### A class of finite $q$-series

Skip to main content (access key 's'),
Skip to navigation (access key 'n'),
Accessibility information (access key '0')

When the parameters are real, the hypergeometric equation defines a Schwarz triangle. We study a combinatorial-topological property of the Schwarz triangle when the three angles are not necessarily less than π.

The maximal operator S⁎ for the spherical summation operator (or disc multiplier) ${S}_{R}$ associated with the Jacobi transform through the defining relation $\widehat{{S}_{R}f}\left(\lambda \right)={1}_{\left|\lambda \right|\le R}f\u0302\left(t\right)$ for a function f on ℝ is shown to be bounded from ${L}^{p}(\mathbb{R}\u208a,d\mu )$ into ${L}^{p}(\mathbb{R},d\mu )+L\xb2(\mathbb{R},d\mu )$ for (4α + 4)/(2α + 3) < p ≤ 2. Moreover S⁎ is bounded from ${L}^{p\u2080,1}(\mathbb{R}\u208a,d\mu )$ into ${L}^{p\u2080,\infty}(\mathbb{R},d\mu )+L\xb2(\mathbb{R},d\mu )$. In particular ${{S}_{R}f\left(t\right)}_{R>0}$ converges almost everywhere towards f, for $f\in {L}^{p}(\mathbb{R}\u208a,d\mu )$, whenever (4α + 4)/(2α + 3) < p ≤ 2.

Mathematics Subject Classification 2010: 35M10, 35R11, 26A33, 33C05, 33E12, 33C20.The paper deals with an analog of Tricomi boundary value problem for a partial differential equation of mixed type involving a diffusion equation with the Riemann-Liouville partial fractional derivative and a hyperbolic equation with two degenerate lines. By using the properties of the Gauss hypergeometric function and of the generalized fractional integrals and derivatives with such a function in the kernel, the uniqueness...