Services & ResourcesWolfram ForumsMathGroup Archive

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

Re: Help with a summation

• To: mathgroup at smc.vnet.net
• Subject: [mg53171] Re: [mg53150] Help with a summation
• From: Andrzej Kozlowski <akoz at mimuw.edu.pl>
• Date: Mon, 27 Dec 2004 06:41:50 -0500 (EST)
• References: <200412250900.EAA18567@smc.vnet.net>
• Sender: owner-wri-mathgroup at wolfram.com

Unfortunately Mathematica seems unable to prove this identity in 
general. At least the best I have been able to do with it is to verify 
the identity for a given pair of integers n and k, with n> k and for 
arbitrary S.

First we slightly redefine your functions, using a singe function name:

PS[n_, k_, S_, S_] =
    FullSimplify[
     Sum[(1/(j + 1))*Binomial[
        n - 1, j]*(1/S)^j*
       (1 - 1/S)^(n - 1 - j),
      {j, k, n - 1}],
     {s > 0, S > 0, n > 0}];


PS[n_, k_, 1, S_] = FullSimplify[
     Sum[(Binomial[n - 1, k - i]*(1 - 1/S)^(k - i)*
        (1/S)^(i - k + n - 1))/(i - k + n), {i, 0, k}],
     {s > 0, S > 0, n > 0}];


PS[n_, k_, s_, S_] = FullSimplify[
     Sum[(Binomial[n - 1, k - i]*(1 - s/S)^(k - i)*
        Binomial[(n - 1) - (k - i), j]*(1/S)^j*
        ((s - 1)/S)^((n - 1) - (k - i) - j))/(j + 1),
      {i, 0, k}, {j, i, n - 1}], {s > 0, S > 0, n > 0}];

Mathematica usually expresses such sums in terms of generalised 
functions, when it can find the sums, which is in the case of the  
first and second definition above but not in the third.

PS[n,k,S,S]//InputForm


((-1 + S)^(-1 - k + n)*S^(1 - n)*Gamma[n]*
   Hypergeometric2F1Regularized[1, 1 + k - n, 2 + k,
    (1 - S)^(-1)])/Gamma[-k + n]


PS[n,k,s,S]//InputForm

Sum[((-1 + s)^(-1 + i - j - k + n)*S^(1 - n)*
    (-s + S)^(-i + k)*Gamma[n])/(Gamma[2 + j]*
    Gamma[1 - i + k]*Gamma[i - j - k + n]), {i, 0, k},
  {j, i, -1 + n}]

Let's now define the function we would like to prove is zero:

f[n_, k_, S_] := Sum[PS[n, k, p, S], {p, 1, S}] - S/n

We can use it to numerically verify your claim for a symbolic S. This 
works even without using FullSimplify:

f[#,Random[Integer,{1,10}],S]&/@Range[20,30]


{0,0,0,0,0,0,0,0,0,0,0}

Unfortunately using FullSimplify on f does not help:

In[10]:=
FullSimplify[f[n, k, S],
   (k | n | S) $B":(B Integers &&
    n > k && n > 0 && k > 0 &&
    S > 0]

Out[10]=
-(S/n) + Sum[
    ((-1 + p)^(-1 + i - j - k +
        n)*S^(1 - n)*(-p + S)^
       (-i + k)*Gamma[n])/
     (Gamma[2 + j]*Gamma[
       1 - i + k]*Gamma[
       i - j - k + n]),
    {p, 1, S}, {i, 0, k},
    {j, i, -1 + n}]


All we get is the identity to be proved in a somewhat different form.
What's worse is that using  FullSimplify with a numerical S and 
arbitrary n and k gives an error:


FullSimplify[h[n, k, S] /. S -> 10,
   (k | n) $B":(B Integers && n > k && n > 0 &&
    k > 0]

Infinite expression 1/0    encountered.

Indeterminate

The same happens with f instead of h.


Andrzej Kozlowski
Chiba, Japan
http://www.akikoz.net/~andrzej/
http://www.mimuw.edu.pl/~akoz/






On 25 Dec 2004, at 18:00, m.elhaddad at gmail.com wrote:

