Proofs of some combinatorial identities

Question

Just wondering if anyone knows any references in the literature to bijections corresponding to the following simple generating function identities. Let $B(z)=\dfrac{1}{\sqrt{1-4z}}$ and $C(z)=\dfrac{1-\sqrt{1-4z}}{2z}$, the generating functions of the central binomial coefficients and Catalan numbers, respectively. I'm looking for bijections corresponding to the following identities: $$B(z)B(-z)=B(4z^2),$$ $$\frac{C(z)+C(-z)}{2}=B(z)(1-2zC(4z^2))=B(-z)(1+2zC(4z^2))$$ and, taking the product of the last two expressions in the second identity and using the first identity and the fact that $1-zC^2=C/B$, $$\left(\frac{C(z)+C(-z)}{2}\right)^2=C(4z^2).$$ Thanks.

Update: In a similar vein, one can try to prove other such identities. For example, let $E(z)=\dfrac{1}{\sqrt{1-2z-3z^2}}$ be the generating function of the central trinomial coefficients, then $$E(4z)=B(-z)B(3z) \quad \text{and} \quad E(2z)E(-2z)=B(z^2)B(9z^2).$$

Answer 1

Partial answer: Your first identity is \begin{equation} \sum\limits_{k=0}^n \left(-1\right)^k \dbinom{2k}{k} \dbinom{2\left(n-k\right)}{n-k} = \left[n \text{ is even}\right] 2^n \dbinom{n}{n/2} , \end{equation} where I am using the Iverson bracket notation. (That is, $[\mathcal{A}]$ denotes the truth value of a statement $\mathcal{A}$.)

This identity is proven in: Michael Z. Spivey, A Combinatorial Proof for the Alternating Convolution of the Central Binomial Coefficients. This paper actually arose from an m.se question.

Disclaimer: I have not read the proof.

Answer 2

It's not the formally published literature, but unless I'm mistaken, the last of your three identities appears in Stanley's list of bijective proof problems (dated 2009) as number 194. He says there that finding a combinatorial bijection for the identity is an open problem. (He suggests some of the open problems in the list as being of particular interest, however, and 194 is not one of those.)

This suggests rather strongly that a bijective proof for the third identity isn't known, or at least wasn't known as of 2009.

I always found it a little surprising that this one was open. Both sides of the identity count natural things: The LHS counts the number of lattice paths from $(0,0)$ to $(2n,2n)$ that do not go above the diagonal, and that return to the diagonal at a distinguished point $(2k,2k)$. The RHS the number of lattice paths from $(0,0)$ to $(n,n)$ that do not go above the diagonal, and whose edges/steps are 2-colored.