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Stockholders' Equity (Tables) | 6 Months Ended
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Mar. 31, 2024
Stockholders' Equity Note [Abstract] |
Summary of accumulated other comprehensive income (loss) | ​
​ | ​ | ​ | ​ | ​ | ​ | ​ | ​ | ​ | ​ | ​ | ​ | ​ | ​ | ​ | ​
​ |  | ​ | ​ |  | Unrealized |  | ​ |  | ​ | ​ |  | ​ | ​
​ | ​ | ​ | ​ | ​ | Gains (Losses) | ​ | ​ | ​ | ​ | ​ | ​ | ​ | ​ | ​
​ | ​ | ​ | ​ | ​ | on Available- | ​ | ​ | ​ | Pension | ​ | ​ | ​
​ | ​ | Currency | ​ | for-Sale | ​ | Gains (Losses) | ​ | Liability | ​ |  | ​
​ | ​ | Translation | ​ | Securities | ​ | on Derivative asset | ​ | Adjustments | ​ |  | ​
​ | ​ | Adjustments | ​ | Net of tax | ​ | Net of tax | ​ | Net of tax | ​ | Total
Balance at September 30, 2022 | ​ | $ | (165,694) | ​ | $ | (10,909) | ​ | $ | 93,020 | ​ | $ | (333) | ​ | $ | (83,916)
Other comprehensive income (loss) before reclassifications | ​ | ​ | 111,264 | ​ | ​ | 4,042 | ​ | ​ | (70,260) | ​ | ​ | — | ​ | ​ | 45,046
Balance at March 31, 2023 | ​ | $ | (54,430) | ​ | $ | (6,867) | ​ | $ | 22,760 | ​ | $ | (333) | ​ | $ | (38,870)
​
​ | ​ | ​ | ​ | ​ | ​ | ​ | ​ | ​ | ​ | ​ | ​ | ​ | ​ | ​ | ​
​ |  | ​ | ​ |  | Unrealized |  | ​ | ​ |  | ​ | ​ |  | ​ | ​
​ | ​ | ​ | ​ | ​ | Gains (Losses) | ​ | ​ | ​ | ​ | ​ | ​ | ​ | ​ | ​
​ | ​ | ​ | ​ | ​ | on Available- | ​ | ​ | ​ | ​ | Pension | ​ | ​ | ​
​ | ​ | Currency | ​ | for-Sale | ​ | Gains (Losses) | ​ | Liability | ​ | ​ | ​
​ | ​ | Translation | ​ | Securities | ​ | on Derivative asset | ​ | Adjustments | ​ | ​ | ​
​ | ​ | Adjustments | ​ | Net of tax | ​ | Net of tax | ​ | Net of tax | ​ | ​ | Total
Balance at September 30, 2023 | ​ | $ | (88,448) | ​ | $ | (5,135) | ​ | $ | 31,487 | ​ | $ | (330) | ​ | $ | (62,426)
Other comprehensive income (loss) before reclassifications | ​ | ​ | 25,725 | ​ | ​ | 3,276 | ​ | ​ | (8,288) | ​ | ​ | (61) | ​ | ​ | 20,652
Amounts reclassified from accumulated other comprehensive income (loss) | ​ | ​ | — | ​ | ​ | — | ​ | ​ | — | ​ | ​ | 46 | ​ | ​ | 46
Balance at March 31, 2024 | ​ | $ | (62,723) | ​ | $ | (1,859) | ​ | $ | 23,199 | ​ | $ | (345) | ​ | $ | (41,728)

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