I sent a Bitcoin transaction a few days ago and it still doesn't have any confirmations. I actually have a ASIC miner. Is it possible to confirm a specific transaction, in this case mine, with my miner?
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You could try to confirm it by mining a block in which you explicitly include said transaction. However, unless you have very powerful mining hardware, it is highly unlikely that you will succeed any time soon to solo-mine a block.
You could either try to double-spend the outputs you used to fund that transaction and for the second one include the correct fee, or just wait until the transaction's priority has risen sufficiently that it will be processed eventually.
It is not possible to confirm a specific transaction, you have to mine a full block that happens to include that transaction.
At current rates you need around 100PH/s to mine a single block within 24 hours. so 2.5EH/s to (roughly!) mine a block in under an hour. That my friend, is a farm that makes USD 250,000 per hour (excluding power mind you).