I can't find information on who actually issues the bitcoins. I understand that miners generate new bitcoins but what party is responsible for issuance?
No one entity overseas the issuance of block rewards. This is one of several revolutionary concepts behind Bitcoin. (There is absolutely no Federal Reserve.) The Bitcoin Protocol and its distributed blockchain consensus mechanism is what effectively awards miners solving a very difficult hashing puzzle. The solution block groups a number of transactions together to be added to the Blockchain, and the preponderance of other "full node" block chains must add that solution block to their records in order for it to be "recognized".
The first transaction in such a block is called the "coinbase transaction." It is paid to the miner that successfully added a block to the Blockchain. It includes the current 25 BTC award and mining transaction fees.
Solo mining is accomplished by a solo miner connecting their miner (e.g., bfgminer, cgminer) to their own bitcoin node to advertise on the Bitcoin Network a solution has been found. Similarly for mining pools, they manage one or more full nodes that advertise when the pool has discovered a solution. Mining pools ensure the multiple outputs of a coinbase transaction pays its members their dues based upon shares shares of work they have successfully completed, hopefully with fair accounting. Miners can't spend their booty until their block is 100+ blocks deep in the Blockchain.
I'll try to give a different answer than the ones already given.
You are responsible for issuing them. You, together with all other users.
Bitcoins are not issued, as they are not tokens that are redeemable for something else. They don't have any promised or guaranteed or even intended value. Bitcoins are simply bitcoins, and the rules of the system determine how many and who can create them.
It is however us all who are allowing that process to take place. The system exists because the result is something which is usable as money, and people choose to accept it, giving bitcoins value.
I suppose the miner does or whoever constructs the block header that's being hashed. When you construct the block, there's a special transaction called a coinbase transaction where you're allowed to assign bitcoins to yourself(or any bitcoin address of your choosing as output).
Other mines and nodes in the network will accept this transaction in the block because you also provide proof-of-work behind it.
The Bitcoin protocol defines rules which outline valid interaction in the network. One of the rules states that each Block must contain a Coinbase Transaction. The Coinbase Transaction may be used to collect the transaction fees, and to create a limited number of new coins.
As the size of this mining reward shrinks over time, there is an overall limited amount to be created altogether.
Therefore, one could argue that the complete amount was issued by Satoshi Nakamoto through the creation of the Bitcoin network, and that new bitcoins are now only discovered.
I think the original question would be answered more efficiently by asking it this way:
I recognize that in mining Bitcoin, the miner is rewarded with a stipend of newly minted Bitcoin. I know how these Bitcoins are thus produced and thereby issued.
What about the other Bitcoins though?
My understanding is that the miners aren't mining the Bitcoin but the blockchain ledger blocks, and are simply rewarded with Bitcoin.
But applying a sort of Regression Theorum, what Bitcoin then, was used in the original, first block that had to be mined?
If mining the ledger is the only way to create Bitcoin, what would there have been to mine on Day 1 with zero produced Bitcoin and no ledger transactions?
I've always assumed that the math behind Bitcoin releases a certain initial issuance of Bitcoin i.e., 5000 coins in year one.
I then assumed year two it would issue 2500; year 3, 1250, and so on, until it reached 21,000,000.
I also assumed during a year, x new amount of Bitcoin was created by mining, capped at the mathematical limit for that year.
It is evident by my reading of various websites that this isn't the case. They make it sound like the miners generate 100% of all coins ever created.
If that is so, what about the first blockchain ledger transaction -- what is there to mine if no one yet has Bitcoin and thus no one has yet used it?