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He was born into a musical family and grew up with jazz. His father Sven Bergcrantz was a well-known jazz pianist. Anders' two older brothers Håkan and Thomas Bergcrantz are both former jazz musicians. After finishing the Music Conservatory in Malmo, Sweden, Bergcrantz became a full-time freelance jazz trumpet soloist playing all over the world both with his own bands and as guest soloist.
Bergcrantz has recorded seven albums as a leader and on many more as a sideman. He has twice been awarded the Gyllene Skivan (the Golden Record) by Sweden's Orkesterjournalen jazz magazine: in 1995 for In This Together, and again in 2007 for About Time.