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