Rendiconti dell’Istituto di Matematica dell’Università di Trieste: an International Journal of Mathematics vol.55 (2023)

We prove three uniqueness theorems for BVPs of ODEs whose unusual assumptions allow to prove existence results for BVPs related to various types of linear and nonlinear boundary conditions.

In this paper we have obtained some novel generating functions of Fn(α, β−α; x) – a modified form of the extended Jacobi polynomial Fn(α, β; x) – by means of Weisner’s group-theoretic method with the suitable interpretation of the parameter α of the polynomial under consideration. Moreover, we have shown that the generating functions involving the extended Jacobi polynomial Fn(α, β; x) derived in [2], obtained by using the same Weisner’s group-theoretic method with the suitable interpretation of (α, β), can be easily obtained from our results. Finally, a group-theoretic method of obtaining general bilateral generating relation from general unilateral generating relation is also discussed.

In this paper we prove that the space of circle expanding maps of degree 2 preserving Lebesgue measure is an arc-connected space homeomorphic to an infinite-dimensional Lie group whose fundamental group is Z. The techniques involved in the proof are rather unexpected and lead to a formulation of a conjecture generalizing this result to higher dimensional infra-nilmanifolds.

We advertise elementary symmetric polynomials ei as the natural basis for generating series Ag,n of intersection numbers of ψ-classes on the moduli space of stable curves of genus g with n marked points. Closed formulae for Ag,n are known for genera 0 and 1 — this approach provides formulae for g = 2, 3, 4, together with an algorithm to compute the formula for any g. The claimed naturality of the ei basis relies in the unexpected vanishing of some coefficients with a clear pattern. As an application of the conjecture, we find new integral representations of Ag,n, which recover expressions for the Weil-Petersson volumes in terms of Bessel functions.