The 65-year-old was convicted in May of possession of indecent images and videos of children, along with indecent behaviour of a sexual nature towards children.
On Friday, he was sentenced to 30 months in prison at the Livingston Sheriff Court.
Detective Inspector Adrian Ure, National Child Abuse Investigation Unit, Police Scotland said:
"Jenkin’s behaviour was deplorable and showed complete and utter disregard for the young victims depicted both in the images he was viewing and the offences he was committing.
“He believed that the internet would provide anonymity and allow him to undertake the online aspect of his offending undetected. This was not the case, and as a result of our investigation he will now rightly face the consequences of his actions.
"Any form of child sexual abuse is a serious criminal offence and is a priority for Police Scotland."