New York State’s top prosecutor is investigating some of the nation’s largest banks in connection with their use of credit-reporting databases that disqualify people seeking to open checking or savings accounts — an inquiry that has gained urgency as the ranks of the unbanked has swelled in the aftermath of the financial crisis.

The New York State attorney general, Eric T. Schneiderman, sent a batch of letters seeking information to six banks on Thursday, including Bank of America, Citibank and JPMorgan Chase, people briefed on the matter said.

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