Shareholders' Equity and Share-Based Payments (Details) - USD ($)	3 Months Ended		6 Months Ended
	Mar. 31, 2024	Mar. 31, 2023	Mar. 31, 2024	Mar. 31, 2023	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	$ 269,338	$ 475,055	$ 733,376	$ 1,016,571
Unrecognized compensation cost, related to non-vested stock options	$ 1,425,821		$ 1,425,821
Amended and Restated 2019 Plan | Maximum
Shareholders' Equity and Share-Based Payments
Number of shares of common stock available for awards	1,950,000		1,950,000
Amended and Restated 2019 Plan | Employee
Shareholders' Equity and Share-Based Payments
Share-based compensation expense	$ 219,748	556,673	$ 375,328	789,798
Amended and Restated 2019 Plan | Non Employee Director
Shareholders' Equity and Share-Based Payments
Share-based compensation expense	$ 49,590	$ 176,703	$ 99,726	$ 226,773