N.H. Rev. Stat. § 82:22
If upon rehearing before the board of tax and land appeals or on appeal, the proceedings result in a decision for the reduction of the tax previously assessed, the board shall certify that fact to the state treasurer and the treasurer shall refund to the taxpayer the amount of any overpayment of the tax upon any funds not otherwise appropriated. And, if upon such rehearing or appeal it shall be found that the tax against such corporation or company was less than the same should have been, the difference between the amount assessed and the amount that should have been assessed shall forthwith be paid by the corporation or company as of the date of the decision.
RSA 82:22
1911, 169:20. PL 69 :21. RL 83:21. 1950, 5, part 8:20. 1953, 107:3. RSA 82:22. 1973, 544:14, IX, eff. Sept. 1, 1985.