When reading "Hitler" by Ian Kershaw, a highly respected historian who has written, and talked, at length about both Adolf Hitler and the Third Reich, I don't recall any mention of Hitler's Jewish ancestry.

However, many claims, which have never been substantiated, have been made about Hitler, such as his Jewish ancestry and only having one testicle.

Just realized I should probably have posted this as a top comment to make sure you see it...

[Ian] Kershaw does make mention of it, but only to dismiss it as nothing more than unsubstantiated rumor. According to him, the rumors only began in the 1920s, and revolved around his surname "Hütter" (Hitler being a corruption of it) being of Jewish origin, which seems to not have any basis in reality.

The more popular 'theory', that his grandmother had become pregnant by a Jewish man in whose house she worked as a servant, Kershaw traces to the memoirs of Hans Frank, which he wrote while awaiting execution following the war. Frank claimed that Hitler commissioned him to look into his ancestry, and found that his grandfather was most likely a Jewish man named Frankenberger, this based on supposed child support the man's father paid to Maria Anna Schicklgruber. Frank claimed that Hitler said that he knew this story, but that his grandmother knew that Frankenberger WASN'T the father, but was just conning him for money.

Regardless, Kershaw dismisses Frank's story as ludicrous. For starters, it is the only source, and was published after the war. Further more, there was no Jewish family named Frankenberger living in Graz at the time Frank claims - Jews were forbidden from living in Graz until the 1860s. Nor did Maria Anna live in Graz even.

The only persons who Kershaw believes likely to be Hitler's grandfather are Johann Georg Hiedler or Johann Nepomuk Hiedler, the former being who traditionally is assumed to be (and the latter was also his mothers grandfather, so would make his family tree... interesting...).

An angle of your question that the other comments haven't adressed is whether the Nazis ever lent credence to or investigated the rumours of Hitler's supposed Jewish ancestry. The answer seems to be that they did, and, surprisingly, they did it at the request of Hitler himself. At least according to Ian Kershaw's Hitler 1889-1936: Hubris, Hitler ordered the Gestapo to delve into his family history on several occasions in the 1930s and 40s. They didn't find any Jewish ancestors, but on the other hand they didn't conclusively establish who his paternal grandfather really was, which is probably why the investigations were repeated so many times. Kershaw cites Robert G.L. Waite, The Psychopathic God: Adolf Hitler (New York, 1977) for this. Waite alleges that the Gestapo investigated in 1932, 1935, 1938, 1940, 1942, and 1943-44. The results are to be found in Gestapo Berichte (Gestapo Memos) that are "now in the main archives of the Nazi party". According to Waite, Hitler was haunted by the fear that he might be tainted by Jewish blood. Waite was a psychohistorian, though, so take such assertions with a grain of salt.