The forms of possessive pronouns vary with person and number. In English, possessive pronouns (except his and its) also have different forms depending on whether they function as determiners in a noun phrase or constitute a noun phrase on their own, i.e. function as "true" pronouns.
The “double” genitive
Possessive pronouns in English occur in prepositional phrases headed by of in examples like the following.
(1) Bill is a good friend of mine. (en god vän till mig)
(2) Mary is a cousin of ours. (en kusin till oss/vår kusin)
(3) Jane met an old friend of hers. (en av sina gamla vänner)
The most common corresponding Swedish construction has a personal pronoun complementing the preposition till, as in the first two examples. Typically, the noun phrase containing the of-phrase has an indefinite determiner (the indefinite article, an indefinite pronoun, a numeral, etc.), but definite determiners also occur, as in (4) and (5), each of which contains an additional postmodifier in the form of a relative clause:
(4) That colleague of his that he was talking about is now dead.
(5) The only friend of hers that I have met is Laura
The construction is often referred to as the double genitive, although there is really only one element that is genitive or possessive in form. The reason the term is used is that the possessive pronoun occurs inside an of-phrase, which is used to encode relations of the same type as the genitive, but which is not in itself a genitive form. The same construction is used with a full genitive noun phrase instead of a possessive pronoun.
(6) Jim is an old friend of my father’s (of my father).
(7) Did you see that painting of Picasso’s? (of Picasso)
As these examples indicate, the genitive is not obligatory. One reason to use the double genitive is to avoid the potential ambiguity of the non-genitive version (the painting was made by Picasso or depicts Picasso). A more common alternative is to use the preposition by instead of of in cases where the prepositional complement carries a subject/agent role (a painting by Picasso).