# Annihilator of the of the generating function not holonomic

The following is a generating function in $$x,h$$ with infinite parameters $$q_1,q_2\ldots,$$ and $$w_1, w_2,\ldots$$.
$$\Psi(x, h)= \sum_{d=0}^{\infty} s_{(d)} (q_1, q_2, \ldots) \exp \bigg( \sum_{r=1}^{\infty} \frac{w_r h^r}{(r+1)!} \left[ \bigg(d-\frac{1}{2}\bigg)^{r+1} - \bigg(-\frac{1}{2}\bigg)^{r+1} \right] \bigg) x^d h^{-d}$$

I want to find the annihilator of the above generating function. Let me list down the cases when it's known So when $$q_1 =q_1$$ and all other $$\{q_i =0\mid i\geq 2\}$$ and $$w_r=w_r$$ and $$\{w_i =0\mid i\geq 2\}$$ $$\big[ h x\frac{\partial}{\partial x}-q_1\hat{x}\mathcal{A} \big] \Psi(x, h)=0$$ where $$\mathcal{A}:=x^{\frac32}exp\Big(w_{r}\frac{x^{-1}\sum_{i=0}^{r}(hx\frac{\partial}{\partial x})^{i}x(hx\frac{\partial}{\partial x})^{r-i}}{r+1}\Big)x^{-\frac{1}{2}}$$ $$\hat{x} = x$$ and $$\hat{y} = h x \frac{\partial}{\partial x}$$. As $$s_{d}(q_1)=\frac{q_1^d}{d!}$$ and $$\mathcal{A}x^d=exp(\left[ \bigg(d+\frac{1}{2}\bigg)^{r+1} - \bigg(d-\frac{1}{2}\bigg)^{r+1} \right])x^{d+1}$$ Hence it annihilates.

Let's fix $$q_1,q_2,\ldots,q_e$$ and $$w_1, \ldots w_{e-1}$$ Is this true that

$$\big[ h x\frac{\partial}{\partial x}-(q_1\hat{x}\mathcal{A}+q_2 \hat{x}^{2}\mathcal{A}^2+\ldots +q_e \hat{x}^{e}\mathcal{A}^e \big] \Psi(x, h) = 0,$$ where $$\mathcal{A}:=x^{\frac32}exp\Big(\sum_{r=1}^{e-1}w_{r}\frac{x^{-1}\sum_{i=0}^{r}(hx\frac{\partial}{\partial x})^{i}x(hx\frac{\partial}{\partial x})^{r-i}}{r+1}\Big)x^{-\frac{1}{2}}$$ $$\hat{x} = x$$ and $$\hat{y} = h x \frac{\partial}{\partial x}$$?

I believe there should be an annihilator as each of the series $$\sum_d s_d(q_1,\ldots q_r)x^d$$ is annihilated by a holonomic function and similarly for $$\sum_d \exp \bigg( \sum_{r=1}^{e-1} \frac{w_r h^r}{(r+1)!} \left[ \bigg(d-\frac{1}{2}\bigg)^{r+1} - \bigg(-\frac{1}{2}\bigg)^{r+1} \right]x^d$$ Though the annihilator is not holonomic if it was then I could be Hadamard product to conclude the existence of the annihilator.