LEAPS-MPS: Dynamic Special Functions

Marshall University Research Corporation, Huntington WV

Investigators

Abstract

This award will support the development of a comprehensive generalized theory of "hypergeometric functions", which are long-studied classical mathematical objects that have appeared in many scientific applications across the last two hundred years. The research focus lies in the theory of time scales calculus, which is an area of mathematics that unifies discrete and continuous analysis and extends them to hybrid domains "in between". Four research projects will be pursued which feed into a primary goal of the unification of special functions on time scales with a secondary goal of the extension of hypergeometric functions to time scales. Hypergeometric functions remain heavily studied by researchers across the world and allow for analytical solutions to problems that otherwise must be solved numerically or with complicated expressions. The research is accessible to both undergraduates and graduate student. This award will create opportunities for Marshall University students to perform and share original research, which is significant for growing and sustaining mathematics education beyond the undergraduate level in West Virginia. Support is included for the Marshall University Differential Analyzer Lab, which houses the only known publicly-accessible differential analyzer in the country. More precisely, the four main research projects are (1) to resolve the longstanding time scales shifting problem, which will allow for the generalization of all polynomial-coefficient differential equations to time scales, which includes hypergeometric differential equations; (2) to develop Bessel functions on time scales, which the discrete calculus literature shows can be a useful step towards the secondary goal; (3) to discover new oscillation results for classical hypergeometric functions, which will be new contributions to classical special functions theory; and (4) to attain new lower bound estimates for matrix-valued hypergeometric functions, which has potential to advance their use in applications. These four projects form a strong basis for achieving the secondary goal and represent a significant step towards the primary goal. In addition to these theoretical projects, support is provided for student lab assistants to fund their training and to work in the Marshall University Differential Analyzer Lab. Differential analyzers were mechanical computers that solve differential equations and were used extensively in the early twentieth century. In the last two decades, replica machines were constructed by the mathematics faculty and students at Marshall University and have been the topic of numerous master's theses, classroom labs, recruitment events, and mathematics outreach activities. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.