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Help with a summation

  • To: mathgroup at smc.vnet.net
  • Subject: [mg53150] Help with a summation
  • From: m.elhaddad at gmail.com
  • Date: Sat, 25 Dec 2004 04:00:56 -0500 (EST)
  • Sender: owner-wri-mathgroup at wolfram.com
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]:="]
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