I originally didn't notice the question was specifically about literary influences so I wrote the thing about Doctor Who below, which I'll leave for completeness, but I did find an interview where he was specifically asked about his literary influences, here was his response:
Other funny writers, of whom the chief is P.G. Wodehouse, who is, in
my opinion, one of the greatest-ever users of the English language --
he's sort of the Mozart of the English language, I think. I
particularly admire funny writers, because I know how incredibly
difficult it is. Evelyn Waugh is very high up there, and Jane Austen.
People have this idea that humor is in some way a sort of lesser
emotion, which I don't accept at all. I think that good, funny writing
is amongst the finest writing of any type, which is why I think that
Wodehouse is one of the finest writers who ever lived.
Later in the interview he also talks about non-literary influences like Python, and he also disavows comparisons to Vonnegut:
Vonnegut is another favorite of mine. I deliberately put him low on
the list, though, because I get embarrassed by people trying to draw
comparisons between him and me -- on one very, very superficial level,
it's an easy comparison: he writes stuff that is a) funny, and b) uses
science fiction to make its points, and I write stuff that is funny
and uses science fiction to make its points.
But that's the only level of comparison. Vonnegut is essentially a
deeply serious writer. Obviously a major part of his world view, if
you like, comes from the experience he describes in Slaughterhouse
Five of being a Prisoner of War in Dresden during the fire-bombing.
And I don't have any experience like that to draw on, you know,
nothing remotely approaching that.
So Vonnegut is essentially a deeply serious writer who uses comedy to
make his points, and I am essentially a comic writer who occasionally
tries to slip a point about something or other "under the counter," so
to speak, and so from that point of view, I find the comparison
embarrassing because he's a great writer, and I think I'm essentially
a frivolous one, I'm afraid.
older comment: The answers above are good, but I just wanted to add that he also was a writer and script editor for Doctor Who, so the style of outlandish sci-fi adventures on that show may have been an influence-as noted at http://tardis.wikia.com/wiki/Douglas_Adams the third novel Life, the Universe and Everything grew out of a Doctor Who story he came up with that was never made (and the plot of the first Dirk Gently novel also closely resembled another unproduced episode called Shada--that one was actually partly filmed so there's a reconstructed version available on DVD).
I should add that looking at the chronology of his career on his wiki page, it seems he didn't start working on Doctor Who until after the first Hitchhiker's radio series was completed, so the show probably had little or no influence on the first one (though he may have watched the show earlier, I don't know).