They power the motor this is the only connection@forkunm, are the central pins essential to the operation of the bike or is it for monitoring unnecessary data on the display. If that is the case and it should be, then you can just use any small 36V battery. I use the smaller batteries every day. Batteries with similar capacity now can be light and highly compact. Here are a couple of them.
The battery is inserted into the frame not on the frame. Interesting, if the pins are the same as in the photo then maybe it could work. Could you send a link to one you thought would work please.Great! Any 36V Li-ion battery will work. Next question: Do you have water bottle cage mounts on the seat tube? That is the portion of the bike's frame that goes from the area near the pedals up to the seat post.
If not that is okay. You will mount the new battery using three Topeak Versamounts. The leads will be connected to the old leads to the motor with automotive heat shrink crimping connectors and shielded with adhesive heat shrink tubbing along their entire length. Red to red, black to black. That is it.
I just saw a couple of batteries that will work for you for under $200 on both eBay and Amazon. These are not top-notch but are a vast improvement for a ten-year old battery and the new one may last five-years. The old battery was made on the 35th day of 2011.
I was hoping others would join in so that you can get second and third opinions. I am happy to help when you have further questions.
@Timpo is saying you should see the prior discussion on your battery issue. And you should. I am suggesting cutting the wires to the old connector and ditching it and moving the battery placement so that you are not stuck with a legacy in frame mount battery. This way you get to pick your battery and are not stuck.The battery is inserted into the frame not on the frame. Interesting, if the pins are the same as in the photo then maybe it could work. Could you send a link to one you thought would work please.