Finding the maximum value of the following equation












2












$begingroup$


For $Nin mathbb{N}$, $Min mathbb{N}$, and $Kin mathbb{N}$, $f(K)$ is given by
begin{equation}
f(K) = sumlimits_{i = 1}^{K} {left( {frac{{left( {M - K} right)!left( {M - i} right)!}}{{left( {M - i - K} right)!M!}}} right)^{N - 1} cdot left( begin{array}{c}
K \
i \
end{array} right) cdot left( { - 1} right)^{i - 1} }
end{equation}
,
where $Mge K$.



We found that $f(K)$ is a convex function by plotting $f(K)$ for varying $K$.
We would like to solve the above equation as a closed-form.
Does anyone help us for solving it or suggesting any ideas?










share|cite|improve this question











$endgroup$












  • $begingroup$
    I think it would help if we had some context as to where this equation came from. Also, why you are convinced there is a nice closed form.
    $endgroup$
    – Don Thousand
    Dec 5 '18 at 16:51








  • 1




    $begingroup$
    Also, what maximum value do we have to find?
    $endgroup$
    – Federico
    Dec 5 '18 at 16:53
















2












$begingroup$


For $Nin mathbb{N}$, $Min mathbb{N}$, and $Kin mathbb{N}$, $f(K)$ is given by
begin{equation}
f(K) = sumlimits_{i = 1}^{K} {left( {frac{{left( {M - K} right)!left( {M - i} right)!}}{{left( {M - i - K} right)!M!}}} right)^{N - 1} cdot left( begin{array}{c}
K \
i \
end{array} right) cdot left( { - 1} right)^{i - 1} }
end{equation}
,
where $Mge K$.



We found that $f(K)$ is a convex function by plotting $f(K)$ for varying $K$.
We would like to solve the above equation as a closed-form.
Does anyone help us for solving it or suggesting any ideas?










share|cite|improve this question











$endgroup$












  • $begingroup$
    I think it would help if we had some context as to where this equation came from. Also, why you are convinced there is a nice closed form.
    $endgroup$
    – Don Thousand
    Dec 5 '18 at 16:51








  • 1




    $begingroup$
    Also, what maximum value do we have to find?
    $endgroup$
    – Federico
    Dec 5 '18 at 16:53














2












2








2





$begingroup$


For $Nin mathbb{N}$, $Min mathbb{N}$, and $Kin mathbb{N}$, $f(K)$ is given by
begin{equation}
f(K) = sumlimits_{i = 1}^{K} {left( {frac{{left( {M - K} right)!left( {M - i} right)!}}{{left( {M - i - K} right)!M!}}} right)^{N - 1} cdot left( begin{array}{c}
K \
i \
end{array} right) cdot left( { - 1} right)^{i - 1} }
end{equation}
,
where $Mge K$.



We found that $f(K)$ is a convex function by plotting $f(K)$ for varying $K$.
We would like to solve the above equation as a closed-form.
Does anyone help us for solving it or suggesting any ideas?










share|cite|improve this question











$endgroup$




For $Nin mathbb{N}$, $Min mathbb{N}$, and $Kin mathbb{N}$, $f(K)$ is given by
begin{equation}
f(K) = sumlimits_{i = 1}^{K} {left( {frac{{left( {M - K} right)!left( {M - i} right)!}}{{left( {M - i - K} right)!M!}}} right)^{N - 1} cdot left( begin{array}{c}
K \
i \
end{array} right) cdot left( { - 1} right)^{i - 1} }
end{equation}
,
where $Mge K$.



We found that $f(K)$ is a convex function by plotting $f(K)$ for varying $K$.
We would like to solve the above equation as a closed-form.
Does anyone help us for solving it or suggesting any ideas?







optimization convex-analysis convex-optimization






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Dec 9 '18 at 13:36







Taehoon

















asked Dec 5 '18 at 16:42









TaehoonTaehoon

112




112












  • $begingroup$
    I think it would help if we had some context as to where this equation came from. Also, why you are convinced there is a nice closed form.
    $endgroup$
    – Don Thousand
    Dec 5 '18 at 16:51








  • 1




    $begingroup$
    Also, what maximum value do we have to find?
    $endgroup$
    – Federico
    Dec 5 '18 at 16:53


















  • $begingroup$
    I think it would help if we had some context as to where this equation came from. Also, why you are convinced there is a nice closed form.
    $endgroup$
    – Don Thousand
    Dec 5 '18 at 16:51








  • 1




    $begingroup$
    Also, what maximum value do we have to find?
    $endgroup$
    – Federico
    Dec 5 '18 at 16:53
















$begingroup$
I think it would help if we had some context as to where this equation came from. Also, why you are convinced there is a nice closed form.
$endgroup$
– Don Thousand
Dec 5 '18 at 16:51






$begingroup$
I think it would help if we had some context as to where this equation came from. Also, why you are convinced there is a nice closed form.
$endgroup$
– Don Thousand
Dec 5 '18 at 16:51






1




1




$begingroup$
Also, what maximum value do we have to find?
$endgroup$
– Federico
Dec 5 '18 at 16:53




$begingroup$
Also, what maximum value do we have to find?
$endgroup$
– Federico
Dec 5 '18 at 16:53










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