Two slightly different polynomial expansions for $\Xi(0)$. Could a connection between these two be derived?

Question

After experimenting with polynomial expansions for the Riemann $\Xi(t)=\xi\left(\frac12+it\right)$-function, I landed on these two equations:

with $M$ the KummerM confluent hypergeometric funcion and $L$ the generalised Laguerre polynomial.

Both equations are equal except for the square in the parameter of $\Xi(x)$.

Question (probably far fetched):

Could it be possible to derive some relation between $\Xi(x^2)$ and $\Xi(x)$ from these equations?

P.S. A similar set of equations exists for $\Xi(1)$.