Doctor Who is a BBC multi-media franchise closely connected to Hitchhiker's.
Douglas Adams was once a scriptwriter for Doctor Who. Life, the Universe and Everything and the Krikkiters were based on a rejected script idea for Doctor Who. Adams also based much of Dirk Gently's Holistic Detective Agency on his then-unmade (though much later completed) Doctor Who serial Shada. Most notably, the character of Professor “Reg” Chronotis debuted in Shada, where his longevity and time travel are explained as his being a retired member of the Doctor's own species, the Time Lords (in Holistic Detective Agency itself, meanwhile, Chronotis claims not to be able to quite remember what it is that he retired from, though he acknowledges that it "must have been something pretty good" — thus skirting around the potential copyright issues).
In the first episode of the Doctor Who serial Destiny of the Daleks, the Doctor reads a book by Oolon Colluphid, a homage inserted by Adams. The Doctor also mentions Betelgeuse. As a nod to Adams' great work on Doctor Who, the 2005 episode The Christmas Invasion, which features the Doctor saving the world in his pyjamas, the Doctor makes reference to himself being very Arthur Dent. He then mumbles that Dent was a nice man. Further details about the Doctor's and Arthur's encounters are given in the novelisation of The Christmas Invasion, where it was revealed that the Doctor was not interested in killing Vogons and that he and Ford Prefect had had spent time together in college. Arthur and Zaphod Beeblebrox also cameo in the 2003 animation of Shada, being imprisoned in the titular prison. Also in the serial, a Nutrimatic Drinks Dispenser is seen in St Cedd's College. In the 'Making of...' documentary of the Hitchhiker's TV series, the Doctor's time machine (the TARDIS) can briefly be seen.
These homages heavily imply that Who and Hitchhiker's exist in the same fictional universe, and many fans believe this as fact, even though there is no concrete statements from the BBC or otherwise to confirm it.
Despite this, Eoin Colfer's And Another Thing... seems to suggest that the opposite. When Arthur watches Earth be destroyed from one of the Heart of Gold's portholes, he remarks on how fake it looks, comparing it to the classic serials of Doctor Who, providing a nice homage to Who, but somewhat contradicting the Hitchwhovian's beliefs. However, as the true Hitchhiker would point out, the universe exists in an endlessly complex and completely non-linear form of space and time, which multiple realities exist and interlock with each other. This results that Arthur has probably seen Doctor Who, being such a person living in the vicinity of Europe. This is scientifically called the Whole Sort of General Mish-Mash. It is also worth noting that many meta-fictional Doctor Who stories have posited that the Doctor Who franchise exists within the Doctor Who universe. As the Tenth Doctor tries to explain the passage of time in "Blink," he refers to the WSGMM but instead calls it "A big ball of wibbly wobbly... timey wimey... stuff."