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. However, he changed topics for his second year to take courses in mathematics and physics. His lecturers at Vienna included the mathematicians Franz Moth and Józeph Petzval, the astronomer Edmund Weiss, and the mathematical physicists Ludwig Boltzmann, Josef Stefan and Viktor von Lang. He graduated in June 1869 with a degree which qualified him to teach mathematics and physics in Gymnasia in Austria. First he spent the year 1869-70 as a probationary teacher in the Gymnasium in Vienna, after which he taught at the Gymnasium in Waidhofen an der Thaya in the north of Austria, but moved to become a teacher at the National Gymnasium in Krems an der Donau, also in Austria. In 1873 he obtained leave from his teaching position in Krems, and was awarded a grant for travel and living expenses, allowing him to undertake research at the University of Berlin. 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. He found Weierstrass's lectures fascinating and spent much time working through the details but he was also influenced by Kronecker's lectures which gave him an interest in number theory which lasted throughout his life. During these years he made a number of close friends and was very active in the Mathematics Society. 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. However, before making his decision, he was offered the position of extraordinary professor at the University of Czernowitz. Now Czernowitz, on the upper Prut River in the Carpathian foothills, was at that time in the Austrian Empire but after World War I it was in Romania, then after 1944 it became Chernovtsy in the Ukraine. Czernowitz University was founded in 1875 and Gegenbauer was being offered the first professorship of mathematics there. Of the two offers, he preferred the professorship at Czernowitz which he happily accepted. He remained in Czernowitz for three years and during this time he married Helene Schuler von Libloy (1861-1924), the daughter of Friedrich Schuler von Libloy who was professor of German Law at the University of Czernowitz. 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. However, he immediately took study leavend spent some of 1878-79 in Rome attending lectures by Luigi Cremona; while there he studied in the Vatican Library. 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. 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, the American James Pierpont, Ernst Fischer, and Lothar von Rechtenstamm.
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. Theses polynomials 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 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. As an example of his work on number theory, let us mention the paper Asymptotische Gesetze der Zahlentheorie (1885) in which he gave the well-known asymptotic estimate 6n/π2 for the number of square-free integers not exceeding n.
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. For example in the journal he published: Einige arithmetische Sätze (1890), Über die Wurzeln der hypergeometrischen Reihe (1891), Über die aus n Haupteinheiten gebildeten complexen Zahlen (1891), Über die G Cantor'sche Zerlegung der reellen Zahlen in unendliche Producte (1892), Über die Tchebychef-de Polygnac'sche Identität (1892), Über eine arithmetische Formel (1892), Über reelle Primzahlen (1893) and another eight papers in 1893. Emil Weyr died in 1894 and Gegenbauer assumed his position on the editorial board. He continued to publish several papers each year in the journal up to the year 1900.
Up to now we have not mentioned one important aspect of Gegenbauer's contributions, namely his work on actuarial science and accounting. He actively contributed to the insurance industry, designed a course on the Theory of Insurance at Vienna University, and was the main driving force behind establishing a chair of actuarial studies at the University of Vienna. Gegenbauer is said to have made the following prophetic statement (see for example ):-
The 20th century is the century of technology: we should orient ourselves toward technology, unless we intend to condemn ourselves to atrophy.Gegenbauer was also involved in educational policy issues and in local politics. He wrote a paper on the regulation of salaries for university professors in which he requested the nationalisation of tuition fees. During 1889-1892, he acted as member of the municipal council of Innsbruck. From 1898 to 1902 he served as a State School Inspector and while in this role he developed the curriculum in mathematics and physics for Gymnasia.
We have already mention the honorary degree awarded to Gegenbauer by the University of Czernowitz in 1879. He received other honours such as elected as a corresponding member of the Austrian Academy of Sciences in 1883. On 6 October 1900 he was elected to the German Academy of Scientists Leopoldina.
In 1901 he began to suffer from a severe nervous disease and had to stop teaching. His health continued to deteriorate until his death in 1903 at the age of 54.
Article by: J J O'Connor and E F Robertson