Leopold Bernhard Gegenbauer (2 February 1849,

Asperhofen Asperhofen is a town in the district of Sankt Pölten-Land in the Austrian state of Lower Austria. Geography Asperhofen lies in the Mostviertel in Lower Austria, 6 km north of Neulengbach, on the edge of the Haspel Forest on the road to Tu ...

– 3 June 1903, Gießhübl) was an

Austrian Austrian may refer to: * Austrians, someone from Austria or of Austrian descent ** Someone who is considered an Austrian citizen, see Austrian nationality law * Austrian German dialect * Something associated with the country Austria, for example: ...

mathematician remembered best as an

algebra Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics. Elementary a ...

ist.

Gegenbauer polynomials In mathematics, Gegenbauer polynomials or ultraspherical polynomials ''C''(''x'') are orthogonal polynomials on the interval minus;1,1with respect to the weight function (1 − ''x''2)''α''–1/2. They generalize Legendre polynomi ...

are named after him. Leopold Gegenbauer was the son of a doctor. He studied at the

University of Vienna The University of Vienna (german: Universität Wien) is a public research university located in Vienna, Austria. It was founded by Duke Rudolph IV in 1365 and is the oldest university in the German-speaking world. With its long and rich histor ...

from 1869 until 1873. He then went to Berlin where he studied from 1873 to 1875 working under

Weierstrass Karl Theodor Wilhelm Weierstrass (german: link=no, Weierstraß ; 31 October 1815 – 19 February 1897) was a German mathematician often cited as the "father of modern mathematical analysis, analysis". Despite leaving university without a degree, ...

and Kronecker. After graduating from Berlin, Gegenbauer was appointed to the position of extraordinary professor at the

University of Czernowitz Chernivtsi National University (full name Yuriy Fedkovych Chernivtsi National University, uk, Чернівецький національний університет імені Юрія Федьковича) is a public university in the City o ...

in 1875. Czernowitz, on the upper Prut River in the Carpathian foothills, was at that time in the

Austrian Empire The Austrian Empire (german: link=no, Kaiserthum Oesterreich, modern spelling , ) was a Central-Eastern European multinational great power from 1804 to 1867, created by proclamation out of the realms of the Habsburgs. During its existence, ...

but after World War I it was in Romania, then after 1944 it became

Chernivtsi Chernivtsi ( uk, Чернівці́}, ; ro, Cernăuți, ; see also other names) is a city in the historical region of Bukovina, which is now divided along the borders of Romania and Ukraine, including this city, which is situated on the upp ...

, Ukraine. Czernowitz University was founded in 1875 and Gegenbauer was the first professor of mathematics there. He remained in Czernowitz for three years before moving to the University of Innsbruck where he worked with Otto Stolz. Again he held the position of extraordinary professor in Innsbruck. After three years teaching in Innsbruck Gegenbauer was appointed full professor in 1881, then he was appointed full professor at the University of Vienna in 1893. During the session 1897–98 he was Dean of the university. He remained at Vienna until his death. Among the students who studied with him at Vienna were the Slovenian

Josip Plemelj Josip Plemelj (December 11, 1873 – May 22, 1967) was a Slovene mathematician, whose main contributions were to the theory of analytic functions and the application of integral equations to potential theory. He was the first chancellor of t ...

, the American James Pierpont, Ernst Fischer, and Lothar von Rechtenstamm. Gegenbauer had many mathematical interests such as

number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic function, integer-valued functions. German mathematician Carl Friedrich Gauss (1777� ...

complex analysis Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates Function (mathematics), functions of complex numbers. It is helpful in many branches of mathemati ...

, and the theory of integration, but he was chiefly an algebraist. He is remembered for the

, a class of

orthogonal polynomials In mathematics, an orthogonal polynomial sequence is a family of polynomials such that any two different polynomials in the sequence are orthogonality, orthogonal to each other under some inner product. The most widely used orthogonal polynomial ...

. They are obtained from the hypergeometric series in certain cases where the series is in fact finite. The Gegenbauer polynomials are solutions to the Gegenbauer differential equation and are generalizations of the

associated Legendre polynomial In mathematics, the associated Legendre polynomials are the canonical solutions of the general Legendre equation \left(1 - x^2\right) \frac P_\ell^m(x) - 2 x \frac P_\ell^m(x) + \left \ell (\ell + 1) - \frac \rightP_\ell^m(x) = 0, or equivalently ...

s. Gegenbauer also gave his name to arithmetic functions studied in

analytic number theory In mathematics, analytic number theory is a branch of number theory that uses methods from mathematical analysis to solve problems about the integers. It is often said to have begun with Peter Gustav Lejeune Dirichlet's 1837 introduction of Diric ...

. The Gegenbauer functions Ρ and ρ (upper case and lower case

rho Rho (uppercase Ρ, lowercase ρ or ; el, ρο or el, ρω, label=none) is the 17th letter of the Greek alphabet. In the system of Greek numerals it has a value of 100. It is derived from Phoenician letter res . Its uppercase form uses the sa ...

) are defined as follows.L. Gegenbauer, Ueber einige zahlentheoretische Functionen. Sitzungber. Mth.-Naturw. Cl. Akad Wiss. Wien 89-2 (1884), 37–79. :

P_(n):=\sum_d^a=:n^a\rho_(n)

In 1973 in Vienna in the district of

Floridsdorf Floridsdorf (; Central Bavarian: ''Fluridsduaf'') is the 21st district of Vienna (german: 21. Bezirk, Floridsdorf), located in the northern part of the city and comprising seven formerly independent communities: Floridsdorf, Donaufeld, Greater Jed ...

(21. Bezirk) a street was named in his honor the ''Gegenbauerweg''.

Selected works

* ''Einige Sätze über Determinanten hohen Ranges'', 1890 * ''Über den größten gemeinsamen Theiler'', 1892

References

{{DEFAULTSORT:Gegenbauer, Leopold 1849 births 1903 deaths Austrian mathematicians Academic staff of Chernivtsi University Mathematicians from Austria-Hungary