Now let us denote by $\Lambda^{*}(n)$ the algebra of polynomials in $x_{1},\ldots,x_{n}$ that become symmetric in new variables $$ x_{i}'=x_{i}-i+c, \ i \in 1,\ldots,n.$$ Here c is a arbitrary fixed number; note that the definition does not depend on it's choice. We call such polynomials shifted symmetric. The algebra $\Lambda^{*}(n)$ is filtered by degree of polynomials, and the specialization $x_{n+1}=0$ is a morphisms of filtered algebras $$\Lambda^{*}(n+1)\rightarrow \Lambda^{*}(n)$$

Let $$\Lambda^{*} = \lim_{\rightarrow}\Lambda^{*}(n), n\rightarrow \infty $$ be the projective limit in the category of filtered algebras, taken with respect to morpshims. We call $\Lambda^{*}$ the algebra of shifted algebra of shifted symmetric functions.

The most important example of shifted symmetric function is shifted schur function $$s_{\mu}^{*}(x_1,\ldots,x_n)=\frac{det\Big((x_i+n-i|\mu_j+n-j)\Big)}{det\Big((x_i+n-i|n-j)\Big)} $$

Let $s_{\mu}$ denote schur function. Then we have the following generating function $$ \sum_{n\geq 0}s_{(n)}(q_1,q_2,\ldots,q_r)x^n$$ in holonomic function in $x$. where $r$ is fixed and all other $\{q_{s}=0|s>r\}$. There is an explicit expression in terms of exponential. This shows that it's holonomic.

We epxress $s_{(n)}(q_1,q_2,\ldots,q_r)$ in power-sum symmetric function. So $$ \sum_{n\geq 0}s_{(n)}(q_1,q_2,\ldots,q_r)x^n$$ becomes $$exp\Big(\sum_{m=1}^{r}(q_m x^m )/m\Big) .$$

I want to know if the following generating series is holonomic.

$$\sum_{n\geq 0 }s_{(n)}^{*}(q_1,q_2,\ldots,q_r)x^n .$$ is holonomic?

$$\sum_{n\geq 0 }s_{(n)}^{*}(q_1,q_2,\ldots,q_r)(x|n) .$$ is holonomic?