Appell system associated with the infinite dimensional Fractional Pascal measure

In this work, we employ a biorthogonal approach to construct the infinite-dimensional Fractional Pascal measure \(\mu ^{(\alpha )}_{_{\sigma }}, 0 < \alpha \le 1\) $\mu _{_{\sigma }^{}}^{(\alpha )},0<\alpha \le 1$ , defined on the tempered distributions space \(\mathcal {E}'\) ${\mathcal{E}}^{\prime }$ over \(\mathbb {R} \times \mathbb {R}^{*}_{+}\) $\mathbb{R}\times \mathbb{R}_{+}^{\ast }$ . The Hilbert space \(L^{2}(\mu ^{(\alpha )}_{_{\sigma }})\) $L^2(\mu _{_{\sigma }^{}}^{(\alpha )})$ is characterized using a set of generalized Appell polynomials \(\mathbb {P}^{(\alpha )}_{\widehat{\sigma }}=\{P^{(\alpha )}_{n, \widehat{\sigma }}, n\in \mathbb {N}\}\) ${\mathbb{P}}_{\widehat{\sigma }}^{(\alpha )}=\{P_{n,\widehat{\sigma }}^{(\alpha )},n\in \mathbb{N}\}$ associated with the measure \(\mu ^{(\alpha )}_{_{\sigma }}\) $\mu _{_{\sigma }^{}}^{(\alpha )}$ . This paper presents novel properties of the kernels \(P^{(\alpha )}_{n, \widehat{\sigma }}\) $P_{n,\widehat{\sigma }}^{(\alpha )}$ in infinite dimensions, offering valuable insights. Additionally, we delve into the discussion of the generalized dual Appell system, broadening the scope of our results.