Shareholders' Equity and Share-Based Payments (Details) - USD ($)	3 Months Ended		9 Months Ended
	Jun. 30, 2024	Jun. 30, 2023	Jun. 30, 2024	Jun. 30, 2023	Mar. 31, 2024	Sep. 30, 2023
Shareholders' Equity and Share-Based Payments
Common stock, shares authorized (in shares)	75,000,000		75,000,000			75,000,000
Preferred stock, shares authorized (in shares)	10,000,000		10,000,000			10,000,000
Amended and Restated 2019 Plan
Shareholders' Equity and Share-Based Payments
Share-based compensation expense	$ 250,901	$ 725,955	$ 214,084	$ 1,230,655
Unrecognized compensation cost, related to non-vested stock options	2,165,328		2,165,328
Amended and Restated 2019 Plan | Maximum
Shareholders' Equity and Share-Based Payments
Number of shares of common stock available for awards					1,950,000
Amended and Restated 2019 Plan | Employee
Shareholders' Equity and Share-Based Payments
Share-based compensation expense	191,623	164,342	566,952	954,140
Amended and Restated 2019 Plan | Non Employee Director
Shareholders' Equity and Share-Based Payments
Share-based compensation expense	$ 59,278	$ 49,742	$ 159,003	$ 276,515