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Stockholders' Equity (Tables) | 3 Months Ended
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Dec. 31, 2023
Stockholders' Equity Note [Abstract] |
Summary of accumulated other comprehensive income (loss) | ​
​ | ​ | ​ | ​ | ​ | ​ | ​ | ​ | ​ | ​ | ​ | ​ | ​ | ​ | ​ | ​
​ |  | ​ | ​ |  | Unrealized |  | ​ |  | ​ | ​ |  | ​ | ​
​ | ​ | ​ | ​ | ​ | Gains (Losses) | ​ | ​ | ​ | ​ | ​ | ​ | ​ | ​ | ​
​ | ​ | ​ | ​ | ​ | on Available- | ​ | ​ | ​ | ​ | ​ | ​ | ​ | ​
​ | ​ | Currency | ​ | for-Sale | ​ | Gains (Losses) | ​ | Pension | ​ |  | ​
​ | ​ | Translation | ​ | Securities | ​ | on Derivative asset | ​ | Liability | ​ |  | ​
​ | ​ | Adjustments | ​ | Net of tax | ​ | Net of tax | ​ | Adjustments | ​ | Total
Balance at September 30, 2022 | ​ | $ | (165,694) | ​ | $ | (10,909) | ​ | $ | 93,020 | ​ | $ | (333) | ​ | $ | (83,916)
Other comprehensive income (loss) before reclassifications | ​ | ​ | 77,414 | ​ | ​ | 1,555 | ​ | ​ | (57,127) | ​ | ​ | — | ​ | ​ | 21,842
Balance at December 31, 2022 | ​ | $ | (88,280) | ​ | $ | (9,354) | ​ | $ | 35,893 | ​ | $ | (333) | ​ | $ | (62,074)
​
​ | ​ | ​ | ​ | ​ | ​ | ​ | ​ | ​ | ​ | ​ | ​ | ​ | ​ | ​ | ​
​ |  | ​ | ​ |  | Unrealized |  | ​ | ​ |  | ​ | ​ |  | ​ | ​
​ | ​ | ​ | ​ | ​ | Gains (Losses) | ​ | ​ | ​ | ​ | ​ | ​ | ​ | ​ | ​
​ | ​ | ​ | ​ | ​ | on Available- | ​ | ​ | ​ | ​ | ​ | ​ | ​ | ​ | ​
​ | ​ | Currency | ​ | for-Sale | ​ | Gains (Losses) | ​ | Pension | ​ | ​ | ​
​ | ​ | Translation | ​ | Securities | ​ | on Derivative asset | ​ | Liability | ​ | ​ | ​
​ | ​ | Adjustments | ​ | Net of tax | ​ | Net of tax | ​ | Adjustments | ​ | ​ | Total
Balance at September 30, 2023 | ​ | $ | (88,448) | ​ | $ | (5,135) | ​ | $ | 31,487 | ​ | $ | (330) | ​ | $ | (62,426)
Other comprehensive income (loss) before reclassifications | ​ | ​ | 46,494 | ​ | ​ | 2,524 | ​ | ​ | (13,368) | ​ | ​ | (35) | ​ | ​ | 35,615
Amounts reclassified from accumulated other comprehensive income (loss) | ​ | ​ | — | ​ | ​ | — | ​ | ​ | — | ​ | ​ | 27 | ​ | ​ | 27
Balance at December 31, 2023 | ​ | $ | (41,954) | ​ | $ | (2,611) | ​ | $ | 18,119 | ​ | $ | (338) | ​ | $ | (26,784)

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