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I have the following series: $$F(l, m) = \frac{(-1)^m}{2^{2l+m}}\sum_{j=0}^l C_l^jC_{m+l}^j(-3)^j, $$ where $l$, $m \in \mathbb N_+$ are positive integers and $C_l^j$ is the binomial coefficient. The problem is to find the largest value of $F(l,m)$.

I tried a lot of calculations and guess that $F(1,1) = 5/8$ reaches the maximum point, but I have no idea about the proof. Could anyone give me some idea to show this?

It seems that for fixed $l$, $|F(l,m)|$ first increases to its maximum and keep decreasing after that. And similar thing happens when $m$ is fixed. If one can find a number $N$ large enough such that the upper bound holds for all $l$, $m > N$, then the other cases can be checked by programming.

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  • $\begingroup$ $C_l^j$ is the binomial coefficent? $\endgroup$ – tired Dec 1 '16 at 12:12
  • $\begingroup$ yes. I edited it. $\endgroup$ – gregarki khayal Dec 1 '16 at 12:16
  • $\begingroup$ Wolfram Mathematica gives the following expression for the $F(l,m)$ $$ F(l,m) = (-1)^m 2^{-2 l-m} \, _2F_1(-l,-l-m;1;-3) $$ $\endgroup$ – uranix Dec 1 '16 at 13:25
  • $\begingroup$ @uranix yes, i also find that, and the problem becomes to estimate the hypergeometric function 2F1[-l,-l-m,1,-3]; do you know where i can find some useful information about such an upper bound? $\endgroup$ – gregarki khayal Dec 1 '16 at 13:29
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The function can be written as $$ F(l,m) = (-1)^m 2^{-2l-m} {}_2F_1(-l, -l-m; 1; -3) $$ where ${}_2F_1(a,b;c;z)$ is the Hypergeometric function. Applying Pfaff's transformation $$ {}_{2}F_{1}(a,b;c;z)=(1-z)^{-a}{}_{2}F_{1}\left(a,c-b;c;{\tfrac {z}{z-1}}\right) $$ we obtain $$ F(l,m) = \left(-\frac{1}{2}\right)^m {}_2F_1\left(-l, 1+l +m, 1; \frac{3}{4}\right) $$ The ${}_2F_1\left(-l, 1+l +m, 1; z\right)$ is a Jacobi polynomial $$ {}_{2}F_{1}(-l,1+m +l;1;z)=P_{l}^{(0 ,m)}(1-2z), $$ thus $$ F(l,m) = (-2)^{-m} P_l^{(0,m)}\left(-\frac{1}{2}\right) = (-1)^{l+m} 2^{-m} P_l^{(m,0)}\left(\frac{1}{2}\right). $$

Now the trick is to estimate the Jacobi polynomial. There are many papers for the different cases, but I've failed to find one with useful bound.

It seems that for fixed $l$ the function $|F(l,m)|$ oscillates when $m \lesssim 2l$ and quickly decays after $m > 2l$. Also $$ \max_{m \in \mathbb N^+} |F(l,m)| < \max_{1 < m < \infty} |F(l,m)| $$ (global maximum is attained just before the decay) is a monotonically decreasing function of $l$, so the maximum is really at $l = m = 1$.

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