Weird binomial identities

In this post, I present some relatively weird binomial identities. I will start by the symmetric case, and then consider the twisted case.

  1. Symmetric case

We prove the following identity
$$C_{l,l}(y) \underset{def}{=} \sum_{i=0}^l \binom{l+i}{l} y^i = \frac{(1+\rho)^l}{1-y} - \frac{y^{l+1}}{1-y}\cdot\frac{\rho^l}{2}\cdot\left( 1+\frac{1}{\rho}\right)^{2l+1}$$
where $y$ is a formal variable, and $\rho = y/(1-y)$. This identity follows from the classical method of coefficient identification.

Indeed, let's introduce a second variable $x$ and consider the polynomial
$$P = \sum_{i=0}^l y^i \cdot (1+x)^{l+i}$$
By applying the binomial formula to $(1+x)^{l+i}$, we see that the coefficient of $x^l$ in $P$ is
$$C_{l,l}(y) = \sum_{i=0}^l \binom{l+i}{l} y^i$$

On the other hand, in $P$, we have a geometric series
$$\begin{align}     P &= \sum_{i=0}^l y^i  (1+x)^{l+i} \\        &= (1+x)^l \cdot \sum_{i=0}^l (y+y x)^i \\        &= (1+x)^l \cdot \frac{1-y^{l+1}(1+x)^{l+1}}{1-y-y x} \\        &= \left( \frac{1}{1-y} (1+x)^l - \frac{y^{l+1}}{1-y} (1+x)^{2l+1} \right) \cdot \frac{1}{1-\rho x} \\        &= \left( \frac{1}{1-y} (1+x)^l - \frac{y^{l+1}}{1-y} (1+x)^{2l+1} \right) \cdot \sum_{s \ge 0} \rho^s  x^s   \end{align}$$

From this expression, the coefficient of $x^l$ is given by the convolution
$$\begin{align}   C_{l,l}(y) &= \sum_{s = 0}^l \left( \frac{1}{1-y} \binom{l}{l-s} - \frac{y^{l+1}}{1-y} \binom{2l+1}{l-s}\right) \cdot \rho^s \\         &= \frac{1}{1-y} \cdot \sum_{s = 0}^l \binom{l}{l-s} \rho^s - \frac{y^{l+1}}{1-y} \sum_{s=0}^l \binom{2l+1}{l-s} \rho^s \\         &= \frac{(1+\rho)^l}{1-y} - \frac{y^{l+1}}{1-y} \rho^l \sum_{s=0}^l \binom{2l+1}{l-s} \left(\frac{1}{\rho}\right)^{l-s} \\   \end{align}$$

Thanks to the symmetry $\binom{2l+1}{l-s} = \binom{2l+1}{l + 1 + s}$, we see that the sum in the last expression is
$$\frac{1}{2} \sum_{s = 0}^{2l+1} \binom{2l+1}{s} \left(\frac{1}{\rho}\right)^s = \frac{1}{2} \left(1+\frac{1}{\rho}\right)^{2l+1}$$

Therefore, writing the two expressions for the coefficient yields the claim
$$\sum_{i=0}^l \binom{l+i}{l} y^i = \frac{(1+\rho)^l}{1-y} - \frac{y^{l+1}}{1-y}\cdot\frac{\rho^l}{2}\cdot\left( 1+\frac{1}{\rho}\right)^{2l+1}$$

In particular, with $y = 1/2$, we get
$$\sum_{i=0}^l \binom{l+i}{l} \frac{1}{2^i} = 2^l$$

  2. Twisted case

We now consider the twisted case
$$C_{a,b}(y) \underset{def}{=} \sum_{i=0}^a \binom{b+i}{b} y^i.$$
The method is exactly the same except that we compute the coefficient of $x^b$ in the polynomial
$$\sum_{i=0}^a y^i (1+x)^{b+i}.$$
We get
$$C_{a,b}(y) = \frac{(1+\rho)^b}{1-y} - \frac{y^{a+1}}{1-y} \cdot \sum_{s=0}^b \binom{a+b+1}{b-s} \rho^s.$$
Unfortunately, the last sum does not simplify as before because of the asymmetry $a \ne b$ (a priori).

However, we can measure this twist by focusing on the quantity $y^b C_{a,b}(y) + y^a C_{b,a}(y)$. Regarding the parts containing the sums, we have
$$\begin{align}             \sum_{s=0}^b &\binom{a+b+1}{b-s} \rho^s + \sum_{s=0}^a \binom{a+b+1}{a-s} \rho^s \\      &=  \rho^b \cdot \sum_{s=0}^b \binom{a+b+1}{s} \left(\frac{1}{\rho}\right)^s + \rho^a \cdot \sum_{s=0}^a \binom{a+b+1}{s} \left(\frac{1}{\rho}\right)^s \\      &= \rho^b \left(1+\frac{1}{\rho}\right)^{a+b+1} - \rho^b \cdot \sum_{s=b+1}^{a+b+1} \binom{a+b+1}{s} \left(\frac{1}{\rho}\right)^s + \dots \\      &= \rho^b \left(1+\frac{1}{\rho}\right)^{a+b+1} - \rho^b \cdot \sum_{s=0}^{a} \binom{a+b+1}{a-s} \left(\frac{1}{\rho}\right)^s + \dots \\      &= \rho^b \left(1+\frac{1}{\rho}\right)^{a+b+1} + (\rho^a - \rho^b) \cdot \sum_{s=0}^{a} \binom{a+b+1}{a-s} \left(\frac{1}{\rho}\right)^s \\  \end{align}$$

Whence the identities
$$y^b C_{a,b}(y) + y^a C_{b,a}(y) \\      = \frac{y^a(1+\rho)^a + y^b (1+\rho)^b}{1-y}       - \frac{y^{a+b+1}}{1-y} \rho^b \left(1+\frac{1}{\rho}\right)^{a+b+1} - \frac{y^{a+b+1}}{1-y} (\rho^a - \rho^b) \sum_{s=0}^a \binom{a+b+1}{a-s} \left(\frac{1}{\rho}\right)^s \\      = \frac{y^a(1+\rho)^a + y^b (1+\rho)^b}{1-y}      - \frac{y^{a+b+1}}{1-y} \rho^a \left(1+\frac{1}{\rho}\right)^{a+b+1} - \frac{y^{a+b+1}}{1-y} (\rho^b - \rho^a) \sum_{s=0}^b \binom{a+b+1}{b-s} \left(\frac{1}{\rho}\right)^s$$

In particular, if $y = 1/2$, we have $\rho = 1$, and the twist is canceled so that
$$\sum_{i=0}^a \binom{b+i}{b} \frac{1}{2^{b+i}} + \sum_{i=0}^b \binom{a+i}{a} \frac{1}{2^{a+i}} = 2$$
which is, somewhat, funny.

Question. Is there a combinatorial proof of these identities ?

pb