Fuchsian differential equations H_j of order j = 3, ..., 6 with three singular points and one accessory parameter are presented. The shift operators for H₆ are studied. They lead to assign the accessory parameter of H₆ a cubic polynomial of local exponents so that the equation has several nice symmetries. The other equations will be studied in the forthcoming papers.

We define a hypergeometric series in m variables with p+(p−1)m parameters, which reduces to the generalized hypergeometric series _pF_p−1 when m=1, and to Lauricella's hypergeometric series F_C in m variables when p=2. We give a system of hypergeometric differential equations annihilating the series. Under some non-integral conditions on parameters, we give an Euler type integral representation of the series, and p^m linearly independent solutions to this system around a point near to the origin. We show that this system is of rank p^m, and determine its singular locus.

Discretely self-similar solutions to Oberbeck-Boussinesq system with Newtonian gravitational field for large discretely self-similar initial data are constructed in this note, extending the construction of Brandolese and Karch on self-similar solutions. It follows the approach of Bradshaw and Tsai and finds an explicit a priori bound for the deviation from suitably revised profiles in similarity variables.