> Hi All,
>
> I'm trying to use mathematica to verify the value of a summation. It
> runs for a long time without returning a solution. I'm not sure my
> input to Mathematica is in  the proper form.
>
> I'm including the mathematica cell expressions below. There I define
> three functions
> PS(N,k,s,S): defined for 1<s<S
> PS1(N,k,S): defined at s=1, and
> PSS(N,k,S): defined at s = S.
>
> Each of these functions is a summation. I'd like to verify that
> for any k,N, and S>2:  PS1+PSS+sum_{2<s<S}PS = S/N, which seems to be a
> correct by numerical evaluation.
>
> Doing it by hand appears to be beyond my reach. I'd appreciate your
> help whether you can see that it holds (or not), or know how to get
> mathematica to work it out.
>
> Thanks,
> --MH
>
> These are the four cell expressions:
>
> Cell[BoxData[
> RowBox[{
> RowBox[{"PS", "[",
> RowBox[{"N_", ",", "k_", ",", "s_", ",", "S_"}], "]"}], " ",
> ":=",
> " ",
> RowBox[{
> UnderoverscriptBox["\[Sum]",
> RowBox[{"i", "=", "0"}], "k"],
> RowBox[{"(", " ",
> RowBox[{
> UnderoverscriptBox["\[Sum]",
> RowBox[{"j", "=", "i"}],
> RowBox[{"N", "-", "1"}]],
> RowBox[{"(",
> RowBox[{
> FractionBox["1",
> RowBox[{"j", "+", "1"}]], " ",
> RowBox[{"Binomial", "[",
> RowBox[{
> RowBox[{"N", "-", "1"}], ",",
> RowBox[{"k", "-", "i"}]}], "]"}], " ",
> SuperscriptBox[
> RowBox[{"(",
> RowBox[{"1", "-",
> FractionBox["s", "S"]}], ")"}],
> RowBox[{"k", "-", "i"}]],
> RowBox[{"Binomial", "[",
> RowBox[{
> RowBox[{
> RowBox[{"(",
> RowBox[{"N", "-", "1"}], ")"}], "-",
> RowBox[{"(",
> RowBox[{"k", "-", "i"}], ")"}]}], ",", "j"}],
> "]"}], " ",
> SuperscriptBox[
> RowBox[{"(",
> FractionBox["1", "S"], ")"}], "j"], " ",
> SuperscriptBox[
> RowBox[{"(",
> FractionBox[
> RowBox[{"s", "-", "1"}], "S"], ")"}],
> RowBox[{
> RowBox[{"(",
> RowBox[{"N", "-", "1"}], ")"}], "-",
> RowBox[{"(",
> RowBox[{"k", "-", "i"}], ")"}], "-", "j"}]]}],
> ")"}]}], ")"}]}]}]], "Input"]
>
> Cell[BoxData[
> RowBox[{
> RowBox[{"PS1", "[",
> RowBox[{"N_", ",", "k_", ",", "S_"}], "]"}], " ", ":=", " ",
> RowBox[{
> UnderoverscriptBox["\[Sum]",
> RowBox[{"i", "=", "0"}], "k"],
> RowBox[{"(",
> RowBox[{
> RowBox[{"Binomial", "[",
> RowBox[{
> RowBox[{"N", "-", "1"}], ",",
> RowBox[{"k", "-", "i"}]}], "]"}], " ",
> SuperscriptBox[
> RowBox[{"(",
> RowBox[{"1", "-",
> FractionBox["1", "S"]}], ")"}],
> RowBox[{"k", "-", "i"}]], " ",
> FractionBox["1",
> RowBox[{"N", "-", "k", "+", "i"}]], "  ",
> SuperscriptBox[
> RowBox[{"(",
> FractionBox["1", "S"], ")"}],
> RowBox[{"N", "-", "1", "-", "k", "+", "i"}]]}], " ",
> ")"}]}]}]], "Input"]
>
> Cell[BoxData[
> RowBox[{
> RowBox[{"PSS", "[",
> RowBox[{"N_", ",", "k_", ",", "S_"}], "]"}], " ", ":=", "  ",
>
> RowBox[{
> UnderoverscriptBox["\[Sum]",
> RowBox[{"j", "=", "k"}],
> RowBox[{"N", "-", "1"}]],
> RowBox[{"(",
> RowBox[{
> FractionBox["1",
> RowBox[{"j", "+", "1"}]], " ",
> RowBox[{"Binomial", "[",
> RowBox[{
> RowBox[{"N", "-", "1"}], ",", "j"}], "]"}], " ",
> SuperscriptBox[
> RowBox[{"(",
> FractionBox["1", "S"], ")"}], "j"], " ",
> SuperscriptBox[
> RowBox[{"(",
> RowBox[{"1", "-",
> FractionBox["1", "S"]}], ")"}],
> RowBox[{"N", "-", "1", "-", "j"}]]}],
> ")"}]}]}]], "Input"]
>
> Cell[BoxData[
> RowBox[{"FullSimplify", "[",
> RowBox[{
> RowBox[{
> RowBox[{"PS1", "[",
> RowBox[{"N", ",", "k", ",", "S"}], "]"}], "+",
> RowBox[{"PSS", "[",
> RowBox[{"N", ",", "k", ",", "S"}], "]"}], "+",
> RowBox[{
> UnderoverscriptBox["\[Sum]",
> RowBox[{"s", "=", "2"}], "S"],
> RowBox[{"PS", "[",
> RowBox[{"N", ",", "k", ",", "s", ",", "S"}], "]"}]}]}],
> ",", " ",
> RowBox[{"S", ">", "2"}]}], "]"}]], "Input",
> CellLabel->"In[6]:="]
>
>
>

• References:
  • Help with a summation
    • From: m.elhaddad@gmail.com