathGroup Archive 2012

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

Search the Archive

Re: reduction/simplification of hypergeometric-function-related formula

  • To: mathgroup at smc.vnet.net
  • Subject: [mg127897] Re: reduction/simplification of hypergeometric-function-related formula
  • From: "Dr. Wolfgang Hintze" <weh at snafu.de>
  • Date: Thu, 30 Aug 2012 04:11:27 -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
  • References: <k1k8np$90f$1@smc.vnet.net>

On 29 Aug., 07:19, Paul Slater <sla... at kitp.ucsb.edu> wrote:
> 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])

I don't see any easy way to simplify your expression but here are two
suggestions to understand it better.

1) Structure

Let us first look at the structure of your complete expression which I
shall call f[a]:

Decomposing as

f[a] = g[a] h[a]

where

g[a_] := 4^(-3 - 2*a)*Gamma[5/2 + 3*a]*(Gamma[2 + 5*a]/(3*Gamma[1 +
a]*Gamma[3 + 2*a]*Gamma[13/2 + 5*a]))

and

h[a] = Sum[ h1[m,a] p[m,a],{m,1,5}]

h1[ m_,a_] := HypergeometricPFQ[{m, 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]

where the arguments can be simplified to

t1 = Join[{m},{2/5 + a, 3/5 + a, 4/5 + a, 5/6 + a, 7/6 + a, 6/5 + a}]
t2 = Table[(2k+1)/10 + a,{k,6,10}]

If there was a typo in t1 (is it?) you can even reduce it to

t11 = Join[{m},Table[k + a,{k,2,7}]]

The degree of the polynomials p[m,a] is 6-m.
The coefficients do not seem follow any simple pattern (you might know
better, because you have calculated them).

2) Graphical and numerical study

Mathematica can help you a lot in this aspect of study.

Use

Plot[{1 - f[a], 1 - 1/(1 + Pi*a^2)}, {a, 0, 2}, PlotRange -> {0, 1.1}]

to see that f[a] has a very simple smmooth and bounded behaviour for
a>=0.
Also I have plottet a suggested approximate formula

fa[a] = 1/(1 + Pi*a^2)

You can look at compley a using e.g.

Plot3D[Abs[h1[3, x + I*y]], {x, -3, 0}, {y, -1, 1}]

Next we can let Mathematica calculate some numerical values

f[0]
1

f[1];
N[%, 30]
0.242424242424242424242424242424

i.e.
f[1] = 24/99 = 8/33

No periodicity is seen in the next example

f[2];
N[%,30]
0.0804953560371517027863777089783281733746130030959752321981424148606811145=
5108359133126934984520123839

but again a rational number is abtained at a=1/2

f[1/2];
N[%,30]
0.4531250000000000000000000000000000000000000000000000000000000000000000000=
000000000000000000000000000

Interesting study (at least for me) which makes use of some of
Mathematica's features.

Regards,
Wolfgang



  • Prev by Date: Re: Luke's list of mathematica questions
  • Next by Date: Re: reduction/simplification of hypergeometric-function-related
  • Previous by thread: reduction/simplification of hypergeometric-function-related formula
  • Next by thread: DSolve for a real function