**Leopold Gegenbauer** was an Austrian mathematician who gave his name to a sequence of orthogonal polynomials.

- In the autumn of 1858 Leopold entered the Piarist Gymnasium in Krems an der Donau in Austria.
- At this school Gegenbauer studied reading, writing, elementary mathematics, grammar, humanities, poetry and rhetoric as well as the specialist Piarist topics of schola principiorum and schola parva.
- Gegenbauer entered the University of Vienna in the autumn of 1866 and in his first year he studied history, Sanskrit grammar, and comparative linguistics of Indo-European langages.
- At Berlin he attended lectures from Karl Weierstrass, Eduard Kummer, Hermann Helmholtz and Leopold Kronecker during the two years he studied there from 1873 to 1875.
- In 1875 he was awarded a doctorate in mathematics for his work on what today are called the 'Gegenbauer polynomials', in particular proving his famous addition formula for these polynomials.
- After graduating from Berlin, Gegenbauer returned to Vienna where, in the autumn of 1875, he received an offer of a professorship at the High School in Wiener Neustadt.
- Czernowitz University was founded in 1875 and Gegenbauer was being offered the first professorship of mathematics there.
- Gegenbauer gave outstanding service to the new university of Czernowitz which they recognised by awarding him an honorary degree in 1879.
- In 1878 Gegenbauer moved to the University of Innsbruck where he became a colleague of Otto Stolz who, like Gegenbauer, was an Austrian who had studied at Vienna but had been strongly influenced by spending two years at the University of Berlin.
- In Innsbruck Gegenbauer again held the position of extraordinary professor.
- After three years teaching in Innsbruck Gegenbauer was appointed full professor in 1881, then he was appointed as full professor of mathematics at the University of Vienna in 1893 filling the vacancy created by the death in September 1891 of his former teacher Józeph Petzval.
- Gegenbauer had many mathematical interests such as number theory, function theory, and the theory of integration, but he was chiefly an algebraist.
- He is remembered for the Gegenbauer polynomials, a class of orthogonal polynomials which he introduced in his doctoral thesis of 1875 and also studied in several papers; they play an important role in potential theory and harmonic analysis.
- The Gegenbauer polynomials are solutions to the Gegenbauer differential equation and are generalizations of the associated Legendre polynomials.
- However, the name of Gegenbauer occurs in many other places, such as Gegenbauer functions, Gegenbauer transforms, Gegenbauer series, Fourier-Gegenbauer sums, Gauss-Gegenbauer quadrature, Gegenbauer's integral inequalities, Gegenbauer's partial differential operators, the Gegenbauer equation, Gegenbauer approximation, Gegenbauer weight functions, the Gegenbauer oscillator, and the Gegenbauer addition theorem published in 1875.
- Around 300 papers appear in MathSciNet whose title includes one of these notions named for Gegenbauer.
- The Monatshefte für Mathematik und Physik was founded by Gustav von Escherich and the emeritus professor Emil Weyr in 1890, and Gegenbauer immediately began publishing papers in the journal.
- Emil Weyr died in 1894 and Gegenbauer assumed his position on the editorial board.
- Up to now we have not mentioned one important aspect of Gegenbauer's contributions, namely his work on actuarial science and accounting.
- We have already mention the honorary degree awarded to Gegenbauer by the University of Czernowitz in 1879.
- On 6 October 1900 he was elected to the German Academy of Scientists Leopoldina.

Born 2 February 1849, Asperhofen (E of Herzogenburg), Austria. Died 3 June 1903, Giesshübl, Mödling, Austria.

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Origin Austria

**O’Connor, John J; Robertson, Edmund F**: MacTutor History of Mathematics Archive