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In this paper, we find new proofs of modular relations for the Göllnitz-Gordon functions established earlier by S.-S. Huang and S.-L. Chen. We use Schröter's formulas and some simple theta-function identities of Ramanujan to establish the relations. We also find some new modular relations of the same nature.

In the 1980's, Greene defined hypergeometric functions over finite fields using Jacobi sums. The framework of his theory establishes that these functions possess many properties that are analogous to those of the classical hypergeometric series studied by Gauss and Kummer. These functions have played important roles in the study of Apéry-style supercongruences, the Eichler-Selberg trace formula, Galois representations, and zeta-functions of arithmetic varieties. We study the value distribution (over large finite fields) of natural families of these functions. For the 2 F 1 functions, the limiting distribution is semicircular, whereas the distribution for the 3 F 2 functions is the more exotic Batman distribution.

The extended transformation method has been applied to the exactly solvable stipulated Morse potential and trigonometric Scarf potential, to generate a set of exactly solvable quantum systems (QSs) in any chosen dimension. Bound state solutions of the exactly solvable potentials are given. The generated QSs are generally of Sturmian form. We also report a system case-specific regrouping technique to convert a Sturmian QS to a normal QS. A second-order application of the transformation method is given. The normalizability of the generated QSs is generally given in Sturmian form.

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