Theorem: The following formula holds: $$n^2(2n-3)Z_n(x)-(2n-1)[3n^2-6n+2-2(2n-3)x]Z_{n-1}(x)+(2n-3)[3n^2-6n+2+2(2n-1)x]Z_{n-2}(x)-(2n-1)(n-2)^2Z_{n-3}(x)=0.$$

Theorem: The following formula holds: $$xZ_n'(x)-nZ_n(x)=-nZ_{n-1}(x)-xZ_{n-1}'(x).$$

Theorem: The following formula holds: $$xZ_n'(x)-nZ_n(x)=-\displaystyle\sum_{k=0}^{n-1} Z_k(x) - 2x\displaystyle\sum_{k=0}^{n-1} Z_k'(x).$$

Theorem: The following formula holds: $$xZ_n'(x)-nZ_n(x)=\displaystyle\sum_{k=0}^{n-1} (-1)^{n-k}(2k+1)Z_k(x).$$

Theorem: The following formula holds: $$\dfrac{1}{1-t} {}_1F_1 \left( \dfrac{1}{2} ; 1 ; -\dfrac{4xt}{(1-t)^2} \right) = \displaystyle\sum_{k=0}^{\infty} Z_n(x)t^n,$$ where ${}_1F_1$ denotes the generalized hypergeometric function.