A 57-year-old man was executed in Oklahoma on Thursday for the murder of his infant daughter after the Supreme Court rejected last-minute appeals that his life be spared on the grounds that he was severely mentally ill.

Benjamin Cole was put to death by lethal injection at the Oklahoma State Penitentiary in McAlester, the Department of Corrections said.

Cole had been on Death Row in the south-central state since 2004 for the murder of his nine-month-old daughter, Brianna. He was accused of killing the crying child to silence her so he could continue playing a video game.

Media witnesses said Cole delivered a rambling, two-minute statement while he was strapped to a gurney in the execution chamber in which he said "Jesus is my personal lord and savior" and urged people to "choose Jesus while you still can."

"I forgive everyone that I have done wrong," the media witnesses quoted him as saying.

Cole's lawyers had appealed to the Supreme Court for a stay of execution, arguing that he suffered from "debilitating mental illness" and that the US Constitution prohibits the execution of someone who is not mentally competent.

Cole had been diagnosed with paranoid schizophrenia and brain damage, his lawyers said, and his condition had deteriorated to the point where he was "largely catatonic."

"(Cole) cannot manage his own basic hygiene, and crawls on the cell floor if without a wheelchair," they said.

Lower courts rejected the claims that Cole was not mentally competent and the Supreme Court denied his last-minute appeals for a stay of execution without comment.

According to Oklahoma authorities, Cole identified as a "Messianic Jew" and "his refusal to speak to certain individuals appeared to be a choice on his part" motivated by his "extreme religiosity."

"Cole's claim of incompetency rests on experts who have not had actual conversations with him concerning his execution," they said.

"In contrast, Cole willingly engaged with a neutral expert at a state-run hospital and very clearly expressed his rational understanding of his punishment," they said.

There have been 12 executions in the United States this year, including four in Oklahoma.