We show that the cohomology intersection number of a twisted Gauss–Manin connection with regularization condition is a rational function. As an application, we obtain a new quadratic relation associated to period integrals of a certain family of K3 surfaces.

We give a new hypergeometric construction of rational approximations to ζ(4), which absorbs the earlier one from 2003 based on Bailey's 9F8 hypergeometric integrals. With the novel ingredients we are able to gain better control of the arithmetic and produce a record irrationality measure for ζ(4).

The aim of this article is to investigate the solutions of generalized fractional partial differential equations involving Hilfer time fractional derivative and the space fractional generalized Laplace operators, occurring in quantum mechanics. The solutions of these equations are obtained by employing the joint Laplace and Fourier transforms, in terms of the Fox’s H-function. Several special cases as solutions of one dimensional non-homogeneous fractional equations occurring in the quantum mechanics are presented. The results given earlier by Saxena et al. [Fract. Calc. Appl. Anal., 13(2) (2010), pp. 177–190] and Purohit and Kalla [J. Phys. A Math. Theor., 44 (4) (2011), 045202] follow as special cases of our findings.

Let $D$ be the $n$-dimensional Lie ball and let $B\text{(S)}$ be the space of hyperfunctions on the Shilov boundary $S$ of $D$. The aim of this paper is to give a necessary and sufficient condition on the generalized Poisson transform ${{P}_{l,\text{ }\!\!\lambda\!\!\text{ }}}f$ of an element $f$ in the space $B\text{(S)}$ for $f$ to be in ${{L}^{p}}\left( S \right)$, $1\,<\,p\,<\,\infty $. Namely, if $F$ is the Poisson transform of some $f\in \,B(S)$$F\,=\,{{P}_{l,\lambda }}f$), then for any $l\,\in \,Z$) and $\lambda \,\in \,C$ such that $Re[\text{i}\lambda ] > \frac{n}{2}\,-\,1$, we show that $f\,\in \,{{L}^{p}}\text{(}S\text{)}$ if and only if $f$ satisfies the growth condition

$${{\left\| F \right\|}_{\lambda ,p}}=\underset{0\le r<1}{\mathop{\sup }}\,{{\left( 1\,-\,{{r}^{2}} \right)}^{\operatorname{Re}\left[ \text{i }\lambda \text{ } \right]-\frac{n}{2}+l}}{{\left[ \,\int_{s}{{{\left| F\left( ru \right) \right|}^{p}}du} \right]}^{\frac{1}{p}}}<\,+\infty $$

We study the genealogy of so-called immortal branching processes, i.e. branching processes where each individual upon death is replaced by at least one new individual, and conclude that their marginal distributions are compound geometric. The result also implies that the limiting distributions of properly scaled supercritical branching processes are compound geometric. We exemplify our results with an expression for the marginal distribution for a class of branching processes that have recently appeared in the theory of coalescent processes and continuous stable random trees. The limiting distribution can be expressed in terms of the Fox H-function, and in special cases by the Meijer G-function.