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Shareholders' Equity and Share-Based Payments (Details) - USD ($)	3 Months Ended		6 Months Ended
	Mar. 31, 2023	Mar. 31, 2022	Mar. 31, 2023	Mar. 31, 2022	Sep. 30, 2022
Shareholders' Equity and Share-Based Payments
Common stock, shares authorized	75,000,000		75,000,000		75,000,000
Preferred stock, shares authorized	10,000,000		10,000,000		10,000,000
2019 Plan
Shareholders' Equity and Share-Based Payments
Common stock, shares authorized	139,691		139,691
Number of shares of common stock reserved for awards	307,905		307,905
Share-based compensation expense	$ 733,376	$ 204,589	$ 1,016,571	$ 250,180
Unrecognized compensation cost, related to non-vested stock options	$ 370,427		$ 370,427
2019 Plan \| Maximum
Shareholders' Equity and Share-Based Payments
Number of shares of common stock reserved for awards	750,000		750,000
2019 Plan \| Employee
Shareholders' Equity and Share-Based Payments
Number of shares of common stock reserved for awards	300,000		300,000
Share-based compensation expense	$ 556,673	44,594	$ 789,798	90,185
2019 Plan \| Non Employee Director
Shareholders' Equity and Share-Based Payments
Share-based compensation expense	$ 176,703	$ 159,995	$ 226,773	$ 159,995